York Mills Collegiate Institute

SPH4U Course Handbook

Fall 2014

Student Name:

SPH4U: Course Syllabus Course Website: http://abelmoodle.abel.yorku.ca/moodle/course/view.php?id=387 This syllabus contains a list of all classes, topics and homework in the Gr. 12 physics course. You are strongly encouraged to explore the simulations and videos listed for each lesson – they are optional but quite interesting! Day Topics Homework Extras

Introduction

1 Course Introduction, How to Answer a Question

Sign-up with course website Have parents sign course outline Handbook: How to Answer a Question Video: Top 10 Amazing Physics Videos

2 Groups Work Homework: Groups Work Video: Dysfunctional Group Video: Functional Group

3 Measurement and Numbers

Handbook: Measurement and Numbers Video: Scientific Notation

4 Fermi Problems Handbook: Fermi Problems Lesson: Fermi Problems 5 Short quiz on

Introduction Read: pg. 18-23 Gr. 11 Review Questions: 1-D Kinematics

Gr. 11 Review Lessons: Kinematics

Kinematics

1 Uniform Acceleration

Read: pg. 24-27 Problems: pg. 27, #20, 22, pg. 38, #15 (Use solution sheets!)

Active Physics: MotionDiagrams Course website: Graphing Kinematics Review Sheets

2 Representations of Motion

Handbook: Representations of Motion Read: pg. 11-13, 20-24

Active Physics: Graphs from Motion Active Physics: Motion from Graphs Video: Converting Between Graphs - Slopes Video: Converting Between Graphs - Areas

3 CGPS: Modeling Problem Solving

Problems: pg. 40 #10, pg. 67 #47 Active Physics: Car Catches Truck Simulation: The Moving Man

4 CGPS Read: pg. 35-8 Problems: pg. 65 #21, 26

Active Physics: Balloonist Drops Lemonade Active Physics: Avoid the Collision

5 2-D Motion Handbook: 2-D Motion Simulation: Maze Game 6 Vector

Components Handbook: Vectors and Components Read: pg. 756-7

Active Physics: Velocity Components Video: Components Video 1 Video: Components Video 2 Video: Components Video 3

7 Projectiles! Handbook: Projectiles! Read: pg.41-45

Active Physics: Changing x-velocity Video: Velocity Vector Components Video: Crazy Motorcycle Jump

8 Projectile Problem Solving

Read: pg. 46-48 Problems: pg. 46 #5, pg.51 # 8

Active Physics: Target Practice I Video: Projectile Problem Solving

9 Projectile Problem Solving (continued)

Problems: pg.51 #5, 7

Active Physics: Target Practice II Simulation: Projectile Motion

10 CGPS Review for test Pg. 171 #50

Projectiles (try #1-14, 35-39, 42-45, 51-53, 66, 71) See sample tests and resources online!

11 Test on kinematics and projectiles

NOTE: Acceleration in Two Dimensions (pg. 28-29) and Section 1.5 not on this test.

Forces

1 Representing Forces

Hand Book: Representing Forces Gr. 11 Review Lessons: Newton’s Laws Gr. 11 Review Questions: Newton’s Laws

1

2 Forces in 2-D Read: pg. 81-83 Problems: pg. 92 #7, pg. 96 #8, pg. 117 #10

Active Physics: Rocket Blasts Off Simulation: Forces in 1 Dimension Video: Simple Force Example

3 Understanding 2-D Forces

Read: pg. 88-9 Problems: pg. 92 #6 a-c, pg.118 #23

Active Physics: Sliding on an Incline Simulation: The Ramp: Forces and Motion Video: Forces with Angles 1 Video: Forces with Angles 2

4 Newton’s 3rd Law Read: pg. 93-94, Handbook: Magnets and the Third Law Problem: pg. 244#1a

5 Weight and Acceleration

Problems: pg. 93#9 Video: Forces and Elevators

6 CGPS Problems: pg. 96 #9 7 Frames of

Reference Read: pg. 108-110 Problems: pg. 110 # 2, 3

Video: Frames of Reference

8 Strings and Composite Objects

Problems: Pg. 94 #10, pg. 95-96 #3, 7 Lesson: Multiple Bodies

9 Tension and Pulleys

Problems: pg. 92 #5, pg. 95 #5

Active Physics: Atwood Machine Video: Atwood Machine

10 CGPS Problems: pg. 96 #10 11 Friction! Read: pg. 97-101

Problems: Pg. 101 #3, 7, pg. 168 #25 Active Physics: Push Crate Up Wall

12 Friction! (continued)

Problems: Pg. 118 # 16, 25(a)

Active Physics: Skier Goes Down Simulation: Forces and Motion Video: Inclines with Friction Video: Pulley, Incline, Friction

13 Going in Circles Problem: pg. 101 #5 Video: Circular MotionIdea Simulation: Ladybug Motion 2D

14 Going in Circles (continued) Centripetal Acceleration

Handbook: Going in Circles Read: 122-126

15 Centripetal Acceleration (continued)

Problems: Pg. 126 #8, 10

Active Physics: Centripetal Acceleration Video: Conical Pendulum Video: Frames of Reference

16 Thinking About Circular Motion

Read: pg. 130-133 Problems: Pg. 133 #3, Pg. 138 #4, 6

Active Physics: Problem Solving Simulation: Ladybug Revolution Video: Tension in Vertical Circle

17 CGPS Problems: pg. 138 #9 Active Physics: Car Circles Track 18 Universal

Gravitation Read: pg. 139-142 Problems: pg. 143 #11, 12

Simulation: Lunar Lander Video: Universal Gravitation

19 Orbits Problems: Pg. 147 #2, 6, use table pg. 776

Active Physics: Satellites Orbit Lesson: Satellites Simulation: My Solar System Video: Gravity in Orbit Video: Why Doesn’t the Moon Fall Down?

20 Orbits (continued)

Lesson: Orbital Calculations Video: Dark Matter

21 Test on Dynamics Test Review: pg. 117 #7, 24 (text answer wrong), 28, pg. 168 #16, 19, 20, 23, 24, 26, 27

Review: 2-D Forces Review: Gravity and Circular Motion (skip Kepler’s Laws)

Energy and Momentum

1 “Oomph” Read: pg. 232, 239-240 Handbook: “Oomph” parts C and D, pg. 243 #5

Active Physics: Save an astronaut Video: Intro to Momentum Lesson: Conservation of Momentum

2 The Idea of Conservation

Read: pg. 233-6 Problems: pg. 237 #8, 10, pg. 248 #6

Active Physics: Momentum and Energy Lesson: Isolated Systems Video: Slow Motion Collision

2

Video: Impulse Lesson: Impulse

2 Types of Collisions CGPS

Read: pg. 246-7 Problems: pg. 252 #12, 13, 14

Active Physics: Elasticity

4 Car Crash! Conservation of Momentum in 2-D

Read: pg. 256-8 Handbook: 2-D Momentum Problem Solving Problems: pg. 257 #3

Video: 1968 Crash Test

5 2-D Collisions Problem: pg. 257 #5, pg. 258 #3

Active Physics: P and E Conservation Simulation: Collision Lab

6 Work and Kinetic Energy

Problems: pg. 181 #7, pg. 183 #7, pg. 186 #4

Active Physics: Work Simulation: The Ramp Video: Roller Coasters

7 Collisions Quiz, Energy and Coordinate Systems

Problems: pg. 197 #6, pg. 201 #5, pg. 227 #17

Lesson: Work

8 Energy Transfers Problems: pg. 200 #13, pg. 202 #8, 10 Handbook: Transfers of Energy

Active Physics: Energy Bar Charts Simulation: Energy Skate Park

9 The Ballistics Pendulum

Problem: pg. 270 #20

Active Physics: Pendulum Bashes Box

10 Spring Force and Energy

Read: pg. 203-206 Problems: pg. 206 #5

Video: Hooke’s Law Video: Energy in Springs

11 Spring Force and Energy, continued

Read: pg. 207-210 Problems: pg. 211 #10, 12a, 13

Active Physics: Inverse Bungee Simulation: Masses and Springs

12 CGPS Problems: pg. 229 #38

Review: Momentum Review: Work and Energy

13 Test Review: pg. 307 #4, 5, 10, 14, 18, 20, 26, 48

Special Relativity

1 Velocity and Frames of Reference

2 The Light Clock

Handbook problems: #1-4, identify the types of intervals only, don’t solve!

Active Physics: Time Dilation Video: Special Relativity

3 The Moving Ruler

Handbook problems: #5, 6

Active Physics: Length Contraction Video: Visualizing Relativity

4 Relativity Problem Solving

Handbook problems: #1-4, 7

5 Energy and Relativity

Handbook problems: #8-10 Video: Einstein Talks

6 Energy and Relativity

Handbook problems: #11,12

Text: pg. 690-691 Video: Large Hadron Collider (LHC) Video: Proton Antiproton Collision

Electric and Magnetic Fields

1 A New Kind of Interaction

Read: pg. 318 – 322

Simulation: Static Electricity

2 Relativity Quiz, A New Kind of Interaction (continued)

Problems: pg. 324 #2 draw charge diagrams to illustrate each answer

3 The Strength of Electrostatic Interactions

Problems: pg. 331 #3, 7 (total thread length = 60 cm)

Active Physics: Coulomb’s Law Video: Charge and Coulomb’s Law

4 Analyzing Electrical Forces

Read: pg. 332-334 Problems: Pg. 334 #4, 8

Active Physics: Combining Charges

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5 Picturing Electric Forces

Problem: pg. 334 #9 (find the magnitude only)

Simulation: Chargesand Fields Lesson: Electric Force and Fields

6 The Electric Field Concept

Problems: pg. 343 #1, 2

Simulation: 3-D Electric Fields Lesson: Electric Field

7 The Superposition of Fields

Read: pg. 342 (sample problem 2) and pg. 353 (sample problem 4) Problems: pg. 347 #3, pg. 344#4, pg. 354 #3, pg.379 #27

Active Physics: Electric Fields Active Physics: Field of Dipole Lesson: Calculating Electric Fields Lesson: Calculating Electric Fields 2 Simulation: Electric Field Hockey

8 Moving Charges in an Electric Field

Read: pg. 369-370 Problem: pg. 371 #1 and finish worksheet / summary

9 CGPS Problem: pg. 371 #2, pg. 380 #35 10 Magnetic Fields

Read: pg. 384-5 and pg. 387-8 (Field of a Straight Conductor)

Active Physics: Field Around Wire Video: Magnetism Review

11 Magnetic Forces on Charges

Read: pg. 392-5 (Sample problem 1b, c only) Problems: pg. 402 #1, 4, 7, pg. 396 #2-4

Active Physics: Magnetic Force Active Physics: Mass Spectrometer Simulation: 3-D Magnetic Fields Video: Force on Moving Charges 1 Video: Force on Moving Charges 2 Image: Hi-res Bubble Chamber

12 Electromagnetic Disturbances

Read: pg. 530-33

Simulation: EM Waves Simulation: EM Waves 2

13 Understanding EM Waves

Simulation: Antenna

14 Light and Polarization

Read: pg. 494-5 Problems: pg. 498 # 4

Active Physics: Polarization Lesson: Polarization Video: Polarizing Filters Video: Polarization and Internal Stress Video: LCD Screens Review: Electrostatic Forces and Fields

15 Properties of 2-D Waves – diffraction, interference

Read: pg. 453-4 Problems: pg. 454 #2, 3

Simulation: 2-D Waves

16 Test

Course Website

1. Go to the website: http://abelmoodle.abel.yorku.ca 2. On the right side of the page click on the “Create new account” button below the login button. 3. Use the initial of your first name followed by your last name for your user name (for example jsmith for

John Smith). Choose any appropriate password. Remember your password!!Follow the rest of the instructions for logging in.

4. A message will be sent to your e-mail address. Follow the instructions in this message to validate your account.

5. Go back to the website: http://abelmoodle.abel.yorku.ca and login with your username and password. 6. Once you have access to the list of courses click on the Toronto District School Board science category

and your course is called: York Mills 12 Physics. Click on the course name. 7. Type your enrollment key in the text box. The enrollment key to get in is: ym12physics

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SPH4U – Physics, Grade 12 University Preparation

Course Website: http://abelmoodle.abel.yorku.ca/moodle/course/view.php?id=387

An Inquiry-Based Course Welcome to the wonderful world of physics! SPH3U is an introduction to the world of physics and a prerequisite for the grade 12 course, SPH4U. This course is designed according to the principles of Physics Education Research which clearly demonstrate the power of learning through inquiry in a collaborative group format. Major Canadian and American universities (U of T, McGill, McMaster, MIT, Harvard, Stanford and more) are transforming their introductory physics courses by reducing or eliminating traditional lectures and replacing them with engaging activities that have a deep conceptual and practical focus. Homework The majority of the class time will be spent doing activities and discussing physics with your colleagues. At home you will be responsible for solving problems using our solution format. You should expect about 45 minutes of physics homework per day on average.Homework problems will be randomly submitted for assessment and must be present at the time of collection. Optional textbook readings, online lessons and resources are listed in the syllabus for each lesson. Assessment and Evaluation Due to the central role of group work in this course, the work you do in in class will account for an important portion of your mark. Daily work will be randomly handed-in and assessed. To help ensure that individual students are pulling their weight in groups, there will be regular quizzes and tests. There is a final exam that covers the entire course material and a major project that will be announced halfway through the course. Mark Breakdown The categories of Knowledge and Understanding(K/U), Thinking and Inquiry(T/I), Communication(C), and Application(A) are a component of most of the assessments used in this course – however some focus on certain categories more than others. The basic mark breakdown for the course is 70% term work and 30% final examination. The term mark is composed as shown in the chart to the right. Attendance and Punctuality Students who are absent or late for class without a valid reason will not be eligible to submit any missed work or write any missed quizzes. Students who are absent are responsible for determining what was missed and making sure that they are caught up before the following class. Missed Tests If you miss a test you must:

• Let me know in advance if it is due to a pre-arranged reason (i.e. appointment for surgery) • Call in to the school so your name goes on the daily “Absent List” in the main office. • Contact me immediately after setting foot in the school upon your return. • Provide a doctor's note if the reason is illness. • Do not discuss the test by any means with your colleagues. • Be prepared to write the test immediately, at my discretion.

Failure to do any of these will result in a zero for the test. Please Read This Document! Please sign below signifying that you have read this course description. ____________________________________ __________________________________ Signature of parent, or student if 18 and over Print name

K/U 28% Tests ~ 7 % each

T/I 14% Daily work and regular quizzes ~ 1.5 % per work check ~ 1.5 % per quiz

C 14% Tests, Summary Notes, and Problem Solutions ~ 1 % per problem solutions ~ 1 % per summary note ~ 3 % per test

A 14% Challenges (CGPS) and Project ~ 1 % per challenge ~ 7% for project

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SPH4U Homework: How to Answer a Question Name: Sign up with the course website and follow the Course Introduction and Kinematics Lessons link. For day 1, click on the link to: Top 10 Amazing Physics Videos. Explore the videos and select one to focus on when you answer the questions below. Remember the four criteria for high quality responses! A: Watch Videos and Earn Marks! What a Deal! 1. Record. What is the title of your chosen video? 2. Describe. Describe what you observed in the video (what you saw or heard, but not any explanations).

3. Reason. Which unit from grade 11 physics does the video connect most closely to? (Kinematics, Forces, Energy, Sound and Waves, Electricity and Magnetism)

4. Explain. What was one interesting or surprising thing from the video? B: Assess Your Responses 1. Evaluate. Use the questions below to help decide if your answers to the video questions are complete. Use the four

criteria to assess your responses to the questions above and give yourself a mark. Be honest! Question Follow-Up Question Mark

A#2 Would your description be useful for someone who did not see the video? Explain.

A#3 Did you cite evidence from the video and connect it to specific ideas from a grade 11 unit? Explain.

A#4 Did you explain why the “thing” you mentioned was interesting to you? Explain.

© C. Meyer, 2014 6

SPH4U: How to Answer a Question? Sign up for your group roles today. Adjust your seating. Go through introduction below together with your group. High quality responses to any physics question must be correct, clear, concise and complete. We will routinely use these terms and the notation explained below for the evaluation of your daily written work. Criteria Description Notation Correct The physics is correctly stated. Conclusions follow logically

from the evidence and the definitions or laws. Technical details are all present and correct.

Incorrect sections are underlined and given an “ X ”. Correct ideas are just checked “ √ ”

Clear The explanation is precisely stated with a good choice of physics vocabulary. The explanation is straight forward with no awkward or unclear phrases. Spelling and grammar are correct.

Unclear sections are underlined with a wiggly line and given a “?“ A poor word choice is indicated by a wiggly line. Spelling errors are cirlced.

Concise There are no extraneous or distracting statements which may or may not be correct.

Phrases that are not relevant are crossed out. Like this.

Complete No important parts of the explanation are missing. The evidence supporting the conclusion is mentioned along with the relevant definitions or laws.

Where an explanation is missing or incomplete we will write “. . . .” or “and …” or “more …” or give a clear hint at what is missing: “force?”

A: Marking-Up 1. Below is a question from Gr. 11 physics. As a group, read each response and use the notation explained above to mark-

up the sample student answers that follow.

Explain. In a road test, the driver of a car slams on the brakes of a fast moving car and brings the car to a stop. An accelerometer in the car measures the magnitude of the acceleration and gives a reading 5.0 m/s2[backwards]. Interpret the acceleration reading to help explain the effect of braking on the car.

Response 1: The car slows down because of the brakes at a rate of 5.0 m/s2in the backwards

direction. The velocity gets slower.

Response 2: The car is slowing down by - 5.0 m/s backwards. So the velocity gets more negative until

it stops because of the brakes force. If it wasn’t for the brakes force, the car would not change its

speed.

Response 3: For every second of time, the car decelerates by 5.0 m/s. This helps us to understand

the effect.

Response 4: The brakes exert a stopping force on the car that acts in the backward direction

(opposite to it’s motion). This is due to the kinetic friction between the rubber of the tires and the

road surface. This causes the car to slow down until it comes to a complete stop.

Response 5: The velocity of the car becomes more negitive, since the acceleration is negitive, which

means it is stopping. Since the definition of acceleration is the change of velocity per second it gets

more negative and slows down.

Recorder: __________________ Manager: __________________ Speaker: __________________

0 1 2 3 4 5

© C. Meyer, Roberta Tevlin 2014 7

Group Check-Up When marking the responses above did your group: (1) Hear from each group member before moving to the next response? Yes ____ No ____ (2) Agree on how to mark-up each response before moving to the next one? Yes ____ No ____

** check with your teacher before moving on ** 2. Record. Write up one response (selected by your teacher) with your markings on a large whiteboard. Write large enough

so that everyone can easily read it.

3. Reason. List the key physics ideas (in point form) that should appear in a high quality responseto the acceleration question.

4. Explain. Work with your group to create a high quality answer to the original question. Make sure it satisfies the four

criteria! (Correct, clear, concise, complete) B: Evaluation Your daily work in physics will be marked based on the four criteria for high quality responses. An overall mark will be assigned on a scale of 0 to 5 depending on how your responses meet the four criteria according to the rubric below.

0-2 3 4 5 Responses are missing, fundamentally incorrect, or challenging to understand. A “yes orno” answer is given.

Response is basically correct, but contains problems or omissions.

Response is correct, but minor details could be improved or clarified.

Response is thoughtful, clear and complete. If another physics teacher saw it they would say, “Wow! A grade 12 student wrote this?”

1. Evaluate. As a group, use the rubric above to evaluate each of the five student responses that you marked above. Record

your mark beside each response. 2. Record. Write your group’s response to the braking question on a big whiteboard. Make sure you write large enough so

everyone can read it from a distance. ** check with your teacher before moving on ** 3. Evaluate. Exchange whiteboards with a neighboring group (your teacher will pick which). Use a different colour to

mark up the other group’s response using the four criteria. Give it a mark. Do this in a respectful, constructive way. When done, return the whiteboard.

4. Evaluate. Do you agreewith the assessment of your group’s response? Explain.

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SPH4U: Groups Work Your group has a challenge: use the elastic attached to three strings (the “flux capacitor”) to stack the cups into a pyramid. There are two rules: (1) only the elastic can move the cups, and (2) each group member must hold only one string by its very end. Start! A: Skills for the 21st Century 1. Reflect. What skills did your group make use of to accomplish (or partially accomplish!) this task? 2. Explain. Highlight one skill that you think was (or should have been!) the most important. Provide a simple rationale.

Record this on your group’s whiteboard and move on.

Here is a reminder of what we learned in grade 11about how groups in physics will work: Roles • Recorder / Checker: The recorder’s copy of the group work receives the detailed feedback. Help the recorder write high

quality responses. The recorder also checks with each group member for understanding before the group continues on. • Manager: Organizes and leads the group through the investigation and helps keep members focused on the same task. • Speaker: Speaks to the class orresponds to the teacher on behalf of the group. If there are four people in a group, the recorder/checker role is broken up. If there are two, both members are also the speaker. Set-Up Sit in a triangle formation, facing one another. Take out a whiteboard and markers and have them ready for the group discussions. Each group member is responsible for writing in their own words high quality responses for the investigations. B: It’s All About Me? Let’s think back to your group work experience in grade 11. 1. Reflect. (individually) Complete the survey below about your experience in gr. 11 physics. If you are new to our

program, think back to any experiences you had with group work. Check off how often each statement applies to you. Think about your personality and try to explain why you think that is the case. You don’t need to share these responses. Often Occasionally Rarely Why I follow the steps and instructions of an investigation carefully.

I stayed focused on each question with the other group members.

I respond to questions thoroughly and thoughtfully.

I offer ideas to the group discussions.

I ask for the ideas of my other group members.

I make sure the group agrees on each response before moving on.

I help explain ideas to the other group members.

Recorder: __________________ Manager: __________________ Speaker: __________________

0 1 2 3 4 5

© C. Meyer, 2014 9

2. Reflect. (individually) Of all the statements listed above, which do you think would be the most important for you to improve on? Explain why and how you might bring about the improvement in yourself.

C: It’s All About the Group 1. Reason. (as a group) Of all the statements listed in the survey, which statement does your group think is the most

important for the good functioning of a group in our physics class? Explain. 2. Reason. If a group is doing poorly at your chosen statement, what could group members do to help the situation? Include

examples of actions and phrases a group member might do or say.

3. Reason. Does a student need to “know all the answers” in order to be a good member of a physics group? Explain.

4. Explain. Emmy, who is in your group, says, “I shouldn’t ask my group members questions because it will slow the group down or it might get annoying.”How should you respond (constructively) to Emmy?

5. Explain. Isaac, who is also in your group, says, “The others in my group figure things out much faster than I do which

makes me feel less smart. I’m not sure if I belong here.” How should you respond (constructively) to Isaac?

6. Explain. Marie says, “I just want to hurry up and get the investigation done. I don’t think all this group stuff helps me much.”How should you respond (constructively) to Marie?

7. Record. On your whiteboard help your Recorder summarize your responses to question C#1 and C#2.

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SPH4U Homework: Thinking About Your Brain! Name:

How Your Brain Learns and Remembers © 2007 Diana Hestwood and Linda Russell

Minneapolis Community & Technical College Permission granted to individual instructors to use and reproduce for their own classroom.

Part 1: What Happens Inside Your Brain When You Learn Something New? Meet Your Brain Brain cells are called neurons. You are born with at least 100 billion neurons. Dendrites (fibers) grow out of the neurons when you listen to/write about/talk about/ practice something. Learning is natural. Neurons know how to grow dendrites, just like a stomach knows how to digest food. Learning is growth of dendrites. New dendrites take time to grow; it takes a lot of practice for them to grow. Connections Form between Neurons When two dendrites grow close together, a contact point is formed. A small gap at the contact point is called the synapse. Messages are sent from one neuron to another as electrical signals travel across the synapse.

Practice Improves Connections Special chemicals called neurotransmitters carry the electrical signals across the synapse. When you practice something, it gets easier for the signals to cross the synapse. That’s because the contact area becomes wider and more neuro-transmitters are stored there. When you practice something, the dendrites grow thicker with a fatty coating of myelin. The thicker the dendrites, the faster the signals travel. The myelin coating also reduces interference. With enough practice, the dendrites build a double connection. Faster, stronger, double connections last a very long time. You remember what you learned! Short-term memory is VERY short! If you learn something new and do it only once or twice, the dendrite connection is very fragile and can disappear within hours. Within 20 minutes, you remember only 60%. Within 24 hours, you remember only 30%. But if you practice within 24 hours, and then practice again later, you remember 80%. Make the Most of Practice Time… You grow dendrites for exactly the same thing you are practicing. If you listen or watch while math problems are solved, you grow dendrites for listening or for watching. If you read over your notes, you build dendrites for reading. If you actually solve the problems yourself, you grow dendrites for solving. Part 2: Brain Friendly Ways to Learn Better

A: Grow Your Intelligence You can grow your intelligence, because your brain knows how to grow dendrites just like your stomach knows how to digest food. Think about a baby who learns to speak in its native language without any special classes or training! B: Do Something Active to Learn You must do something active to learn, like explaining, solving, drawing, or writing. That’s because dendrites grow ONLY when you are actively doing something. No one else can grow dendrites for you!

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C: Grow Off of What You Know Dendrites cannot grow in a void. New dendrites can only grow off of what is already there. New skills must connect to, and grow off of, previously learned skills. If you do not have the necessary dendrites in place, new material will seem to go “right over your head”. D: Give It Time and Practice Learning takes time, because it takes a lot of practice for dendrites to grow. This is why you do homework. This is why trying to cram everything into your brain the night before a test doesn’t work. E: Mistakes Are Essential Making mistakes, and getting feedback so you can correct them, allows you to check the accuracy of the connections in your brain. Be sure to get feedback quickly so you don’t practice the wrong thing and build a strong, but wrong, connection! Emotions Affect Learning and Memory Anxiety floods your body with adrenaline (“fight or flight”).Adrenaline makes it hard for the neurotransmitters to carry messages across the synapses in your brain. That causes “blanking out” on a test. How can emotions help you? Endorphins make you feel calm. Your body produces endorphins when you relax, exercise, laugh, or learn new things. If you practice producing calming hormones, it will help when you are under stress. Part 3: What Does All This Mean For Me? Use your understanding from this article to answer the following questions. (Remember to give a 5/5 response!) 1. Explain. Marie says, “I understand what’s going on in the class just fine. But when I get home and start on the

homework assignment, why am I lost?” Explain to Marie why. 2. Explain. Isaac says, “I attend class and do all the homework and feel like I understand everything. Then why do I just

‘blank out’ on the test and can’t do anything?” Help Isaac understand why. 3. Explain. Emmy says, “Why should I do all this homework? It’s just the same thing over and over.” Respond to Emmy. 4. Explain. Albert says, “I’ve been absent for a week and there’s a test tomorrow. I’ll be fine if I cram tonight.” So What Should You Do? 1. Do some of the homework as soon as possible after class, before you forget. 2. Try to practice new skills every day. 3. To manage anxiety, set aside regular study-time in your schedule, get lots of sleep and exercise, and learn simple

relaxation techniques such as slow, deep breathing. 4. Make sure you are actively DOING something when you study: draw pictures or diagrams, solve lots of problems, check

your answers 5. Check your understanding by explaining how to do a problem to another student. 6. Create a practice test for yourself. Work it in the same amount of time you’ll be given in class.

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SPH4U: Measurement and Numbers Measurements form the backbone of all science. Any theory, no matter how slick, is only as good as the measurements that support it. Without careful measurements, science becomesguess work, hunches and superstition. A: The Meter Stick Our most basic scientific tool is the meter stick. But, do you know how to use it? Really?For this investigation you will need one meter stick 1. Observe. Each member of your group will independently (and secretly!) measure the height of your table. Don’t share

the results until everyone has made the measurement. Record everyone’s measured values here. The number we read from a measurement device is the indicated value. The readability of a device is the smallest increment or change in a quantity that you can discern from the measuring device. The readability is sometimes called the reading error, but the term “error” is very problematic so we will avoid it. When we record a measurement, we should also record the readability with a statement like: “scale readable to ± 1.0 cm”. The reported readability may vary from person to person. If you think you can estimate a reading between the lines of a scale, do so. Always record a measurement as carefully as you can.

2. Reason. What is the readability of your measuring device? Your estimation of the readability may be different from the

others in your group, and that can be OK as long as you are using the device appropriately. Explain how you decided on your readability.

3. Reason. Now think about the height measurements your group has made. How do they compare with one another?

Would you say, roughly speaking, that there is a lot of uncertainly or little uncertainty in your group’s measurements? Explain.

The true value is the actual, ideal value for a quantity that we are trying to measure. The true value of a quantity is usually never known: in science this is simply not possible (welcome to science)! Through hard work and ingenuity, we try to get our measurements (our indicated values) closer to the true value, but there is always some uncertainty in this since we never get to know the true value! 4. Reason. Some groups find differences of ± 0.1 cm amongst their height measurements. What are some suggestions for a

future group to reduce the differences in their height measurements? (This is the ingenuity we mentioned.)

B: The Stopwatch Now we will examine another common measuring device. You will need one stop watch

1. Observe. Measure the amount of time for the pencil to drop from a 1 m height. Write this reading as a number in

decimal notation with units of seconds.

2. Reason. What is the readability of the stopwatch? Explain.

Recorder: __________________ Manager: __________________ Speaker: _________________

0 1 2 3 4 5

© C. Meyer 2014, adapted from John Denker 13

3. Observe. Perform the pencil drop seven times and recordyour data below.

4. Reason. Examine the individual measurements in your data above. You probably notice quite a bit of variation in them.

What might be responsible for the spread in these values? All repeated measurements (except simple counting) will produce a range of values or a distribution of values. The size or width of this distribution is a measurement of the uncertainty in our result. One technique to find the width of the distribution is the standard deviation, but we will not use this in grade 12 physics. Instead,we will define the uncertainty(σ) by making avery crude estimation of the width (really the half-width) of the distribution:

σ = (high value – low value)/2

5. Reason. What is your best estimation of the true value for the time for the pencil drop? What is the uncertainty in this result?

When we present a calculated result based on our measurements we typically report two numbers, the mean value (m) and the uncertainty (σ) written as: m ± σ. This expression represents a distribution of values centred at the mean value m and with a width of σ on each side – it is not an ordinary number. We interpret this expression by saying, “m is our best estimation of the true value, but we wouldn’t be surprised if the true value was as high as m + σ or as low as m – σ.”

6. Reason. When you computed the mean value, the calculator likely displayed many digits. Use your uncertainty to help

decide how to write the digits of your final result in the form m ± σ.

The uncertainty in a result determines which digits are useful to keep. In the past you may have learned rules about significant digits. These rules are flawed and unhelpful. Only the uncertainty in a value determines which digits have significance (usefulness). In many situations we don’t have information about the uncertainty, but we should not pretend that it does not exist. Here are our rules about uncertainty and significant digits for grade 12 physics. • We will often estimate the uncertainty in simple situations rather than ignore it. • When recordingresults, use three significant digits to avoid too much rounding error (3 digits determined by how it

would be written in scientific notation, ex. calculator reads: 1 056 428, you write: 1 060 000 or 1.06x106). • For middle steps in calculations, keep one or two extra guard digits tohelp reduce the amount of rounding error. • When we are given a quantity like 5 km, we will assume it has 3 significant digits unless we know its uncertainty 7. Calculate.Make a calculation to predict the time for the pencil to drop (use the equation ∆y = v1∆t + ½a∆t2 and a = 9.8

m/s2). A fundamental process in science is to decide whether a measured result “agrees” with a result calculated from a theory or model. We will adopt a simplified decision rule for this. Two results agree with one another if one lies within the uncertainty distribution of the other. In more advanced studies, you will refine and greatly strengthen this rule. 8. Evaluate.Does this calculated value agree with your measured result? Explain.

14

SPH4U Homework: Measurement and Numbers Name: A: The Pebble Drop It is sunset. You and a friend walk on to a bridge that passes over a river. After gazing off into the distance and into each other’s eyes you both arrive at the same question: How high are we above the water? Luckily you have your smartphone with a timer app. Your friend finds a few rocks which he releases (v1 = 0). You time the fall until you see them splash in the water below. Your data is shown below.

1.73 s 1.79 s 1.82 s 1.69 s 1.81 s 1.77 s 1.74 s

1. Calculate. Find the time for the rock to fall. Express your result in the form m ± σ with an appropriate number of digits.

2. Calculate. Use your time result to calculate the distance the rocks fell (use ∆y = v1∆t + ½a∆t2). 3. Reason. How certain are you in the result of this calculation? Explain. The calculations we routinely make in physics are based on values that have uncertainties – we have just ignored this in the past. However, when we use measured results with an uncertainty we know (like the time result above) and perform a calculation with them, we would like to see the uncertainty reflected in our calculated result. To do this we will use the crank three times technique. We repeat the final step of the calculation three times, inputting the high, middle and low values from our measured result. This yields a range of values for the calculated result which we can write in the form m ± σ. 4. Calculate. What are the high and low values for your time result? Use these in your calculation to find the range of

distance values. Show the work here. Note: We emphasize showing work in our physics program, but the crank three times technique is meant to be quick and not onerous. In your calculation steps you should have done the work to simplify your equation as much as possible before the substitution of your values (∆y = ½ a∆t2). In your calculator, plug in the high and low values and record the results. We don’t need to see this work in the future – just report the resulting uncertainty (the next step below). 5. Interpret. Report your results for the distance in the form m ± σ. Provide an interpretation for this distribution of values. In our gr. 12 physics work, we will always tell you when to use the crank three times technique. If we don’t, you should simply estimate the uncertainty in a final result. This is quite crude, but sometimes is desirable to avoid too many complications.

© C. Meyer2014, adapted from John Denker 15

SPH4U Homework: Fermi Problems Name: 1. Write a complete solution (don’t skimp on the steps) for the Fermi problem: How many postage stamps would it take to

completely cover a soccer field? Physicists think about numbers in different ways compared with most people and even mathematicians. For example, if you think about the idea of the size of numbers (quantity), there are really three kinds: BIG numbers (greater than one), unity (equal to one), and small numbers (less than one). It is important to know how these numbers behave under mathematical operations. All your work on this page should be done without a calculator! 2. Estimate whether the result of each expression is BIG, small, or close to one.

a) 1 / BIG b) 1 × BIG c) 1 / small d) 1 × small e) BIG + small f) BIG – small g) BIG × small h) BIG / small i) small / BIG j) BIG / (small + BIG) k) BIG ×BIG

Physicists are often interested in the general patterns illustrated by numbers rather than their specific values. Students and even some teachers rely too much on calculators to do their thinking about numbers. As a physicist you should feel comfortable thinking about and using numbers in scientific notation without a calculator in sight! 3. Describe an easy way to compute: 6×106 + 5×106 without a calculator. 4. Compute these expressions. No calculators!

a) 6.5×105 + 3×105 = b) 6.4×1012 +2.9×1012=

5. Describe an easy way to compute: (3×102)(6×106) without a calculator. 6. Compute these expressions.

a) 3×104×2×104 = b) 6×102× 8×102 = c) 4.9×10340÷ 7×1090=

7. Describe how to use estimations and scientific notation to easily compute: 2 168 222 × 4 937 without a calculator. 8. Estimate the results of these expressions.

a) 1 168 222 × 6 900 000= b) 0.0529 × 8.0036=

© C. Meyer, 2011 16

SPH4U: Fermi Problems Enrico Fermi is a legend in the world of physics. He had a remarkable ability to find rough but reliable answers to complex problems use simple reasoning and skilful estimations.We want to be like Fermi and solve “Fermi Problems”! A: Feeling Hungry? Here is your first Fermi problem: What mass of food do Torontonians eat in one year? 1. Reason. Imagine you had a truly smartphone that would allow you to look up the information you need to solve this

problem. What information would be helpful to know for your solution? Record these on your whiteboard. You will share these with the class.

2. Record. We will call these pieces of information our key ideas for the Fermi problem solution. Record them here. In Fermi problems, we don’t usually know the values for our key ideas so we will need to estimate them. Because of this, we won’t have many reliable or significant digits. Use just one digit to write number in scientific notation whenever it is helpful. You should be able to do all the math without a calculator! When you write the values for the key ideas, assign a variable for each. 3. Reason. One key idea is the number of days per year. A sample is shown below of how you should write this. Write the

value for this key idea using one digit in scientific notation.

Number of days per year: d = days/year (common knowledge) Every number you use in a Fermi problem needs to be justified or explained. If a number is well known, indicate it as common knowledge. If it is not, you need to explain how you come up with it. The starting point for all estimations should be some number that you do know. You work from that number to get the value of your key idea.

4. Reason.Another key idea is the mass of food eaten by one person each day. You have seen many numbers associated

with the food you eat. Use this to start your estimation. Explain your estimation using words or simple calculations.

Mass of food per person per day: m = 5. Calculate. Create an equation that will give a solution to the problem. Substitute the values (including units!) into the

equation. State a final answer with one digit in scientific notation.

Recorder: __________________ Manager: __________________ Speaker: _________________

0 1 2 3 4 5

© C. Meyer2014 17

B: YourTurn! 1. How many litres of water are used for drinking purposes each year in Canada? 2. How many breaths would Julius Caesar have taken if he were still living today? 3. A car travels on the highway and collides with the concrete pillar of an overpass. What is the acceleration of the

passenger (buckled in) during the collision? Estimate the quantities you will need in order to calculate the acceleration. Show your reasoning and justify any quantities you estimate.

Fermi estimating the size of his bald spot.

Fermi about to press a button 18

t

v

SPH4U: Uniform Acceleration A:Gotta Love a Physicist in Uniform (Acceleration) 1. Reason. Describe how you could use a stopwatch and a police radar gun to

decide whether a car was moving with uniform acceleration. 2. Explain. Use the word “net force” to help explain in a simple way to decide whether an object’s acceleration is uniform. 3. Reason. A cart is on a frictionless incline. Consider the following series of events: (1) The cart is at rest and you begin to

exert a steady force on the cart up the incline. (2) The cart leaves contact with your hand as it rolls up the incline. (3) It reaches its highest point. (4) The cartreturns to the bottom of the incline. (a) Albert comments, “The cart is accelerating uniformly from moment (1) until moment (3).” Do you agree? Explain. (b) Marie says, “The acceleration of the cart changes at moment (3). This makes sense because it changes direction.” Do you agree? Explain. (c) (as a class) Sketch a v-t graph for the cart’s motion based on the motion detector. Label events 1 through 4.

B: The BIG 5 The equations in the table to the right are affectionately known as the “BIG 5” equations of uniform acceleration. When an object is accelerating uniformly, the five kinematic quantities, d∆ , iv , fv , a , and t∆ , for that time interval are related by these equations. For the BIG 5 to give correct values, you must make sure that the object is accelerating uniformly during the entire time interval between the two events you choose – otherwise you need to choose new events!

d∆ iv fv a t∆

tavv if ∆+=

221 )( tatvd i ∆+∆=∆

tvvd fi ∆+=∆ )(21

221 )( tatvd f ∆−∆=∆

davv if

∆⋅+= 222

Recorder: __________________ Manager: __________________ Speaker: _________________

0 1 2 3 4 5

© C. Meyer, 2013 19

1. Find a Pattern. Place an “X” in the column of any quantity that is not found in each equation. Describe the pattern you observe in the chart.

Note that iv and fv are the instantaneous velocities at the initial and final moments of an interval of time. We will always use a numerical subscript corresponding to an event to help label different velocities and positions (for example v2, v3 for thevelocities at moments 2 and 3). If you ever need to distinguish between two different intervals, you can write Δt12 compared with Δt23, otherwise we just write Δt (similarly for Δd and a). For displacements in the x- or y-directions, you may write Δx or Δy instead of Δd. 2. Apply. You have some data for thecart problem we discussed in question A#3. At moment 3, v3 = 0 m/s. During an

interval, a23 = 1.6 m/s2 and∆x23 = 0.80 m. Is v3 the initial or final velocity for this interval? 3. Apply. Describe how you can use the chart to help choose the best equation to solve a problem where v3 = 0 m/s, a23 =

1.6 m/s2, ∆x23 = 0.8 m, and ∆t23 is the unknown. 4. Reason. The BIG 5 equations, how many quantities (pieces of data) do you need to know in order to be able to solve for

any unknown kinematic quantity? 5. Summarize. (as a class) What is the magic saying for solving problems using the “BIG 5”? C: The Solution to All Your Problems Solution writing is like writing an essay: not only must you have the right ideas, but they must be convincingly presented using proper grammar and form. The BIG 5 are vector equations. This means they take into account the direction of the kinematic quantities. A simple way to handle direction information is to use a sign convention and write down the BIG 5 as scalar equations.Each exampleshown to the right is correct, but the scalar version is often simpler and quicker to write down. This will be our preferred notation. When we work out solutions for our homework, and on tests and quizzes we will use a very careful solution process.The rationale is this: the more carefully we think about a single problem, the deeper we will understand it. We will learn more by doing a few problems very carefully than by doing many problems carelessly. This helps us to learn how to explain what is happening in a problem using many different techniques. On the next page you will find the solution sheet that we will use for all our problem solving. You must use this process for your homework problem solving. 1. Evaluate.A world’s land speed record was set by Colonel John P. Stapp when in March 1954 he

rode a rocket propelled sled that moved along a track at 1020 km/h. The brakes were activated and the sled was brought to a stop in 1.4 s. What acceleration, in m/s2 did he experience while stopping?A sample solution is provided on the next page. Its format is ideal, but there are minor errors in the math or physics. Circle all the errors,describe the error, and correct each one.

Vector notation x∆ = 53 m [E]

1v = 12.4 m/s[E] a = 0.59 m/s2 [W]

t∆ = ? 2

21

1 )( tatvx ∆+∆=∆

Scalar notation

E +x v1 = 12.4 m/s

∆x = 53 m ∆t = ?

∆x = v1∆t + ½a(∆t)2

a = - 0.59 m/s2

20

Motion Homework Sheet Problem:Colonel Stapp! A: Pictorial Representation Sketch, coordinate system, label givens & unknowns using symbols, conversions, describe events

B: Physics Representation Motion diagram, motion graphs, velocity vectors, events

C: Word Representation Describe motion (no numbers), explain why, assumptions Colonel Stapp is initially moving rapidly and begins to slow down to the sled’s brakes. He comes to rest. We assume his acceleration is constant. D: Mathematical Representation Describe steps, complete equations, algebraically isolate, substitutions with units, final statement Colonel Stapp is slowing down for a time interval: ∆t = t2 – t1=1.4 s – 0 = 1.4 s Find his acceleration while slowing down. We know v1, v2and∆t.

v2 = v1 + a ∆t. ∴a = v1 /∆t, since v2 = 0 = (3672 m/s) / (1.4 s) = 2622 m/s

He slowed down at a rate of 2600 m/s in the backwards direction. E: Evaluation Answer has reasonable size, direction and units, why? Since he was travelling very fast and slowed in a very small time interval, we expect his acceleration to be quite large. His acceleration was negative while his velocity was positive which is correct for an object slowing down. The units are m/s which are appropriate for acceleration.

1 ●

●

●

● + x

2 ●

1v

v∆

Event 1 = he begins to slow down Event 2 = he comes to a stop

x1 = 0 v1 = 1020 km/h = 3672 m/s t1= 0

x2 v2 = 0 t2= 1.4 s

+ x

a= ?

v

t

x a

t

t

1 ●

1 ●

2 ● 2

● 2 ●

● 1

Hint: Did you find seven unique errors?

21

SPH4U Homework: Representations of Motion Name: A cart moves along a straight line track starting at rest at the origin. You only know acceleration information. 1. Interpret. What does the area between an

acceleration graph and the axis represent? (Hint: the width is a time interval and the height is an acceleration. For extra hints look in the text!)

2. Reason. During the first 30 s, what is

happening to the cart? Explain. 3. Reason. According to the acceleration

graph and important event takes place at 30 s? What might have happened?

4. Reason. How will the velocity value just

before and just after the event at 30 s compare? What does this imply about the slope of the position graph at those times?

5. Reason. We know the cart started from rest. How can we decide based on the acceleration graph when next it is at rest?

(Hint: Use some simple calculations) 6. Calculate. Complete the calculations necessary to reconstruct the velocity graph. Then complete the calculations

necessary to reconstruct the position graph.

© C. Meyer, 2014 22

SPH4U: Representations of Motion A cart travels along a dynamics track in a variety of situations. Use the information provided to complete all the other representations of motion. For all cases, the positive direction is to the right and the origin is at the left end of the track. The sample equation is one possible equation that represents the desired motion – the actual values should be reasonable for our dynamics track. Begin by looking through all the examples shown on these two pages so you have a good sense of the format for each part. A: Comparing Velocity Vectors In grade 11 we have been drawing velocity vectors to provide another way of understanding motion. Now we will learn a special way to compare two velocity vectors and find the change in velocity. 1. Reason. A car is travelling west and slowing down. What is the direction of its acceleration? What is the direction of its change in velocity? 2. Represent. The car was initially moving at 10 m/s [W] and was later travelling at 3 m/s [W]. Draw each velocity vector

and draw a vector representing your guess at the car’s change in velocity. The change in velocity can be found form the expression: ∆v = v2 – v1. We can draw a vector diagram to represent this equation. In gr. 11 we learned that when we add vectors we draw them tip-to-tail. Here, the two vectors are subtracted. We represent this by drawing them tail-to-tail. Another way of thinking about this is that the two vectors start at the same point. The change vector is a new vector going from the tip of the first to the tip of the second vector. This representation will be our velocity-vector diagram. 3. Represent. Redraw your velocity-vector diagram so your three vectors are connected together as described above. . 1 Description of Motion

The cart starts at rest atthe origin, moves in the positive direction and speeds up.

Motion Diagram

Velocity Vectors ∆v = v2 – v1.

Sample Equation

∆x = (0 m/s)∆t + ½ (0.50 m/s2)∆t2

Real-Life Situation

Graphs

x v

t

a

t t

Recorder: __________________ Manager: __________________ Speaker: _________________

0 1 2 3 4 5

+x

© C. Meyer, 2014 23

2 Description of Motion

Motion Diagram

Velocity Vectors

SampleEquation

∆x = ( )∆t + ½ ( )∆t2

Real-Life Situation A car passes a police cruiser and begins slowing down (1) until the proper speed limit is reached (2).

Graphs

|

3 Description of Motion

Motion Diagram

Velocity Vectors

SampleEquation

∆x = ( )∆t + ½ ( )∆t2

Real-Life Situation

Graphs

4 Description of Motion

Motion Diagram

Velocity Vectors

SampleEquation

∆x = ( )∆t + ½ ( )∆t2

Real-Life Situation

Graphs

d v a

t t t

1v2vv∆+

d v a

1

2

t t t

+

1 ●

●

●

2 ●

+

x v a

t t t

24

SPH4U: 2-D Motion How does a force affect the motion of an object that is free to move in two dimensions? Let’s find out! A: The Hoverpuck The hoverpuck creates a cushion of air that lifts it above the ground. When it moves, friction is so small that we can safely ignore it. Imagine you give the puck a gentle push along the level ground and release it with a velocity 1v as shown below. 1. Predict. Describe how the puck will move after it leaves

your hand. 2. Represent. Draw three images of the puck after you release it after equal intervals of time (much like our motion

diagrams). 3. Test. (as a class) Describe the motion of the puck along the level ground after it is released. Does this agree with your

prediction? 4. Reason. Marie says, “The force from your push is carried by the puck as it travels along the floor. That’s why it keeps

moving forwards after your push.” Do you agree or disagree with Marie? Explain. B: One Tap Now imagine you release the puck and then your friend gives it one short ‘tap’ in a direction that is perpendicular to its motion. 1. Predict. How will the puck move after the tap? What will

the shape of its path be? 2. Predict. Describe how the overall speed of the object will

change. 3. Test. (as a class) Describe the motion of the puck after the tap and describe the shape of its path. Does this agree with

your prediction? 4. Represent. Draw three images of the puck three equal intervals of time after the tap. 5. Explain. Why does the puck gain speed in the y-direction? 6. Speculate. Did the puck gain or lose speed in the x-direction?

Recorder: __________________ Manager: __________________ Speaker: _________________

0 1 2 3 4 5

S N (+x)

W (+y)

E

tap 1v

1v

© C. Meyer, 2013 25

C: Off to the Races! It is difficult to decide from our previous observations whether the speed in the x-direction changed after the tap. To find out, we will have a race between two pucks. Each puck starts with the same initial velocity in the x-direction. Puck A travels on an incline tilted to the east. The puck B moves beside puck A along the level ground. 1. Predict. What is the shape of the path of puck A after it is

released? Explain. 2. Predict. Which puck will reach the finish line first? Explain. 3. Test. (as a class) Describe the motion of Puck A after it is released. Does this agree with your prediction? 4. Evaluate. Did the velocity of Puck A in the x-direction change? Explain how you decided based on your observations of

the race.

5. Represent. Draw three images of each puck after three equal intervals of time after they are released. Explain how you

chose the x- and y-positions for the images of the pucks. D: Rematch! Consider another race. Both pucks are on the incline tilted east. Puck A starts from rest, while puck B has an initial velocity in the x-direction. 1. Predict. Describe how you think puck A will move when it

is released. 2. Predict. Which puck will reach the bottom of the incline

first? Explain. 3. Test. (as a class) Describe the motion of Puck A after it is released. Which won the race? Do these results agree with

your predictions? 4. Represent. Draw three images of each puck after three equal intervals of time after they are released. Explain how you

chose the x- and y-positions for the images of the pucks. 5. Summary. How does a force in one direction affect the motion of an object in the perpendicular direction?

A 1v

1v

finish line

B

1vA B

finish line

26

SPH4U Homework: 2-D Motion Name: When we represent motion in two dimensions, we can imagine we are drawing a motion diagram that can produce dots anywhere in the x-y plane. Study the motion diagram for the trip of a very interesting hover puck that is shown to the right. A few of the positions have been marked with event numbers. 1. Interpret. Is the puck accelerating between moment 1 and 2? Explain how

you can tell.

2. Interpret. Is the puck accelerating between moments 4 and 5? How can you tell?

When we draw an instantaneous velocity vector, we start at the moment in time (the dot) we are interested in. Its length corresponds to the speed and its direction is tangent to the path of dots. 3. Represent. Draw an instantaneous velocity vector for the puck’s motion at

moments 1, 2, 3 and 5. (v4 has been drawn as an example.) Their exact length is not important, only their relative length.

4. Interpret. Did the puck’s velocity change between events 2 and 3? Did it accelerate between events 2 and 3? Explain

how you can tell. When vectors are subtracted, we draw them tail-to-tail. The vector that represents the difference goes from the tip of the first vector to the tip of the second. 5. Interpret. Examine the velocity vector diagram representing the vector equation:

2323 vvv −=∆ The vector that represents the difference (the change ∆v23) goes from the

tip of the first vector (v2) to the tip of the second (v3). Use this diagram to help describe the direction of the force experienced by the puck.

6. Reason. Would you describe the force acting on the puck between events 2 and 3 as a short, sudden force or a long

lasting force? Use the pattern of dots to help explain.

7. Represent and Reason. Draw a vector diagram for the equation: 4545 vvv −=∆ . Use the

diagram to help explain the direction of the force acting on the puck between moments 4 and 5. Would you describe this force as a short, sudden force or a long lasting force? Explain.

1

2

3

4

5 +x

+y

Vector Diagram

© C. Meyer, 2014

2v3v

23v∆

27

SPH4U Homework: Vectors and Components Name: 1. Reason. Three vectors are drawn on the grid shown to

the right. Rank the six components of these three vectors in order of increasing size. For example: |Ay| > |Bx|

2. Reason. A vector makes a 30o angle with the x-axis. Which of its two components has the larger size? Explain.

3. Reason. Two vectors are shown on the grid to the right.

Rank the four components of these two vectors in increasing order according to size.

4. Represent and Calculate. A vector has the

components: ∆dx = - 46 m and ∆dy = 30 m. Sketch a component triangle for the resultant vector (not a scale

diagram!). Calculate the complete vector d

∆ .

5. Represent and Calculate. A plane is travelling with a velocity of 340 km/h [N40o W]. Sketch a component triangle and calculate the components of this vector. (Don’t draw a scale diagram!)

6. Reason. Vector equations have components too! You

were doing this last year without realizing it when you wrote separate expressions for Fnetx and Fnet y. Imagine we are adding two force vectors: tfnet FFF

+= where

fF

= 17 N [N 30o E] and tF

= 9 N [N 50o E]. (The force vectors are not drawn to scale!)

(a) Draw a component triangle attached to each of the

three vectors in the diagram above. Calculate the components of the vectors you know.

(b) The x-component of the net force can be found from taking the x-component of the entire equation! Fnet x = Ff x + Ft x. Fill in this equation and solve for Fnet x.

(c) Do the same for the y-component.

(d) Use the component triangle for Fnet to find netF

.

+y

+x

fF

tF

netF

+y (N)

+x (E)

+y

+x

A

B

+y

+x

A

B

C

© C. Meyer, 2014 28

SPH4U: Vector Components How do we analyze the curving motion of the puck on the ramp? We need to develop new vector techniques for two-dimensional motion. Here we go! A: The Component Triangle In yesterday’s experiments, careful measurement were made by a student who found that the puck moved through a displacement of 4.0 m [N 30o E]. 1. Represent. Draw this vector on the grid starting from the image of the puck and label it d

∆ . The grid represents the

position-space of the puck. This means that the lengths of vectors in this diagram represent the magnitude of an object’s displacement. Use the scale of 1.0 cm = 1.0 m for the position-space.

2. Represent. During the 2.0 seconds of its motion, the puck

moved both in the x- and y-directions. Draw a right-angle triangle on the grid with d

∆ as the hypotenuse. Draw the

smaller sides using a different colour or dashed lines. We will call this the component triangle. Draw arrowheads on each side of the triangle.

3. Measure. Use the scale diagram and a ruler to measure

how far the puck travelled in each direction. Don’t forget the uncertainties!

a) x-direction: b) y-direction:

A vector is a quantity with two parts: a magnitude (size) and direction. It is usually notated with a vector sign on top ( d

∆ , v

). When we want to refer to the magnitude of the vector only, we write v in absolute value signs. However, out of convenience, if it is understood that the quantity involved is a vector, we usually write just v to indicate the magnitude. Note that the magnitude of a vector is always a positive quantity. A vector can always be determined from the values of its components. The two small sides of the triangle you drew in question 2 represent the components of that vector which we call

xd∆ and yd∆ . The components are written as scalar quantities without a vector sign. Be sure to use a sign convention to show the direction of the components: make the values of the components either positive or negative. 4. Write down the components you found for d

∆ .

xd∆ = yd∆ = B: Constructing Vectors Now we will draw a different kind of diagram. The grid now represents the velocity-space of the puck. This means that the lengths of vectors in this diagram represent the magnitude of an object’s velocity. It no longer indicates anything about the object’s position or displacement. 1. Represent. Draw the velocity vector, v = 8.0 m/s [S 20oW]

on the grid using the scale 1 cm = 2.0 m/s. Label the vector and construct its component triangle.

2. Measure. Use your diagram to measure the values of the

components of v . Don’t forget uncertainties! vx = vy =

3. Reason. Use the Pythagorean Theorem to write an equation that relates the magnitude of v to its components vx and vy. Don’t include any numbers yet.

S N (+x)

W (+y)

E

Recorder: __________________ Manager: __________________ Speaker: _________________

0 1 2 3 4 5

© C. Meyer, 2014

W (+y)

N (+x)

E

S

29

4. Calculate. Use the measured values of vx and vy to calculate the magnitude of vector v . Estimate an uncertainty for this result. Does your calculated magnitude based on your measurements agree with original magnitude you were given?

5. Reason. Use the sine and cosine functions to write an expression for vx and vy when we know v and the angleθ inside the

component triangle at the tail of the vector v . You must include a negative sign for the sign convention when appropriate. (Note: Do not memorize these expressions! Always deduce them from your component triangle.)

6. Calculate. Use the angle at the tail of v and the given magnitude of v to calculate the values of vx and vy. How do

these values compare with those you measured directly for vx and vy? 7. Calculate. How we can find the angle θ from the values |vx| and |vy| and the inverse tangent function? Find θ using your

measured values. Estimate an uncertainty. 8. Reason. Does your calculated value for the angle θ agree with the original value you were given?

9. Reason. If a vector was reversed – flipped to the opposite direction – how would the values of the components change? C: Summary 1. Describe. What procedure would to you use to find a complete vector if you were given its two components? 2. Calculate. The same displacement vector d

∆ is shown to the right along with two different coordinate systems A and B.

Complete the chart below using measurements from the diagrams. Scale:1.0 cm = 10.0 cm Coordinate System

A Coordinate System B

d

∆ (magnitude + direction)

Angle with respect to the x-axis

x-component

y-component

W E (+x)

N (+y)

S

d

∆

A

W (+x)

E N (+y)

S

d

∆

B

30

SPH4U: Projectiles! Now it’s time to use our new skills to analyze the motion of a golf ball that was tossed through the air. Let’s find out what is special about the motion of a projectile. A: Tracking a Projectile 1. Observe. Choose a convenient reference point on the golf ball to help track its motion. Measure the x- and y-components

of the position of the golf ball at each moment in time. The coordinate system for your measurements is drawn on the picture. The strobe light for the photo flashed at 10 Hz. Complete the chart below.

B: Horizontal Motion 1. Represent. Plot a graph of x vs. t 2. Interpret. Based on the pattern of data in the graph, describe the motion of

the projectile in the x-direction. When we analyze data graphically, we are looking for a relationship or pattern between the two related quantities. Never connect the dots on a graph and create a zig-zag pattern. Decide if the relationship fits a straight line or a smooth curve. 3. Calculate. Use an appropriate graphical technique to find the horizontal component of the velocity, vx. Do not use

values from the data table, use readings off the graph!

4. Represent. Use the graph to help create an equation that represents the horizontal position of the ball. Be sure to include

units!

Image No.

t(s) x (cm) y(cm)

1

2

3

4

5

6

7

8

9

10

+x

+y

Recorder: __________________ Manager: __________________ Speaker: _________________

0 1 2 3 4 5

© C. Meyer, 2014

0 0.2 0.4 0.6 0.8 1.0 t (s)

x (cm)

8.0

6.0

4.0

2.0

0

31

5. Reason. Albert says, “This graph shows that there are no significant forces acting in the horizontal direction”. Do you agree or disagree? Explain. What can we conclude about the effects of air resistance?

** call your teacher over ** C: Vertical Motion 1. Represent. Plot a graph of y vs. t 2. Reason. Emmy says, “This graph shows evidence of a downwards

acceleration.” Do you agree or disagree with Emmy? Explain. 3. Represent. Sketch a vy-t and an ay-t graph based on the y-t graph. Line up the

graphs’ features with the y-t graph above. Label the regions in time when the ball is speeding up, slowing down and has a vertical speed of zero.

4. Reason. Marie says, “At its highest point the acceleration of the golf ball is

zero since it’s turning around.” Emmy says, “No. At the top it’s still accelerating.” Who do you agree with? Explain.

D: Projectile Motion 1. Summarize. Use the observations you have developed to create a model of projectile motion. Your model should begin

by explaining that we treat the object as a point particle. Describe the characteristics of the particle’s vertical and horizontal motions and mention any assumptions the model relies upon.

Projectile Motion:

2. Apply. According to your model, will the golf ball ever be found falling straight down? Explain. 3. Predict. Two projectiles are launched with the same initial speeds but different angles. Marie launches hers with an

angle of 60o to the horizontal and Albert launches his at an angle of 30o to the horizontal. According to your model, whose projectile will spend more time in the air? Explain. We will test this using a simulation.

y (cm)

6.0

4.0

2.0

0

0 0.2 0.4 0.6 0.8 1.0 t (s)

32

SPH4U Homework: Projectiles! Name: The Orthogonality Principle: Forces acting in one direction have no effect on an object’s motion in the perpendicular (orthogonal) direction. You are using a toy truck to conduct some physics experiments that will explore the relationship between the truck’s speed, the table’s height and the time it takes the truck to fall and reach the floor. You plan on using the six situations shown in the diagram below.

1. Reason. Isaac has a hypothesis: “The truck’s speed

affects the time to fall” and his prediction is “The greater the speed of the truck, the more time it will take to fall.” Which of the six situations could you use to best test his prediction? Explain.

2. Evaluate. Your results for the three different situations

using the same height tables are 0.29 s, 0.32 s, 0.27 s each with an uncertainty of ±0.05 s. Do your results support or refute Isaac’s prediction? Explain.

3. Explain. Isaac’s hypothesis has been proven wrong. If he understood the orthogonality principle better he would have

realized that his hypothesis didn’t have a chance. Help explain to Isaac why his hypothesis was doomed! 4. Reason. Marie has a hypothesis: “The height of the table affects the time to fall” and she predicts “The greater the

height, the greater the time to fall.” Which of the six situations could you use to best test her prediction? Explain. 5. Evaluate. Your results from the three different situations using the same speed are 0.28 s, 0.35 s and 0.40 s each with an

uncertainty of ±0.05 s. Do your results support or refute Marie’s prediction? Explain. 6. Explain. Use the orthogonality principle to explain why Marie’s hypothesis was correct. 7. Reason. Rank the six situations according to how much time the trucks spend in the air before hitting the ground, from

smallest to largest. Explain your ranking.

33

34

SPH4U: Projectile Problem Solving The key idea which allows us to solve projectile problems is the relationship between the horizontal and vertical motions. Since the vertical physics does not affect the horizontal physics, we can treat a single projectile problem as two related kinematics problems – one for each direction. A convenient way to show the direction of the velocities used to describe projectile motion is to simply indicate the angle and use a sign convention with positive for above the horizontal and negative for below. For example: 12 m/s [32o] or 150 km/h [-12o]. The Ski Jump – One Giant Leap … The ski jump is an exciting and death-defying event that turns human beings into projectiles! Let’s study the physics of the craziest winter sport as featured at the Vancouver Winter Olympics in 2010. A typical ski jumper will be launched with a velocity of 26.1 m/s [-11.25o]. What is hard to notice from TV and photos is that the launch angle is below the horizontal (downwards)! A jumper makes her leap and we note three events: (1) leaving the ramp, (2) part way down after 1.8 s, and (3) just before landing, 35.8 m below the starting position.

A: Pictorial Representation Sketch, coordinate system, label givens & unknowns, conversions, describe events

B: Physics Representation Motion diagram, motion graphs, velocity vectors, force diagram, events

For two-dimensional motion we will draw a special kind of motion diagram. Start by drawing a motion diagram along the x-axis for the motion in the x-direction.Next, do the same for the y-axis. Finally, use the two motion diagrams to help draw a third one using the grid which shows the complete, two-dimensional path of the object. 2. Represent. Complete the 2-D motion diagram for the ski jumper. Be sure to

label events 1, 2 and 3. Don’t worry about being too precise, as long as the correct ideas are shown. Describe how you chose to draw the dot patterns along the x- and y-axes.

Recorder: __________________ Manager: __________________ Speaker: _________________

0 1 2 3 4 5

1v

Horizontal Vertical v1x = v1y =

= =

ax = ay =

∆y13 =

∆t12 =

1. Represent. Begin all your projectile motion problems by drawing a sketch and creating a chart listing what you know about the horizontal and vertical motion. Draw a component triangle for any known vectors and include the components in the chart.

© C. Meyer, 2014

+x

+y 35

3. Represent. Complete motion graphs for the x- and y-components of the ski jumper’s motion.

A velocity vector diagram shows the relationship between a pair of velocity vectors and shows how the velocity has changed. This is summarized by the equation: ifif vvv

−=∆ . Draw the vectors vf and vi tail-to-tail in order to subtract them. The change in velocity, ifv∆ , points from the tip of vi to the tip of vf. When drawing your 2-D velocity vectors, make sure that both the lengths and directions seem right for the moments in time you are considering. 4. Represent. Draw the velocity vector diagram for events 1 and 2. Draw a force

diagram for the ski jumper. 5. Reason. How do the directions of ∆v, a and Fnet compare? Why? 6. Represent. Complete the word representation describing the physics below.

C: Word Representation Describe motion (no numbers), explain why, assumptions

7. Solve. Find her complete velocity vector (magnitude and direction) at moment 2. Complete the math representation

below.

D: Mathematical Representation Describe steps, complete equations, algebraically isolate, substitutions with units, final statement

Vectors 1212 vvv

−=∆

Force Diagram

vy t

t y

t vx

x t

iv

fv

ifv∆

36

8. Evaluate. Imagine you erased all the math in part D above. Would your descriptions of the steps be good enough to help a struggling student work their way through this problem? If not, go back and improve them! Then complete part E, the evaluation, below.

E: Evaluation Answer has reasonable size, direction and units?

** Call your teacher over to check your work. Then you are ready to check your results against the simulation** 9. Reason. Isaac says, “To find how far the ski jumper travels horizontally when she reaches the ground I just need to

construct a triangle with an 11.25o angle and a 35.8 m height.” Albert says, “I don’t think this will help.” Who do you agree with? Explain.

10. Solve. How far has the skier travelled horizontally between moments 1and 3? Answer this in Part D below.

D: Mathematical Representation Complete equations, describe steps, algebraic work, substitutions with units, final statement

11. Reason. Emmy says, “I was wondering about this. Imagine I toss a blob of playdoh which lands on the floor. When it

hits the floor its final velocity is zero. Do you think I could use v2 = 0 in my equations?” Can she? Explain. The Great Jumper Sondre Norheim (1825 – 1897) was a ski jumping champion and the designer of the modern ski used for ski jumping. The modern ski acts like a wing, providing the jumper with an upwards lift force. In gr. 12, we ignore all effects of the air and this upwards force. The story goes that Sondre wowed a group of spectators by jumping over a very tall rock. Let’s explore the physics of this daredevil event. We will suppose that he launched from a ramp with a speed of 18.0 m/s at an angle of 28o above the horizontal. The edge of the ramp was 1.5 m above the ground level. The tallest point of the rock was located 13.8 m horizontally from the edge of the ramp and was 5.0 m above the ground. The ground in this area is quite level. Complete the three parts of the solution below.

37

A: Pictorial Representation Sketch, coordinate system, label givens & unknowns, conversions, describe events

1. Reason. Our goal is to determine whether or not he will make it over the rock. What kinematics quantity would it be

helpful to find (and compare with the given information) that would allow you to decide? Explain carefully. B: Physics Representation Motion diagram, motion graphs, velocity vectors, force diagram, events

Vectors

Force Diagram

2. Calculate. Solve this problem in part D below. D: Mathematical Representation Describe steps, complete equations, algebraically isolate, substitutions with units, final statement

vy t t

vx

t y x

t

+y

+x

Horz Vert

38

39

SPH 4U: Representing Forces Forces make the world go round! They are the real reason you get up in the morning. To help us understand forces we use a variety of tools which we will explore today. An interaction diagram helps us to visualize the interactions between the system objects and the environment. Start by writing the name (or an abbreviation) for each object involved in the interactions. Draw circle around the system objects to highlight them. Draw an interaction line between each pair of objects for each interaction. Label these lines with a letter similar to the ones you use to label forces. A: The Handy Ball 1. Interpret. What does the interaction diagram tell us about this situation?

2. Represent. The ball is moving downwards and is slowing down. Draw the

motion diagram for the ball. A force diagram uses our understanding of the interactions to construct a picture of the forces acting on a system. When you construct a force diagram: Identify the object or objects of interest – we will call these objects the system. Objects outside the system are parts of

the environment. Represent the system as a point-particle where we imagine all its mass is compressed into a single point. Draw a vector arrow representing each force acting on the system associated with each interaction. The force vectors do

not need to be drawn to scale, but should be drawn roughly according to their relative magnitudes. Include a separate wiggly acceleration vector when appropriate. Draw a coordinate system with a sign convention. In most cases, it is convenient to choose the direction of the

acceleration as positive; otherwise choose the direction of the velocity as positive. 3. Reason. Examine the force diagram for the system. Do the lengths of the vectors

correctly represent this situation? Explain. The total effect of all the forces acting on the object is called the net force, Fnet. To calculate the net force, we must add up all the force vectors found on the FD. To do this, we construct two scalar equations that represent the x- and y-components of Newton’s second law. Make sure you do this even if the net force is zero. Always write the scalar equation for the 2nd law using the force sign convention. The Force Sign Convention: In gr. 12 we will write scalar equations for forces (no vector arrows!). When we do, all force symbols, such as Fg, are magnitudes meaning they are positive quantities. Show the directions of the forces by using a sign convention and adding or subtracting the appropriate force symbol. 4. Explain. Examine the scalar equation for Newton’s 2nd law in the box above. Explain what the implicit (hidden) positive

sign in front of the Fg and the minus sign in front of the Fn tell us about the two forces.

Sketch Interaction Diagram

Motion Diagram

Force Diagram

Newton’s 2nd Law

Fnet y = may

Fg – Fn = may

gF a

+y

● nF

ball

n g hand

Earth

Recorder: __________________ Manager: __________________ Speaker: __________________

0 1 2 3 4 5

© C. Meyer, 2014

+y

40

5. Reason. Consider two similar situations where the ball is (1) at rest and (2) moving upwards while slowing down. Are the interactions in either of these situations different compared with the first one above?

6. Represent and Explain. Why are the results (the acceleration) different in these

cases compared with the first one? Draw the force diagram for each situation to help explain.

Whenever a person touches another object they interact. This interaction can give rise to a normal force or both a normal and friction force. To keep things simple, when a person pushes on something we often call the resulting force an applied force. It is important to understand that this applied force may actually be a normal force or a combination of normal and friction forces. If you include an applied force on a force diagram, be careful that you don’t also include the normal or friction force which is already includes as part of your applied force. 7. Predict. When you place a heavy object in your hand, you feel pressure against your hand. Which interaction, gravity or

normal is your hand feeling? 8. Test. Place a heavy mass (~1.0 kg) in the flat palm of your hand. How does the sensation change when (i) the mass is at

rest, (ii) when you suddenly accelerate it upwards, and (iii) when you suddenly accelerate it downwards? Match each sensation to one of the previous FDs.

9. Evaluate. Which force in your three force diagrams above have the same characteristics as the sensation you feel? Which

interaction does your hand actually feel? (Look back to your interaction diagram to confirm!) B: The Skydiver! A skydiver is now falling at a constant speed because her parachute has deployed. 1. Represent. Complete the chart to the right for the

system of the skydiver and parachute.

2. Explain. There is an interaction between the skydiver and the parachute. (Make sure this is clear in your ID). Should we include a force on the FD due to this interaction? Explain.

3. Reason. Emmy says, “Since the skydiver is moving downwards, the net force should be downwards.” Do you agree or

disagree with Emmy? Explain.

FD at rest FD slowing down

Interaction Diagram

Motion Diagram

Force Diagram

Newton’s 2nd Law

+y

41

The Air Resistance Rule: In Gr. 12 physics, we will always assume there is no force due to air resistance unless it is mentioned or the situation does not make sense without it. C: The Trampoline You are bouncing very high up and down on a trampoline. We will describe the force the trampoline exerts on you as an elastic force (Fe). 1. Represent. Draw a

motion diagram and a force diagram for each moment during your bounce.

2. Reason. Isaac says, “On the force diagram for moment 3 we should draw an upwards force due to

the trampoline since he is still traveling upwards. We should make it smaller than gravity since that force is running out.” Do you agree or disagree with Isaac? Use an interaction diagram to help explain.

3. Reason. If there is no upwards force acting on you at moment 3, why do you continue moving upwards? Explain without

using the word inertia. The Inertia Principle: All objects made of ordinary matter have the property of inertia. This property means that it takes time (an interval of time) in order for the velocity of any object to change. The time interval in some cases may be very small, but it is never zero. The amount of inertia is the object’s mass. Inertia is not a force- it is a property of matter. D: Book Learnin’ A friend is pushing a heavy textbook towards you across the rough surface of a table. While your friend is pushing, you help to slow it down with your hand before it falls off the table. When multiple forces point in the same direction, draw those forces tip-to-tail, just like you do when adding vectors in a vector diagram. That way we can easily see the total force in each direction. 1. Represent. Complete the

chart for the system of the book while the two of you are pushing on it.

2. Explain. How did you

decide on the length of the force vectors in your force diagram?

(1) Traveling downwards and

slowing down due to the trampoline

(2) Travelling upwards and

speeding up due to the trampoline

(3) Moving upwards after leaving contact with the trampoline

(4) Moving downwards before landing back on the

trampoline

Interaction Diagram Force Diagram Newton’s 2nd Law

Motion Diagram

+x

+y +y +y +y

ID

42

SPH4U Homework: Representing Forces Name: Complete the chart for each situation. Description Sketch Interaction Diagram Force Diagram Newton’s 2nd Law

1 A ball was thrown straight upwards and

hits the ceiling.

System = ball

2 A tasty chocolate in your hand is moving

upwards and is slowing down as it approaches your

mouth.

System = chocolate

3

Fnet x = max

Ff = max Fnet y = may Fn – Fg = 0

4 A tennis ball is struck by a racquet. System = ball

5

6 A book is sitting on top of a cart. You pull the cart such that the cart and the book move with a steady speed. System = cart + book

7 A book is sitting on top of a cart. You pull the cart such that the cart and the book move with a steady speed. System = cart only!

book

book

●fF

aF

nF

gF

a

v

© C. Meyer, 2014

43

SPH4U: Forces in 2-D A: Tilted Forces The cart experiences a force, 1F

= 5 N [Left] and a force 2F

which is directed

[Right 30o Up]. The cart remains at rest on a level surface. 1. Represent. Draw the ID and FD for this situation. Construct a component triangle for the force 2F

. (Reminder: an ID

only shows the objects and interactions)

ID

FD Component Triangle for 2F

2. Explain. How are the forces or components of forces balancing in this situation? 3. Predict. How do you think the magnitude of

2F compares with

1F ? Predict a value for F2. Explain your reasoning.

4. Reason. Now it’s time to get two 10-N spring scales, one dynamics cart, and one protractor. What is the readability of

your spring scale? If you make a series of measurements of one quantity with your scale and compute an average value, what do you think the uncertainly of that result might be? Provide an estimate.

Spring Scale Tips: 1) Hold the scale in the vertical or horizontal position in which it will be used. Calibrate it to read 0 N with no forces applied. 2) Do not twist it when forces are applied. The internal pieces may bind and give a false reading. 5. Test and Evaluate. Create this situation using two spring scales for 1F

and 2F

, and a protractor. Measure the size of 2F

and record it here with its uncertainty. Does this result agree with your prediction?

6. Represent. Write an equation for Newton’s 2nd Law in the x- and y-directions. Get in to the habit of always showing the

sin or cos functions for your components.

7. Reason. Isaac says, “Finding the size of the normal force is easy! It is the same size as the force of gravity.” Do you

agree or disagree? Explain.

x

y

Recorder: __________________ Manager: __________________ Speaker: _________________

0 1 2 3 4 5

© C. Meyer, 2014

44

B: Forces on a Tilt The cart is at rest on a surface inclined at an angle to the horizontal. It is held in place by a force,

aF

,that is provided by your spring scale and is parallel to the incline. Later, you will use the incline set up at the front of the room. 1. Represent. Draw an ID and FD for the cart. Show the components of

any forces that make an angle with this special choice of coordinate system.

Draw the components of a force that is already shown in a FD using dashed lines or a different colour so the components won’t be mistaken for another, new force. 2. Reason. Which forces, or components of a force, act in each

direction?

+ x - x + y - y

3. Reason. Which forces, or components of a force, balance each other? How can you tell? 4. Reason. Which force or component of a force pulls the cart down along the incline? 5. Represent. Draw the component triangle for the force of gravity

gF

relative to the tilted coordinate system. Locate angle in the triangle. What are the magnitudes of the components of

gF

? Write these components using sin or cos.

6. Predict. Measure the angle of the incline and the weight of your cart. Use

Newton’s 2nd law in the x-direction to predict the size ofaF

. 7. Test. Use the incline at the front of the class to measure the magnitude of

aF

. How does this compare with your prediction? (Did you remember the uncertainty?)

8. Predict. Explain how Fa would change if the angle of the incline is increased. Test this (lift up the incline a bit). 9. Calculate. Use Newton’s 2nd law in the y-direction to find the magnitude of

nF

. 10. Reason. Explain how Fn would change if the angle of the incline is increased. The normal force exists because the two

surfaces are being pressed together. What has happened to the amount of pressing as the angle increases?

FD

x

y

ID

y

x

45

SPH4U: Understanding 2-D Forces A: Rocks on Inclines For each situation, draw the FD and ID. Write out a complete expression for Newton’s 2nd Law in the x- and y-directions. Make sure you show your trig! A rock is sliding without friction.

ID FD Newton’s 2nd Law

Friction prevents the rock from sliding.

ID FD Newton’s 2nd Law

1. Explain. What force or component of a force causes the acceleration in the first example? 2. Explain. How did you choose your coordinate system for these two examples?

B: The Rough Incline A rock slides up and then down an incline. There is a force of friction with a constant size, but not quite enough to prevent the rock from sliding back down. 1. Represent. Complete the chart below.

Sketch

Motion Diagram

Force Diagram Newton’s 2nd Law

Sketch

Motion Diagram

Force Diagram Newton’s 2nd Law

2. Predict. Will the magnitude of the rock’s acceleration be the same when it is going up the incline as when it is going

down? Justify your prediction. We will test this as a class – move on for now.

+

+

Recorder: __________________ Manager: __________________ Speaker: _________________

0 1 2 3 4 5

© C. Meyer, 2014

46

C: The Floating Ball A ball is tied to a string. Emmy says, “Watch this. I can pull the ball with the string at an angle, like this, so it moves horizontally through the air.” Isaac replies, “I don’t understand how that’s possible. The forces simply don’t work out properly.” 1. Reason. Use the

force diagram to carefully explain how the ball must be moving.

D: The Magic Eraser Albert says, “Look! I can make my eraser stay in place against my vertical hand without me holding on to it!” Marie says, “Nonsense! Gravity is guaranteed to pull it down.”

1. Reason. Complete the chart above. Who do you agree with? Explain.

C: Washing the Windows Albert is making a bit of money for university by washing windows. He pushes a sponge upwards along a window to get a dirty spot that is above his head. 1. Represent. Complete the

chart for the system of the sponge at the moment Albert’s arm makes a 30o angle with the window. The sponge is at rest. You may assume friction is small enough to ignore.

2. Reason. Rank the size of the three forces in this problem. Explain your ranking.

Can the ball travel horizontally?

ID FD Newton’s 2nd Law

Can the eraser “stick” to a person’s hand?

ID FD Newton’s 2nd Law

Sketch

ID FD Newton’s 2nd Law

v

v

47

SPH4U: Newton’s Third Law A: Forces as Interactions Throughout our unit on forces, we have been making use of the term interaction. When two objects affect one another, we say that they interact. We have also noticed that these interactions come in the form of a push or a pull on the objects which we call forces. This brings us to a very important idea. Whenever two objects interact, they each exert a force on the other. These two forces are really just two parts of a single interaction. We will call the two forces a 3rdlaw force pair. The forces in a 3rd law force pair share some important characteristics: they have the same magnitude they point in opposite directions they are the same type (gravitational, normal, etc.) they arise and act simultaneously they involve the same pair of objects This understanding of interactions is known as Newton’s 3rd Law. Please never use the words action or reaction when describing forces. To do so is simply wrong. B: Book Learnin’ One of the exciting things about studying physics is that as your understanding grows, the physics of very simple situations becomes much more nuanced and subtle. Let’s think about your physics book at rest upon a table. 1. Represent. Draw an ID and FD for the system of the book.

To show better the details of an interaction, we can use a more specific force notation which we will call 3rd law notation. For example the earth interacts with the textbook gravitationally and we can symbolize this as:

BEgF

, which reads: “the force of gravity of the earth acting on the

book”. Using this notation we can write Newton’s 3rd Law as: ABBA FF

with the understanding of the characteristics of a force-

pair. 2. Represent. Label the forces in your force diagram using this new

notation. Note that if you have done this correctly, all the subscripts for the forces on one object will end with the same letter. Double check this every time you use this notation. 3. Reason. Isaac says, “I think gravity and the normal force make up a third-law pair in this situation. Just look at the size

and direction of the forces.” Do you agree or disagree with Isaac? Explain. 4. Reason. Emmy says, “Do gravity and the normal force always have the same size? Why do they here?” Explain to

Emmy. 5. Reason. Albert says, “So if gravity and the normal force are not members of the same 3rd-law pair, is there a 3rd-law

partner forBEgF

?” Explain what it is and why it doesn’t appear on this force diagram.

Sketch Interaction Diagram

Force Diagram

book

Recorder: __________________ Manager: __________________ Speaker: _________________

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© C. Meyer, 2013

48

C: The Stack O’ Books Your homework is piling up – your slender physics text is sitting on the table and your massive biology text is on top. 1. Represent. Draw a single

interaction diagram for this situation. We will consider the bio book and physics book as two separate systems: circle each.

2. Represent. Draw a force diagram

for each text. (Hint: you should draw five forces in total!) Whenever we study the forces of more than one object, use the 3rd law notation introduced above.

A contact force is one that is evident only when two objects are in contact. A non-contact force is evident even when the two objects are not in physical contact. 3. Reason. Which forces are contact and non-contact forces? 4. Reason. Are there any 3rd-law force pairs appearing in the two diagrams? If so, indicate these with a small “” on each

member. If there is a second pair use a “” on each member, and so on. 5. Reason. Rank the magnitude of the forces appearing in the two force diagrams from smallest to largest. Explain your

ranking. 6. Represent. Based on your ranking of the magnitudes of the forces, do you need to make any modifications to your two

force diagrams? Explain why or why not. 7. Reason. Compare the force diagram for the physics text from parts B and C. Which forces changed when the biology

text was added and which remained the same? Explain. 8. Test. Marie says, “In this situation, the weight of the biology text acts on the physics text.” Test this assertion. Use your

hand in the place of the physics text. Lower the biology text (or other, hefty book) onto your hand. Are you feeling a contact force, or a non-contact force?

9. Summarize. Based on today’s investigation and the rest of our work in gr. 12 physics, what can we conclude about the

size of the normal force compared with the force of gravity? What is the best way to determine the size of the normal force?

Sketch Interaction Diagram Force Diagram Force Diagram

Biology Physics

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51

SPH4U: Weight and Acceleration

A: The Elevator You may have noticed a curious sensation while travelling in an elevator. At certain times, it feels like your weight is changing. Since you are such a curious student, you decide to investigate this. You step into an elevator at the ground floor of a tall building and stand on top of a bathroom scale that gives readings in newtons. You haven’t pushed any buttons yet and you look down at the scale. 1. Represent. Draw an ID and FD for the system of you in the

elevator while it is at rest on the ground floor. 2. Reason. Marie suggests, “There should be another force on

the FD showing the upward effect of the cable.” Do you agree with Marie? Explain.

In the world of physics, weight is a synonym for the force of gravity: Fg = mg, where the gravitational field strength g = 9.8 N/kg. Our physical sensation of weight corresponds not to the force of gravity, but to the force supporting us (often a normal force). The reading of a bathroom scale measures this supporting force which we call our apparent weight. 3. Solve. Use Newton’s 2nd law (Fnet = ma) to determine your apparent weight in this situation. (If you like, assume your

mass is 65 kg). 4. Evaluate. Is your weight different from your apparent weight in this situation? B:Going Up! Now you press the button in the elevator and go for a ride! The elevator starts speeding up as it begins your trip to the 20th floor. 1. Reason. Isaac says, “In this situation we need to add another upwards force to the

force diagram since you are now accelerating upwards.” Do you agree or disagree with Isaac? Explain and draw a revised ID and FD for this situation.

2. Predict. Will the reading of the scale increase or decrease compared to the when you were at rest? Explain without using

any math. 3. Test. Hang a heavy mass from a spring scale and gently accelerate it upwards. Compare the scale’s reading (apparent

weight) with the weight of the object (force of gravity). Does this agree with your prediction?

Sketch ID FD

ID FD

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Scale

52

4. Solve. The elevator is accelerating at a rate of 1.5 m/s2. Use Newton’s 2nd law to determine your apparent weight. How does this compare with your weight?

C: The Trip Up As you are going up, somewhere around the 2nd floor you notice the scale reading returns to normal. 1. Explain. How has the motion of the elevator changed? Use your spring scale and mass to help explain. The elevator is near the 19th floor and continues to move upwards, but you notice another change to the scale reading as the elevator is slowing down. 2. Reason. Emmy comments, “I think the upwards force must still be larger than the

downwards, or else the elevator would not be moving upwards.” Respond to Emmy and draw a new ID and FD.

3. Predict and Test. How will the reading of the scale will change as the elevator slows

down? Test this with your mass and spring scale. Describe your observations. 4. Solve. The elevator slows at a rate of 3.4 m/s2. Determine your apparent weight. 5. Summary. How did the interaction and forces change in the different situations you have explored? 6. Summary. In general, how is apparent weight related to the acceleration of an object? D: A Strange Elevator You wake up to find yourself in a very strange elevator with no buttons, lights or windows. You are floating just above the scale and have lost the sensation of weight. The scale itself reads zero. Offer two possible explanations for this very curious situation. 1) 2)

ID FD

53

SPH4U: Frames of Reference Your friend is standing on a bus that is travelling east and speeding up at a constant rate along a level road. While this is happening she holds up a rope with a ball attached to the end of it. The ball is allowed to hang freely. Assume east is to the right. Answer the following questions while the bus is accelerating and the ball hangs in a steady way (not swinging around!) 1. Represent. Draw a motion diagram for the ball

relative to each frame of reference while the bus accelerates in a steady way.

2. Represent and Explain. Draw an interaction

diagram for the ball according to an observer in each frame of reference. Are there any differences between the two interaction diagrams?

3. Represent. Draw a force diagram for the system

of the ball according to an observer in each frame of reference. Do not include fictitious forces yet. Include an acceleration vector!

4. Reason. There is a contradiction within one of

the force diagrams. Explain carefully. The rules we have learned for forces and motion breakdown in accelerating frames of reference producing contradicting results like the one above. To “patch things up” and help our rules work again, we introduce a convenient fiction - a fictitious force. A fictitious force is not a part of any known interaction and thus is not a real force, but we may work with it as if it was. Such a force is only ever noticed or needed in an accelerating frame of reference and should never appear in a FD by an observer who is not accelerating. 5. Represent. Draw a revised FD for the bus frame of reference. Add a fictitious

force, Ffict, such that FD agrees with the motion of the ball according to an observer on the bus.

6. Explain. How does the direction of the fictitious force compare with the

acceleration of the bus, ab, as measured in the earth frame?

Earth frame Bus frame Sketch

Sketch

Motion Diagram Motion Diagram

ID

ID

FD

FD

Revised FD

θ θ

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7. Summarize. Think of other situations where you have been accelerated. Except when in free fall, we feel as if we are somehow being pushed. In all these situations, how does the direction of your acceleration compare with the direction of the sensation you feel?

8. Analyze. From the Earth frame of reference, write an expression for the x- and y-components of Newton’s 2nd law. Solve

for the acceleration ab in terms of θ and g. B: Fictitious Forces and Acceleration 1. Summarize. How does the direction of your fictitious force compare with the direction of the acceleration? 2. Summarize. What do the two observers agree on? Consult your table on the first page. 3. Reason. Will the two observers agree on the size of any of the forces in their FDs? Explain. 4. Solve. Your friend on the bus measures an angle of 13o with a 127 g ball. What is the size of the fictitious force

according to your friend? 5. Apply. Describe a situation where you have experienced a fictitious force. Explain what was happening with your new

understanding.

55

SPH4U: Strings and Composite Objects How to strings do their thing? Let’s find out! A: String Theory You need three identical spring scales. Connect all three spring scales into a chain. Each scale represents a piece of “string material” and the spring connections (hooks) represent the forces that act between the pieces of string material. 1. Observe. Hold on to each end of the chain of scales and stretch them horizontally along your table. How do the readings

of each scale compare? What happens to the readings when you pull harder?

2. Reason. Is it possible to have tension in the string (your chain of scales) by pulling on only one end (don’t attach the

other!)?

3. Reason. Your hands are pulling on each end of the string and the ends of the string are pulling on your hands (what law

was that?). How hard does each end of the string pull on your hands? Physics String has a few special properties: (1) it is massless, (2) it does not stretch, and (3) it exerts equal and opposite forces on the objects it connects. Consequently, we can think of a string as producing a single tension interaction, between the two connected objects, an interaction which obeys Newton’s 3rd law. In our interaction diagrams we do not consider string as a separate object. B: Composite Objects Most objects are made up of millions of smaller parts, all interacting together. With so many interactions, why can we actually do simple physics and not get bogged down? Consider the situation shown below. Three blocks are connected by strings and pulled along a frictionless table by a steady force from your hand. The masses of the blocks are indicated. 1. Reason. How does the motion of blocks A, B, and C

compare? A composite object is any object whose parts all move together with the same acceleration. 2. Represent. We will start by considering each

block as a separate system (systems A, B and C). Complete the interaction diagram for the three blocks.

3. Represent. Draw a separate FD for each of

systems A, B, and C. Don’t forget to: use the 3rd law notation, draw an acceleration vector, and draw the vector lengths carefully!(next page!)

String R String Q String P

Block C mass = 2m

Block B mass = 2m Block A

mass = m

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System A mA=

System B mB=

System C mC=

4. Reason. Rank in increasing size the magnitudes of the net forces on systems A, B, and C. Explain. 5. Represent and Explain. Use a new colour to draw a new system boundary

(circle) in the interaction diagram consisting of blocks B and C together. Use the words “internal” and “external” to help explain how to use the interaction diagram to decide which forces should appear on a new force diagram for the new system, which we will call system D.

6. Represent. Use another colour to draw another new system in the interaction

diagram on the previous page consisting of blocks A, B and C together. Draw the force diagram for system E. Indicate the mass of the system.

7. Reason. How does the acceleration of system D and E compare with that of A,

B, and C? Explain. 8. Reason. Which system has the simplest force diagram for the purpose of finding the force of tension in string P? What

about for string Q? for string R? 9. Summary. Why can we do simple physics with complex objects that are made up of many, many parts? (Why aren’t

there millions of forces on our FDs?) 10. Solve. Consider a situation where m = 1.0 kg and the hand exerts a force of 15 N. Find the magnitude of the other tension

forces. (Tip: Always make sure that the mass used in the 2nd law is the mass of the system you are analyzing!)

System D mD =

System E mE =

57

SPH4U: Tension and Pulleys A: Physics Pulleys You will need two pulleys, some string, 500 g masses and two spring scales. Set up your equipment as shown in the diagram. 1. Reason. What is the readability of your spring scale? Estimate the uncertainty in the results you

will get.

2. Observe. Change the angle of the string that passes over the pulley. Describe what you observe. 3. Reason. Devise a rule describing how a pulley affects the magnitude and direction of the force of tension in a string. Physics Pulleys: B: Testing String Theory 1. Predict. Don’t set this up yet! Consider the two pulley situation shown in the next

diagram which uses two 500 g masses. Suppose we measure the tension in the string at the three positions indicated. Predict the values of those three tension measurements. Provide a rationale for each. Prediction Rationale (1)

(2)

(3)

2. Test and Evaluate. Set up the materials as shown (just hold the pulleys in the air). Record your results along with

uncertainties. Do you results agree with your predictions? Offer an explanation for the results you found. Result Explanation (1)

(2)

(3)

3. Reason. Create a definition of how tension works that explain what it means when we say, “there is 4.9 N of tension in a

string”. Tension

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m

m m

1

2

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C: The Atwood Machine An Atwood machine consists of two masses tied together and suspended over a pulley. For this part of the investigation, you will use the equipment set up at the front of the class. 1. Predict. The two masses are identical. You give a gentle push to one mass. Predict the motion

of the mass after it leaves contact with your hand. Justify your prediction. 2. Test. Try it out. Describe your observations. Do they agree with your prediction? 3. Predict. The mass of A is now less than B. Predict the motion of mass A after you release it. How will the motion of the

two objects compare? Justify your prediction. 4. Test. Try it out. Describe your observations. Do they confirm your predictions? 5. Represent. Draw a FD for mass A and for mass B. (mA<mB).Choose your sign

conventions so they agree with the acceleration of each object! 6. Reason. For the FDs, do you need to use a different symbol for the magnitude

of the acceleration of mass A and mass B? What about the forces of tension and gravity acting on each mass? Explain.

7. Predict. How will the magnitude of the force of tension compare with FgA and

FgB? Explain. 8. Represent. Write a complete expression for Newton’s 2nd Law for each mass. Be very careful with your notation! 9. Solve and Test. Algebraically eliminate FT from the above two equations and solve for the acceleration of the masses.

Use the masses from the Atwood machine set up in the classroom and solve for the acceleration. Then test this result using the motion detector.

10. Evaluate. Now go back and solve for the force of tension. How does this result compare with your prediction? Explain.

Mass A

Mass B

mA

mB

59

SPH4U: Friction! We normally think of friction as the force that stops things from moving. This is in many respects still true, but we must also realize that friction is the force that is usually responsible for starting things moving too! A: Rubbing the Wrong Way? Find a convenient, hand-sized object. Place the object on the flat palm of your hand. Keep your hand completely horizontal and cause the object to move horizontally and speed up with your hand (no slipping). 1. Reason. Why does the object accelerate? 2. Reason. Marie says, “At first I thought a horizontal normal force causes it to accelerate. But then I remembered that a

normal force is always perpendicular to the surfaces. It doesn’t make sense to have a horizontal normal force here.” Do you agree with Marie? Explain.

3. Reason. Emmy says, “In this situation a friction force must act in the forwards direction.” Albert responds, “That’s not

right. Friction always opposes an object’s motion. It should point in the backwards direction.” Who do you agree with? Explain.

4. Predict. What would happen to your object if there was no friction at all between it and your hand? Explain. 5. Test. Try this out with the equipment at the front of the class. Record

your observations. Do they support your prediction? Do you need to revise any of your earlier answers?

6. Represent. Draw an ID and FD for the object while it is speeding up. Friction is a contact force that tries to prevent two surfaces from sliding relative to one another. If there is no slipping of the two surfaces, static friction may be present. If the two surfaces are slipping, kinetic friction is present. B: Two Kinds of Friction 1. Represent. Place your object on the table. Exert a force on the object using a 5N

spring scale such that it slides at a constant velocity. Draw a FD for it while sliding.

2. Explain. Why does the spring scale reading equal to the force of kinetic friction?

ID FD

FD

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3. Explain. You might have noticed that reading on the spring scale jerked lower as soon as the box began to slide. (If you didn’t notice this, apply a small and gradually increasing force or find a heavier object.) What does that higher initial reading tell us about the friction between your object and the table? (Hint: Remember there is another type of friction!)

The force of friction depends on the characteristics of the surfaces involved (as represented by the coefficient of friction, μ), and how hard the surfaces are pressed together (the normal force, Fn). The two forces of friction can be found from the two expressions, Ffk = μkFn and 0 <Ffs ≤Ffsmax, where Ffsmax =μsFn. 4. Calculate. Make measurements and determine the coefficient of kinetic friction for your object. 5. Reason. Imagine you repeat this experiment in an elevator accelerating upwards. Emmy says, “I bet the value for μ will

get smaller since μ depends on Fn”. Albert says, “The equation written as μ = Ff / Fn makes it look like μ depends on Fn, but is actually doesn’t.” Who do you agree with? Explain.

C: Friction Causing Motion Place your object in the middle of a dynamics cart (make sure your object doesn’t touch the front or back of the cart). Use the spring scale to pull on the cart and cause both the cart and object to move together. We will assume there is no friction between the cart and the table. 1. Observe and Explain. Exert a force on the cart, causing both the cart and object to move together. Describe the motion

of the object. What force was responsible for your object’s motion? Be specific. 2. Represent. Draw an interaction diagram for the cart and object

while you pull on the cart. 3. Represent. Now we will draw three FDs while the cart and object

are moving together (no slipping). Use the 3rd law notation for the forces. Indicate any 3rd law force pairs that appear. Hint: The cart FD has five(!) forces.

Your object The cart The cart + object system

ID

61

4. Explain. Write an expression for the net force experienced by your object. Why can this net force take on a range of values? Is there any limit to these values? What does this imply about the possible values of the acceleration?

5. Reason. If we pull too hard on the cart, the object will begin to slip. We want to determine the largest force of tension

that we can exert on the cart without any slipping of your object. Which FD is easiest for finding the acceleration? Measure the mass of your object and cart.

6. (Homework) Solve. Determine the largest force that we can exert on the cart before the object starts to slip. (Hint: Use

your reasoning from the two previous questions. Assumes = 0.6) D: Friction and the Ramp You need a track. 1. Observe and Calculate. Determine the coefficient of static for your

object on the horizontal track surface. Draw an ID and FD for this situation. There is a lot of error in this measurement, so record a high and low possible value for your measurement and calculate a high and low possible value for the coefficient. Show your work.

2. Reason. Now draw a FD for the box at rest on an incline. Let the x-axis be parallel to the incline. How will each force

change when the angle of the incline increases? Explain.

3.

ID FD

Fgx Fgy

Fn Ffs max

FD

62

Predict. Analyze the situation using Newton’s 2nd law. What angle do you predict the box would start moving at? (Find a high and low value.)Do all your work algebraically and you will get a very simple expression.

4. Test. Call your teacher over to test your prediction. Record the angle at which it begins to slide. Do your observations

confirm your prediction? Note that there are large sources of error in this investigation! E: Friction and You What is the cause of our motion when we walk, drive, or ride a bike? 1. Observe. Have one group member walk slowly. Watch carefully: does their back foot

slide against the ground? 2. Reason. Consider the system of the back foot while walking. Have that person freeze in place as their back foot is

pushing against the ground. What is the back foot interacting with? 3. Represent. This is a very difficult situation to model with a

FD, but being fearless, we will try! Draw an ID and a FD for the system of the back foot at the moment pictured above. Include a force at an angle for the effect of the leg on the foot.

4. Explain. What force is pushing forward on the foot?

Ultimately, this is the external force responsible for making the system of your body accelerate forward.

5. Reason. If this forward force disappeared, what would happen as we try to walk? 6. Reason. Isaac says, “According to my third law, there is an equal size friction force of the foot on the earth. Why don’t

we notice this?” Explain to Isaac why we don’t notice the effect of this force.

ID FD

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SPH4U: Going in Circles? What makes the world go ‘round? Let’s find out! A: Observing Circular Motion Let’s find out how forces in different directions affect the motion of the intrepid physics buggy! A reminder - velocity has two parts: a magnitude (speed) and a direction. 1. Observe. Describe the velocity of the buggy when it drives along the level floor with no additional forces. 2. Observe. Attach a piece of string to the buggy – if it tends to flip, adjust the string angle. In the chart below, describe

how the velocity (speed and direction) of the buggy changes when you exert a constant, horizontal force on the buggy in a direction: (a) parallel to its motion, and (b) perpendicular to the direction of the buggy’s motion.

Forces Parallel Forces Perpendicular

3. Reason. In which case(s) above is the buggy accelerating? Explain. 4. Reason. Create a principle describing the effect of forces on an object’s speed and direction.

Speed and Direction Principle: 5. Observe and Reason. Swirl a marble around inside a roll of masking tape. Once it’s going, hold the roll of tape still and

observe. We will neglect the small amount of friction which slows down the marble after your stop “swirling”. a) Draw an ID for the system of the

marble.

b) Draw a force diagram for the marble at two moments in time: when it is moving towards you and when it is moving away from you. Draw it from a head-on point of view (as if your eye was at table level.

c) What forces or components of forces

balance?

d) What is the direction of the net force experienced by the marble?

ID FD (head-on, towards) FD (head-on, away)

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6. Observe and Reason. Tie an object to a length of string and swing it in a slow, horizontal circle. (Don’t hit anyone!)

a) Draw an ID and a FD for the system of the object.

b) What forces or components of forces balance?

c) Isaac says, “I think the net force point up at an angle, along the string.” Do you agree or disagree with Isaac? Explain.

B: Making Rules and Breaking Rules 1. Reason. Devise a provisional rule for the direction of the net force of an object moving in a circle at a constant speed.

Provisional Rule #1:

2. Describe. Think about your experience turning a corner in a car. How does that feel?

Provisional Rule #2: When an object moves in a circle there is a radial force acting outwards, away from the centre.

Provisional Rule #3: When an object moves in a circle, there must be a tangential force forwards, in the direction the object is moving.

We don’t know yet which of these rules are correct. In science, we make observations, develop theories (rules) and then test them out. In the following experiment we will look for evidence which supports or refutes these three rules. 3. Predict and Test. In a moment you will attach a hover puck to a

string. The hover puck is given a short push to start it moving. The end of the string is held fixed on the ground. (a) Predict how the puck will move after the push.

(b) Test your prediction and describe the motion of the puck.

(c) Draw an ID and a FD for the puck after the push from a bird’s eye view (from above). (d) Each rule makes a prediction about this situation (after the push) which is shown below. Based on your observations

or your FD, which rules are confirmed and which are refuted? If you think a force is present, describe the interaction it is a part of. (next page!)

ID FD

Sketch ID FD

push

65

Rule #1 Rule #2 Rule #3 Prediction: There is a net force in the radial inwards direction.

Prediction: There is a force in the radial outwards direction.

Prediction: There is a force in the forwards direction.

Conclusion: Conclusion: Conclusion:

4. Reason. Based on your observations, elevate one of the provisional rules to an official rule. Mention what uniform

circular motion means. Net Force and Uniform Circular Motion:

Centripetal is an adjective that describes any force or component of a force that points towards the centre of curvature of an object’s path of motion. A centripetal force changes an object’s direction without changing its speed. It is responsible for keeping an object moving in a circle. There is no special symbol for a centripetal force – never write Fc, anywhere! Your textbook does use this symbol, but it is a faulty habit – don’t! Centripetal is simply an additional label for already familiar forces (Ft, Fg, Fn, etc.) and is a helpful term when discussing circular motion. 5. Reason. Which forces in the examples we have studied so far have acted as centripetal forces? C: Turning a Corner You are a passenger in the back seat of a car. You placed your water bottle on the seat beside you and then the car makes a sharp turn to the left. Your bottle slides across the seat until it hits the door on the other side of the car. It seems to be pressed against the door. 1. (as a class) Observe and Explain. Watch the simulation and complete the chart below.

Describe Motion

in Car Frame Describe Motion in Earth Frame

ID System = bottle

FD in Earth Frame (top view)

While sliding

While pressed against the door

66

SPH4U Homework: Going in Circles Name: A: Turning a Corner 1. Reason. Marie says, “When we turn the corner in a car, there must be an outwards force that pushes you across the seat

and presses you against the door. That’s what I feel.” Do you agree or disagree with Marie? Explain. 2. Reason. A car turns a corner. Draw an ID and FD for the car. Which

force acts as the centripetal force in this situation? 3. Reason. Isaac says, “I don’t see how friction could be the centripetal

force in this situation. It just doesn’t seem right.” Explain to Isaac what would happen if there was no friction interaction. So what force is responsible for keeping the car moving in a circle?

4. Reason. When we investigated frames of reference a few classes ago, we decided that when we accelerate, we feel as if

we are being pressed in the opposite direction. Is that conclusion valid for circular motion as well? Explain.

5. Reason. Use your new force rule to explain how to cause a hover puck to move in a circle simply by using gentle taps from a metre stick (no string!) Illustrate the force of your taps in the diagram.

6. Reason. The hover puck moves in a very mysterious manner due to unknown forces. A motion diagram is shown below

with dots produced after equal intervals of time. Draw a vector that represents the centripetal force at the four moments in time indicated. Can you guess which net force is greater? Offer a rationale for your answer (our next lesson will help us clearly decide which is greater).

ID FD

1

2

3

4

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67

SPH4U: Radial Acceleration What is special about the motion of an object moving in a circle? Only vectors will tell! When we discuss circular motion, there are often two important directions: (1) the radial direction which points inwards, to the centre of curvature of the object’s path, and (2) the tangential direction which points forwards, tangent to the object’s curving path. In many cases it may be convenient to choose one these as our x- or y-directions for our coordinate systems. A: Instantaneous Velocity and a Curving Path Attach a piece of string to a hover puck and swing it in a circle. 1. Observe. Release the string when the puck is in the 9 o’clock position. In what direction

will the puck travel? This observation gives us an idea about the puck’s instantaneous velocity at the moment it was released.

2. Represent. Draw an instantaneous velocity vector for the hover puck at the four moments in time shown in the diagram. When drawing instantaneous velocity vectors, they should appear tangent to an object’s path with the tail starting at the position of the puck.

3. Explain. When an object moves in a circle at a steady rate, does the speed change? Does the velocity change?

B: Velocity Vector Diagrams The technique explained below will help you estimate the average acceleration of an object during two-dimensional motion. To find the acceleration, we need to find a change in velocity, v2 – v1. Subtracting two vectors is done by drawing them tail-to-tail.

Notice that when the velocity vector diagram is complete, it follows our familiar pattern: v2 = v1 + Δv (since Δv = v2 – v1!) C: Acceleration and Circular Motion 1. Represent. An object moves at a constant speed in a circle. Use the vector subtraction technique to estimate the

acceleration vector at the four moments shown in the diagram. Neatly show your vector work for each example (the first example is almost complete). Draw a wiggly acceleration vector inside the circle for each moment.

a

The acceleration, a, is in the same direction as the change in velocity, v, and has a magnitude v/t.

2v

1v v

Redraw the two velocity vectors tail to tail. Draw the change in velocity vector going from the head of v1 to the head of v2.

2v

1v

We want to estimate the acceleration for an object when it passes the middle point (). Draw the initial velocity vector (v1) just before the point. Draw the final velocity vector (v2) just after the point.

Adapted from The Physics Active Learning Guide by A. Van Heuvelen, Pearson, 2006

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An object moving in a circle experiences an acceleration (ar) in the radial direction pointing towards the centre of its circular path. This acceleration is called the radial acceleration (in the textbook this is called the centripetal acceleration). 2. Reason. How does the pattern for the acceleration vectors compare with the rule you developed for the net force in the

previous investigation? What law of physics is this a result of? 3. Represent. A physics buggy moves at a constant speed across the top

of a circular hill, as shown in the illustration. In the chart below, use the vector subtraction technique to find the acceleration, draw a FD, and explain how the two results agree according to Newton’s 2nd law. Velocity Vectors

FD Agreement?

4. Reason. What would change if the physics buggy was at the bottom of a valley instead of the top of a hill? 5. Apply. Have you ever been in a situation like the ones described above? What was the situation and what did it feel

like?

D: The Pendulum of Fate In our classroom we have a 0.50 kg mass hanging on the end of a 1.0 m string. The string is connected to a spring scale. 1. Predict. The mass is at rest, hanging from the scale. Draw a FD. Predict the spring

scale reading. Justify your prediction. 2. Predict. Imagine you pull the mass to the side and release it such that it swings like

a pendulum. Even though the mass does not move in a complete circle, this is still circular motion! Apply your new understanding of circular motion when the mass is at its lowest point. Predict how the spring scale reading will change.

3. Test. Perform the experiment. Explain how the result agrees with our understanding of acceleration and force in circular

motion.

FD – Rest FD - Swinging

69

E: Acceleration, Speed and Radius Two identical physics buggies travel at a constant speed along two identical circular paths. Buggy A moves with a speed v and Buggy B with a speed 2v. Buggy A completes a ¼ trip in a time ΔtA and experiences a change in velocity vA. 1. Represent. Use the vector subtraction technique to compare the radial accelerations of the two buggies.

Illustration Change in Velocity Time Interval Acceleration

Bug

gy A

For a ¼ trip:

ΔtA

aA = ΔvA / ΔtA

Bug

gy B

Hint: How does ΔvB compare with ΔvA?

How does ΔtB compare with ΔtA?

2. Reason. How does the magnitude of the acceleration depend on the speed? Write out a mathematical expression for this

relationship. (Use the proportionality symbol “”)

Two identical buggies travel with the same constant speed along two circular paths with different radii. Buggy A moves in a circle of radius r and buggy B moves in a circle with radius 2r. Buggy A completes a ¼ trip in a time ΔtA. 3. Represent. Use the vectors subtraction technique to compare the radial acceleration of each buggy.

Illustration Change in Velocity Time Interval Acceleration

Bug

gy A

For a ¼ trip:

ΔtA

aA = ΔvA / ΔtA

Bug

gy B

4. Reason. How does the magnitude of the acceleration depend on the radius? Write out a mathematical expression for this

relationship.

•

•

1v

2v

1v

2v

Av

•

•

1v

2v

•

•

1v

2v

1v

2v

Av

•

•

1v

2v

70

5. Speculate. Combine the two results above and write equation for the magnitude of the radial acceleration during circular

motion. **Check your result with your teacher ** F: Test the Acceleration Expression When we experiment it is often simplest to measure the frequency (f) or the period (T) of the circular motion rather than the speed. We can create two other handy equations for the radial acceleration: ar = 42rf 2 = 42r/T2. Recall that f = 1/T. 1. Derive. Start with your equation for the radial acceleration. Use your knowledge of circles(a distance of 2rand a time

T) to eliminate v from your equation and create the two other equations shown above. 2. Predict. According to our understanding of acceleration and net force, explain what happens to the magnitude of the

centripetal force if an object spins faster with the same radius. 3. Predict. Explain what happens to the magnitude of the centripetal force if the object spins with the same frequency and

the radius increases. 4. Test. Find the circular motion kit: a small tube, string (at least 1 m), rubber

stopper, and spring scale (5 N). Attach the string to the stopper and thread it through the tube. Have one person hold the tube and another person hold a spring scale connected to the bottom of the string. Find some space where you can swing the stopper in a horizontal circle in the air above your heads. Don’t hit people! Practice swinging it at a steady rate.

5. Test and Evaluate. Test your two predictions above. Note that keeping the

same frequency is quite tricky! Were your predictions consistent with the observations? Do the new equations work?

spring scale

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SPH4U: Thinking About Circular Motion A: The Balancing Act? Marie swings a rock in a vertical circle at a constant speed. The rock reaches the lowest point in its trip. Emmy says, “I can feel the string pulling down on my hand. There must be a net force downwards.” Isaac says, “I think at the bottom the forces balance.” Albert says, “Since the object is moving to the left at this moment, the net force must be to the left.” Who do you agree with? Explain. Sketch

ID

Velocity Vectors

FD Explanation

B: Changing Tension? Now Emmy swings the rock in a vertical circle at a steady rate. Emmy says, “I can feel that the tension in the rope is different when the rock is at the top compared with at the bottom of its trip. I guess the force of tension is changing.” Marie says, “No, the force of tension can’t be changing since the speed and acceleration of the rock is constant.” Who do you agree with? Explain. Sketch

FD – top

FD - bottom Explanation

Test. Swing a ball in a vertical circle and verify whether you can feel the changing tension force. C: Rocks at 33½ Albert places a rock on a rotating platform, just like an old fashioned record turntable. It does not slip on the platform surface. Albert says, “At this moment in time the instantaneous velocity of the rock is downwards, from this point of view, so friction will act in the opposite direction, upwards.” Marie says, “I’m not sure. Since the rock is moving in a circle, I think the force of friction must act towards the centre.”

Top view

ID FD (side view) Explanation

Recorder: __________________ Manager: __________________ Speaker: _________________

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© C. Meyer, 2014

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D: The Ball and the Funnel Albert is visiting the Ontario Science Centre and sees an exhibit where a marble rolling in a funnel glides around in horizontal circles with no noticeable vertical motion (friction is negligible). Isaac says, “That is so cool!” Albert says, “But I don’t understand–shouldn’t gravity pull it down to the bottom pretty quick? I don’t see how it could travel in a circle.” Explain to Albert how this is possible. Sketch

ID Newton’s 2nd Law (line up your coordinate system with the acceleration!)

Explanation

FD

E: Evaluate the Problem Identify any errors in the solution to the following problem. Provide a corrected solution if there are errors. Problem: 80-kg Samuel rides at a constant 6.0-m/s speed in a horizontal 6.0-m radius circle in a seat at the end of a cable. Determine the tension in the cable. Proposed Solution: The situation is pictured above. We simplify by assuming that Samuel, the system, is a particle. A FD for Samuel is shown at the right along with the acceleration direction.

Ft = m(v2/r) = (80 kg)(6.0 m/s)2/(6.0 m) = 480 N

The tension is 480 N. F: Design a Problem You are given the second last step in a problem where quantities have been substituted in an equation. You task is to reverse engineer the problem from this step. Draw a sketch, a force diagram and describe the situation just like a textbook problem! Sketch

FD Description 200 N + (50 kg)(9.8 N/kg) = (50 kg)v2 / (12 m)

ca

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SPH4U: Universal Gravitation Good old Sir Isaac Newton determined the relationship between the force of gravity, mass (m) and the separation between the centres of the two objects (r):

221

r

mGmFg

Where G = 6.67x10-11 Nm2/kg2. Your friend, who is now an astronaut (mA = 100 kg), is currently standing on the Earth (mE = 5.98x1024 kg, rE = 6.38 x 106 m). 1. Represent. Your astronaut friend

jumps into the air. Draw a sketch of your friend and Earth during the jump. Label the quantity, r. Draw an ID and a FD for your friend and Earth.

2. Calculate. Use the new equation

for universal gravitation to find an algebraic expression for the ratio of force per unit of mass for the astronaut (Fg/mA). Substitute the appropriate values into your expression and evaluate it.

3. Interpret. The result you found in the previous question from the ratio Fg/mA tells us something important about the Explain the meaning of this result.

4. Reason. In your FDs you identified two forces of gravity. Compare the size and direction of these two forces. 5. Reason. Isaac says: “I understand from my third law that the astronaut exerts an upwards force on the earth, but this just

seems strange. How come we never notice this upwards force?” Explain to Isaac.

Sketch ID Astronaut FD

Earth FD

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© C. Meyer, 2014

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6. Explain. Calculate the size of the force of gravity the earth exerts on the astronaut and the force the astronaut exerts on the earth using the expression for universal gravitation. Even if you never heard of Newton’s Third Law, why is it not necessary to perform two separate calculations?

7. Calculate. Your friend blasts off and travels away from the earth. Complete the table showing the size of the force of

gravity due to the earth on the astronaut at different distances from the centre of the earth. Plot a graph of Fg vs. r

8. Find a pattern. Describe in words how the size of the force of

gravity varies with the separation of the objects. 9. Reason. How far does Earth’s gravitational force extend into the universe? Explain. 10. Reason. Anything with mass will interact gravitationally with anything else that has mass. This means there is a

gravitational interaction between you and the person sitting beside you. Why don’t we notice this interaction? Use a sample calculation to support your reasoning.

11. Reason. Imagine a hole is dug straight through the centre of the Earth. Describe your motion if you

were to fall in. Ignore any air resistance!

Location Distance Fg

Earth’s Surface 6.38 x 106 m (0.0638 x 108 m)

Space Shuttle Orbit

6.39 x 106 m (0.0639 x 108 m)

Geosynchronous Orbit

4.22 x 107 m (0.42 x 108 m)

Lunar Transfer Orbit

2.17 x 108 m

Moon’s Orbit 3.85 x 108 m

Mars’ Orbit 7.9 x 1010 m

0 |

4 x 108 |

1 x 108 |

2 x 108 |

3 x 108 distance (m)

500

1

000

|

|

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SPH4U: Orbits Consider an object (mo) like a satellite or a moon travelling in a circular orbit around the earth. 1. Represent. Label the altitude (h), the radius of the circular orbit (r), the radius of Earth (rE), matching them with the

letters X, Y, and Z. 2. Reason. Which quantity, X, Y, or Z, does r in the expression for universal

gravitation represent? Explain.

221

r

mGmFg

3. Reason. Which quantity, X, Y, or Z, does r in the expression for the centripetal acceleration represent? Explain.

2

24T

rac

4. Represent. Draw a force diagram for the object at the moment shown. 5. Solve. Use Newton’s 2nd Law, universal gravitation and an expression for centripetal

acceleration to create an equation that relates the radius of the orbit and the period of the orbit. Solve this for T2. Be careful with the labels for the masses!(Don’t memorize this equation.)

This result is known as Kepler’s law, named after the brilliant mathematician who spent 25 years working out this result. How long did it take you? (Thanks, Newton!) 6. Solve. Complete the chart below which compares orbital velocities, altitude, radii and periods.

mE = 5.98x1024 kgrE = 6.38 x 106 mG = 6.67x10-11 Nm2/kg2

Location Altitude r (m) T (s)

Space Shuttle Orbit 600 km

Geostationary Orbit (satellite stationary above Earth’s surface)

24 h =

Moon’s Orbit 27 d =

Earth’s Orbit around Sun (msun = 1.98 x 1030 kg)

FD

X

Y

Z

Recorder: __________________ Manager: __________________ Speaker: _________________

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© C. Meyer, 2014

mA

mB

76

7. Reason. Albert says, “I understand that there still is gravity out in space, sometimes lots of it, but why do we see astronauts floating in the Space Shuttle when it is in orbit around the earth? They seem weightless.” Explain to Albert why. (Consult your chart from the previous investigation.)

8. Reason. Earth’s gravity is always pulling objects towards the centre of the earth. Why don’t objects in orbit fall straight

down and crash into Earth? 9. Represent. Sketch and label a scale diagram of the radial distance of the first two orbits from question 6 in terms of

earth radii. Draw the earth itself. The given point represents the centre of the earth. 4 boxes = 1 rE.

10. Represent. Using this scale, where would the Moon be located? Find a golf ball and marble. Carefully position these to

model the earth and moon. Call your teacher over. B: Twin Stars Consider a binary star system (YM alpha and YM beta) consisting of two equally massive stars which orbit one another. 1. Reason. How do you decide which star will orbit around which? Explain. 2. Represent. Label the radius of the orbit (ro) and the separation between the centres of

mass of each star (d), matching them with the letters A (radius) or B (diameter). C is the common centre around which both stars orbit.

3. Represent. Use Newton’s 2nd law to write a complete expression relating the period

and radius for the stars’ orbits. Be sure to use the given symbols!

C X

Y

α

β

77

SPH4U: “Oomph” When you catch a heavy object you feel a lot of “Oomph”. What is this mysterious quantity that we all kind of know? Let’s find out. A: Figuring Out the Formula for Oomph! The more oomph something has, the harder it is to stop, and the more ability it has to knock other things over. Let’s figure out the formula for oomph.

1. Reason. A small pebble and a larger rock are thrown at the same speed. (a) Which one has more oomph? Why?

(b) The rock is twice as massive as the pebble. Intuitively, how does the rock’s oomph compare to the pebble’s oomph? Is it twice as big? Half as big? Three times as big?

2. Reason. Picture two identical bowling balls, one of which is rolling faster than the other.

(a) Which ball, the faster or slower one, has more oomph? Why?

(b) The faster ball is exactly 7 times as fast as the slower one. Intuitively, how does the faster ball’s oomph compare to the slower ball’s oomph?

3. Find a Relationship. The physics concept corresponding to oomph is momentum. Building on your above answers, figure out a formula for momentum (oomph) in terms of mass and velocity. Explain how the formula expresses your intuitions from parts A and B above. (For nutty historical reasons, physicists use the letter p for momentum.).

** check with your teacher at this point ** B: Testing Our Momentous Intuitions You will need a dynamics track and two carts. In the previous section, your intuitions about oomph led to a formula for momentum. Now let’s see if your ideas hold true for collisions. Cart A (1 kg) is rolling with negligible friction at 3 m/s and collides with and sticks to cart B (identical to cart A). So, after colliding, the carts roll together as a single unit.

1. Predict. Don’t do it yet! Using your intuitions, guess the post-collision speed of the two carts. Briefly explain your reasoning.

Moment 2?

3 m/s

1 kg 1 kg

Moment 1

Recorder: __________________ Manager: __________________ Speaker: _________________

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© C. Meyer, 2014, adapted from University of Maryland Physics Education Research Group, Fall 2004

78

2. Test. Use the carts and dynamics track to test your prediction. Roughly speaking, do your observations agree with your prediction? Did the first cart gain or lose momentum? What about the second cart? What about the system of two carts, did it gain or lose momentum?

To help represent the momenta of a system before and after a process like a collision we can construct an impulse-momentum bar chart. In our notation, we use letters to denote different objects and numbers to indicate different moments in time. We draw a bar that represents the size and direction of the momentum of each object in the system. The exact heights are not important, but the bars must clearly show the correct ideas. The change in the momentum of the system is called the impulse and is represented by the symbol J. 3. Represent and Explain. Complete the momentum-impulse bar chart for the system of

the two carts before and after the collision described above. For convenience, you can think of the cart A as having 31 = 3 units of momentum. Once complete, explain two ways in which the graph visually represents the fact that the impulse on the system was zero.

Here is another situation to consider: in a similar experiment, cart A collides with cart B magnetically. The two carts don’t actually touch – the magnets act like a perfect spring between the two carts. After the collision, cart A is at rest. 4. Predict. Again using intuitions, predict the post-collision speed of cart B.

5. Test. Use the carts and dynamics track to test your prediction. Roughly speaking do your

observations agree with your prediction? Did the first cart gain or lose momentum? What about the second cart? What about the system of two carts, did it gain or lose momentum? Complete the momentum-impulse bar chart.

6. Summarize. Based on your work above, state a general rule about how the total

momentum of a system changes during a collision. Here is one last situation to try out. The two carts are initially moving at 3 m/s in opposite directions. They collide and stick using Velcro. 7. Predict and Test. Intuitively, after the collision, how fast do the carts move and in what

direction? Test your prediction.

pA1 pB1 J pA2 pB2

+ 0 -

Moment 2?

3 m/s

1 kg 1 kg

Moment 1

stopped

AFTER

?

3 m/s

1 kg 1 kg

BEFORE3 m/s

pA1 pB1 J pA2 pB2

+ 0 -

79

pA1 pB1 J pA2 pB2

+ 0 -

8. Reason. In all cart collisions explored above, momentum was conserved; it was the same before and after the collision. Because conserved quantities are useful in problem-solving, it would be cool if we could define momentum in such a way that it’s always conserved in collisions (between objects that are free to move). Is there some way to modify or clarify the momentum formula you figured previously so that momentum is conserved in the head-on collision between the two carts? Explain. (Hint: Maybe oomph “cares” about direction.)

9. Represent. Complete a momentum-impulse bar chart for this collision. Explain how the

idea of direction is visually represented in the chart. ** check with your teacher ** C: The Conservation of Momentum Conservation of momentum is a fundamental physical law. Among other things, it says that when two objects collide, the total momentum of the system immediately after the collision equals the total momentum of the system immediately before the collision:

Conservation of momentum: 2211 BBAABBAA vmvmvmvm

Since vmp

, and since velocity “cares” about direction, so does momentum. So, a negative oomph (momentum) can partially or fully cancel a positive oomph, as the Velcro™ carts demonstrated. Problem. Let’s practice using momentum conservation. A 500 g cart moves east at 1.0 m/s and collides with a 1000 g cart that is at rest. The carts bounce magnetically off each other. After the bounce, the 1000 g car is moving east at 0.65 m/s. 1. Represent. A good, first step in any conservation problem is sketching the initial and final states of the process. These

are two key events. Complete Part A below.

A: Pictorial Representation Sketch showing “before” and “after”, coordinate system, label givens and unknowns with symbols, conversions, describe events

2. Predict. Without doing calculations, make a quick guess for the final direction of motion of the 500 g car. Briefly

explain your reasoning. 3. Reason. Are there any important interactions between the system objects and the external environment? Will these affect

the momentum? Explain.

80

+ 0 -

pA1 + pB1 + J = pA2 + pB2

pA1 + JA= pA2

+ 0 -

FD

PB1+ JB = pB2

+ 0 -

FD

4. Represent. Complete part B for the process. Draw the diagrams for the two carts during the collision. B: Physics Representation momentum bar chart, interaction diagram, force diagram

5. Calculate. Now calculate the 500 g car’s speed and direction of motion after the collision. Use the bar chart to help

construct your momentum equation (leave out any terms you know are zero).

D: Mathematical Representation Complete equations, describe steps, algebraic work, substitutions with units, final statement

6. Evaluate. Is your final answer reasonable (size, magnitude and direction)? Does it agree with your prediction?

E: Evaluation Answer has reasonable size, direction and units? Why?

7. Test. Set up this situation on your track and try it out. Does our momentum law work?

D: A Further Look

1. Represent. Some of you are probably wondering about impulse. We didn’t see any examples where there was an impulse. Try this: complete a momentum-impulse bar chart for the system of cart A and for the system of cart B in the first collision example (B#1). Draw a FD for each cart below the bar chart.

2. Explain. What does each chart and FD tells us about the impulse experienced by the individual carts?

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SPH4U: The Idea of Conservation There are some situations where momentum seems to appear or disappear. Let’s study one of these situations carefully. A: The Slowing Block A 1.0 kg block initially sliding at 1.5 m/s along a rough surface comes to a stop. 1. Represent. Complete a momentum-impulse chart for the system of the block. How much

momentum did the system lose? 2. Reason. Are there any important interactions between the system objects and the external environment? How does this

help to explain the loss of momentum? The Momentum Conservation Condition: The momentum of a system is conserved when the net force the system experiences is zero. The total momentum of a system can change if the system experiences a net force from its environment (objects outside the system). This change, also known as the impulse, is related to the net force and the amount of time the force acts: tFJp netsystem

. When Fnet (or J) = 0 the system’s momentum is conserved. Use this as the criteria to decide whether to use momentum to solve a problem. 3. Represent. Draw an ID and FD for the system of the block.

Write an expression for the net force. 4. Speculate. Consider the force responsible for slowing the block. What is the other force in a 3rd law pair with that force?

Use that other force to help you guess where the block’s momentum went. Make a guess and move on! B: The Block on a Track The block is moving at 1.5 m/s, just like before, and is gently lowered on to a level track that is supported on wheels and is free to move (no friction). The track has a mass of 2.3 kg and is initially at rest. 1. Predict. What will happen after the block is released? 2. Test and Observe. Use the equipment at the front of the class to test our two predictions. Describe how your

observations help to confirm your predictions.

ID FD

Recorder: __________________ Manager: __________________ Speaker: _________________

0 1 2 3 4 5

p1 + J = p2

+ 0 -

© C. Meyer, 2014

82

3. Represent and Reason. Complete a momentum-impulse bar chart for the block-track system. What is the net force on the block-track system? Explain.

4. Calculate. What is the final velocity of the track? 5. Reason. Imagine the mass of the track was increased enormously to equal that of the earth. Describe what would be

different. How does this relate to the situation of part A? C: The Process of a Collision Collisions often occur very quickly and we don’t usually notice what is actually happening during a collision. In this example, cart A (1.0 kg) collided with a smaller cart B (0.5 kg) using an uncompressed spring. The velocity of each cart was recorded at 9 moments in time and used to calculate the momentum and kinetic energy. A third line on each graph represents the total momentum and total kinetic energy of the system of two carts.

Initial

Final

1. Represent. Draw a vertical line on the graph

labelled “A” to indicate the moment in time when the collision begins and one labelled “C” to indicate when the collision ends. What is the duration of the collision?

2. Reason. What would we observe about the spring at

moments “A” and “C”? 3. Represent. Draw a single ID and two FDs for each cart during the collision.

ID FD – Cart A FD – Cart B

Cart A Cart B Cart A Cart B

pA1 + pB1 + J = pA2 + pB2

+ 0 -

0

2

4

6

8

10

12

14

16

18

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

Mo

men

tum

(kg

m/s

)

Time (s)

Momentum during a Collision

P A

P B

Ptotal

83

4. Reason. Based on the momentum graph, is there any evidence for a small net force acting on this system? Explain. 5. Find a Pattern. Calculate the impulse and

average net force experienced by each cart during the collision. How do the impulses and net forces compare (try to disregard the small variations due to friction).

The Idea of Conservation: A conserved quantity is one whose total for a system remains the same at every moment in time. 6. Reason. Carefully study the graphs

showing the total momentum and total kinetic energy of the system. Ignoring the small losses due to friction, which of these quantities is a conserved quantity? Explain.

7. Reason. Notice how the total kinetic energy dips down during the collision. This indicates a transfer of energy. Where

has been transferred? 8. Represent. Draw a vertical line on the kinetic energy graph labelled “B” to indicate the moment in time when the spring

was at its maximum compression. Approximately how much energy was stored in the spring at this moment? D: The Buggy Challenge 1. Predict. Your teacher will turn on a physics buggy, gentle lower it on to the sliding track used earlier today, and release

it. The mass of the buggy is less than the track. How will the buggy and track move? Use appropriate physics diagrams to support your prediction.

Cart A Cart B Impulse

Fnet

0

5

10

15

20

25

30

35

40

45

50

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

Kin

etic

En

erg

y (J

)

Time (s)

Kinetic Energy during a Collision

Ek A

Ek B

Ek total

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SPH4U: Types of Collisions A collision may roughly fall in to three categories based on how the system’s total kinetic energy (Ek = ½mv2) before and after the collision compares. If the total kinetic energy decreases due to the collision, it is called an inelastic collision. If the total kinetic energy remains is the same, the collision is called elastic. If the total kinetic energy increases, the collision is called superelastic. Our goal today is to study collisions that fall in to these three categories. You need a dynamics track and two carts. We will not be making precise measurements. We will simply estimate speeds or changes in speeds and try to making some decisions based on that. Make sure that carts don’t hit the floor! Turn them upside-down when you are not using them! Place obstacles at the ends of the track! A: The “Sticky” Collision 1. Observe and Represent. For the collision below, sketch the before and after situations.

2. Reason. Is the total momentum of the system constant during this collision? Explain.

3. Represent. Complete a momentum-impulse bar chart for the two cart system. Use your understanding of momentum conservation to help estimate the velocities just before or after the collision (for example, vA1 = 1 unit).

Situation: Sticky Type of Collision:

Before - Moment 1

After - Moment 2

Momentum Bar Chart Energy Bar Chart

A work-energy bar chart shows the amount of energy stored in different mechanisms of a system (motion, gravitational field, etc.) at two moments in time. The shaded column represents the flow of energy in or out of the system due to external forces, also known as the external work. A new column has been included to account for the internal energy: energy stored in other mechanisms of this system (for example heat, vibrations and many more). The heights of the bars in the graph do not need to be exact – we typically draw these charts before we make any calculations. What is important is that bars illustrate the correct ideas. 4. Interpret. What does this bar chart tell us about the amount of kinetic energy in the system before and after the

collision? What about the total energy? Which quantity is conserved? 5. Reason. What type of collision is this an example of: elastic, inelastic, or superelastic? Record this above the chart. A sticky collision is not the only kind of inelastic collision. Technically, any collision that loses kinetic energy is an inelastic collision. A sticky collision is special because is produces the greatest loss of kinetic energy for the given starting conditions. For this reason it is also called a completely inelastic collision.

EkA1 EkB1 Wext EkA2 EkB2 Eint

+ 0 -

+ 0 -

pA1 + pB1 + J = pA2 + pB2

vA1 = 1 unit vB1 = 0

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© C. Meyer, 2014

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B: The “Bouncy” Collision 1. Represent and Reason. For the collision below, sketch the before and after situations. Estimate the speeds involved.

Complete a momentum-impulse bar chart and energy bar chart. Does the system gain or lose kinetic energy? Decide what type of collision it is.

Situation: Bouncy Type of Collision:

Before - Moment 1

After - Moment 2

Momentum Bar Chart Energy Bar Chart

C: The Explosion 1. Represent and Reason. Sketch the before and after situations. Estimate the speeds involved (for example, vA1 = 1 unit).

Complete a momentum-impulse bar chart and a new kinetic energy bar chart. Does the system gain or lose kinetic energy? Decide what type of collision it is.

Situation: Explosion Type of Collision:

Before - Moment 1

After - Moment 2

Momentum-Impulse Bar Chart

Energy Bar Chart

In this final example energy was stored in the spring (elastic energy, Ee). This could have been counted as another storage mechanism for internal energy, but since we will soon be able to measure and describe energy stored in springs carefully, we gave spring energy its own bar. 2. Reason. Isaac drew his bar chart for the previous example with EkA2 as a positive bar and EkB2 as an equal-sized positive

bar. Albert drew EkB2 as an equal-sized negative bar. Who do you agree with? Explain.

3. Reason. Is energy a vector or scalar quantity? Use the bar charts you drew in this investigation to help explain.

EkA1 EkB1 Ee1 Wext EkA2 EkB2 Ee2

+ 0 -

+ 0 -

pA1 + pB1 + J = pA2 + pB2

vA1 = 0 vB1 = 0

Compressed spring is released.

Get ready to catch!

EkA1 EkB1 Wext EkA2 EkB2 Eint

+ 0 -

+ 0 -

pA1 + pB1 + J = pA2 + pB2

vA1 = 1 unit vB1 = 0

Carts collide magnetically

86

SPH4U: 2-D Momentum Homework Name: A typical problem involving the conservation of momentum in 2-D is often challenging, but usually due to lack of organization and careless mistakes. Follow the solution format and these following suggestions carefully! Note that parts B and C have been omitted here, but continue to do them whenever you can. Problem Two hover pucks glide towards each other, collide and then glide away. Puck A (5.0 kg) was initially travelling at 2.0 m/s [E 25o N]. Puck B (3.0 kg) was initially travelling at 4.0 m/s [E 30o S]. After the collision, puck A travelled at 1.6 m/s [E 30o S]. Determine the velocity of Puck B after the collision. A: Pictorial Representation Sketch showing “before” and “after”, coordinate system, label given information, conversions, unknowns, key events

D: Mathematical Representation Complete equations, describe steps, algebraic work, substitutions with units, final statement

Use the conservation of momentum to find the x-component of the velocity of puck B after the collision:

mava1x +

E: Evaluation Answer has reasonable size, direction and units?

vA1x = vA1y =

vB1x = vB1y =

vA2x = vA2y =

vB2x = vB2y =

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SPH4U: Momentum in 2-D Momentum is a vector quantity and the Law of Conservation of Momentum is a vector equation. An object’s momentum can be broken up into components and so can the law, meaning that momentum is conserved in each component direction. A: The Collision Consider an example where a small, fast moving hover puck (ma = 2.0 kg) collides with a large stationary hover puck (mb = 4.0 kg). Friction is small enough to be negligible.

1. Predict. Use your intuition to predict the direction mb will travel after the collision. Draw a vector arrow in the diagram

to show this. Briefly explain your reasoning. Do this quickly and move on! 2. Solve. For simplicity we choose a coordinate system that lines up with va1. Determine the x- and y-components of all the

known velocities. Record these in the table above. Indicate which components are unknown. Show your work and be sure to use the sign convention!

B: The x-Momentum 1. Solve. Find the x-components of all the three momentum vectors we have velocities for. Show your work.

pa1x = pb1x = pa2x =

2. Represent. Draw a bar in the momentum bar chart for the three x-momenta you calculated above.

3. Reason. Are there any external forces acting on the system of two pucks? Does the

system experience a net force from its environment? What does this imply about Jx? 4. Predict. Draw a fourth bar in the momentum bar chart that represents the x-momentum of puck B after the collision.

Explain your prediction. 5. Represent. Write an equation showing the relationship between the four momentum values in the bar chart. Use the

symbols in the bar chart.

va1x = va1y =

vb1x = vb1y =

va2x = va2y = ma

va1 = 6.0 m/s

va2 = 2.0 m/s

mb

vb1 = 0

60o

Ex

N y

Recorder: __________________ Manager: __________________ Speaker: _________________

0 1 2 3 4 5

© C. Meyer, 2014

pa1x + pb1x + Jx = pa2x + pb2x

+ 0 -

88

Since there is no net force in the x-direction, the x-momentum is conserved and: pa1x + pb1x = pa2x + pb2x

6. Solve. Use your new equation to explicitly solve for pb2x. 7. Reason. Isaac remarks that the magnitude of pb2x is quite large compared with the other components. “It must be going

really fast.” Do you agree or disagree? Explain. C: The y-Momentum 1. Solve. Find the y-components of all the three momentum vectors we have velocities for. Show your work.

pa1y = pb1y = pa2y =

2. Represent. Draw a bar in the momentum bar chart for the three y-momenta you calculated above.

3. Predict. Draw a fourth bar in the momentum bar chart that represents the y-

momentum of puck B after the collision. Explain your prediction without math. Since there is no net force in the y-direction, the y-momentum is conserved and: pa1y + pb1y = pa2y + pb2y

3. Solve. Use the conservation of momentum in the y-direction to explicitly solve for pb2y. 4. Reason. Looking back at your work so far, why was it helpful to choose a coordinate system that lined up with va1? D: The Final Result 1. Explain. Describe how we can use the results found so far to find the missing momentum 2bp

.

2. Solve. Find the momentum vector 2bp

. Be sure to draw the component triangle.

3. Solve and Test. Find 2bv

. Compare this with your prediction and the simulation, if possible.

4. Summarize. What does it mean to say, “Momentum is conserved in two dimensions”?

pa1y + pb1y + Jy = pa2y + pb2y

+ 0 -

89

SPH4U: 2-D Collisions The collision on the next page has had the path of one object removed! Your challenge is to reconstruct the path. Examine the collision and read the details describing it, then follow the steps below. 1. Observe. Label the points where the collision appears to begin and end. Explain how you can tell. 2. Reason. Marie says, “Let’s choose a coordinate system that lines up with the page – that will be the most helpful.”

Emmy says, “I think we should choose one that lines up with the initial velocity of puck A. That will be easiest.” Who do you agree with? Explain.

3. Observe and Reason. Find va1x on the opposite page and record the result on your whiteboard. Isaac says, “Let’s use

units of cm/s for our result – that will be convenient.” Albert says, “Hmm … maybe we should use m/s, even though the numbers will be very small – they are S.I. units.” Who do you agree with? Explain.

4. Observe and Calculate. Write an equation for the conservation of momentum in the x-direction where vb2x is the

unknown. Make the necessary measurements from the first page (show these) and record your calculated results in the chart. Solve the equation for vb2x. Note: the equation will simplify. Explain why.

5. Observe and Calculate. Write an equation for the conservation of momentum in the y-direction where vb2y is the

unknown. Make the necessary measurements from the first page and record these in the chart. Solve the equation for vb2y. 6. Calculate and Test. Determine the vector 2bv

. Carefully draw the vector beginning at the circle under the “B” (Bonus:

draw the circles too!)

** Check this with your teacher! ** 7. Calculate and Reason. Use kinetic energy calculations to help explain what type of collision this is.

Recorder: __________________ Manager: __________________ Speaker: _________________

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© C. Meyer, 2014

90

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91

SPH4U: Work and Kinetic Energy How can the energy of a system change? When an external force acts on a system while a system object moves through a displacement, energy flows into or out of the system. This process is called work and the amount of work done (or energy that has flowed) due to the force can be found from the equation: W = |F||∆d|cosθ, where θ is the angle between the force and displacement vectors. A: The Flow of Energy Your teacher has a track with two carts set up. A large cart, cart A (1.0 kg) is moving right and a small cart, cart B (500 g) is at rest. They collide magnetically and travel on a frictionless track. 1. Predict. (individually). Describe how each cart will move after the collision. 2. Observe (as a class). Describe the motion of each cart after the collision. An energy-flow diagram is similar to an interaction diagram, but with two differences: we only draw the lines connecting objects if there is a flow of energy between them and if there is an energy flow, we add an arrow showing the direction of the flow. 3. Reason. Which object has gained energy? Which has lost energy? Is energy lost to the environment? 4. Represent. Draw an energy flow diagram for the system of the two carts.

5. Represent. Moments 1 and 2 occur at the beginning and end of the collision. Follow the instructions and complete the

chart below for the system of cart A and then for the system of cart B. (a) Draw a vector arrow for the displacement of the cart and the force acting on the cart during the collision. (b) Complete a motion diagram for the cart between moments 1 and 2. Describe its motion: is the cart speeding up,

slowing down, or moving with a constant speed? (c) What is happening to the energy of the system of the cart? Draw an energy flow diagram for the system of the cart. (d) According to the equation for work, is the work done by this force positive, negative or zero? Give a sample

“calculation” for the sign. (e) Complete the work-energy bar chart for the system of the cart.

System

= Cart A

Displacement of cart Motion Diagram Energy-Flow Diagram

Work-Energy Bar Chart

Sign of Work Force of B on A Description of Motion

Ek1 Wext Ek2

+ 0 -

Recorder: __________________ Manager: __________________ Speaker: _________________

0 1 2 3 4 5

© C. Meyer, 2014

92

6. Reason. We have been assuming (consciously or not) that the gravity and the normal force do not cause energy to flow

into or out of the system. Use the equation for work to help explain why this was correct. (Hint: ) 7. Reason. Does the result of the work equation depend on your choice of sign convention? Is work a scalar or vector

quantity? Explain. B: Work from Multiple Forces 1. Represent. In a new experiment, two hands push horizontally on the same cart. Hand 1 pushes to the left on the cart and

Hand 2 pushes to the right. The cart is moving to the right and is now speeding up. (a) Draw a motion diagram for the cart. (b) Draw a complete force diagram for the system of the cart. (c) Decide whether the work done by each force is positive, negative, or zero. Indicate this by writing a +, –, or 0 along side the force vector. (d) Draw an energy-flow diagram for the cart (e) Is the net work on the system above positive, negative or zero?

Motion Diagram Force Diagram Energy-Flow Diagram Net Work

When a system has multiple interactions with its environment it may gain or lose energy due to a number of forces. The total change in energy of the system is the net work. The net work can be found in two different ways: by adding up the work from the individual forces or by finding the work due to the net force.

Wnet = W= W1 + W2 + … or Wnet = |Fnet||d|cos A system that experiences a net force will accelerate and gain or lose kinetic energy. This idea is called the net work - kinetic energy theorem, Wnet = Ek. This theorem is closely related to Newton’s 2nd Law. 2. Represent. Two hands continue to push on opposite ends of the cart, but with different magnitudes than before. For each

situation, complete the chart below like in question #1 above.

3. Summarize. What is the physical significance of positive, negative and zero net work? What happens to the system?

System

= Cart B

Displacement of cart Motion Diagram Energy-Flow Diagram

Work-Energy Bar Chart

Sign of Work Force of A on B Description of Motion

Description

The cart is moving to the left and is slowing

down.

Motion Diagram Force Diagram Energy-Flow Diagram Net Work

Ek1 Wext Ek2

+ 0 -

93

SPH4U: Energy and Coordinate Systems Gravitational energy (Eg = mgy) is a special kind of energy that depends on an object’s vertical position. But what happens if we choose coordinate systems that have different vertical origins for our measurements? Let’s find out! On a hot day, you look out a third story window and hold in your hands a 2.0 kg water balloon (a big one!). Your friend walks below the window, not noticing you 10 meters above her. You release the water balloon. We will analyze what happens next from the frame of reference of you and also of your friend. 1. Represent. Draw an energy flow diagram for the Earth-balloon system while the balloon

falls. A: Coordinate System One You set the vertical origin at your hand. Calculate the energies involved. Use a kinematics calculation to help find v2. Add up the total energies at each moment. Complete the work-energy bar chart for the earth-balloon system.

Sketch

Gravitational Energy Eg1 = Eg2 =

Kinetic Energy Ek1 = Ek2 =

Total Energy ET1 = ET2 =

Work-Energy Bar Chart

B: Coordinate System Two Now you set the vertical origin at your friend’s head. Calculate the energies involved. Use a kinematics calculation to help find v2. Add up the total energies at each moment. Complete the work-energy bar chart.

Sketch

Potential Energy Eg1 = Eg2 =

Kinetic Energy Ek1 = Ek2 =

Total Energy ET1 = ET2 =

Work-Energy Bar Chart

C: Thinking About Energy 1. Reason. How does Eg, Ek and Et change as measured in each coordinate system? 2. Reason. The two observers do not agree on the total energies, but this is not a problem. What is important in physics are

energy changes. What does each observer conclude about the amount of energy that transfers from gravitational to kinetic within the system? Explain.

Ek1 Eg1 Wext Ek2 Eg2

+ 0 -

y

y1 = 10.0 m v1 = 0

y2 = 0 v2 < 0

Ek1 Eg1 Wext Ek2 Eg2

+ 0 -

y

y1 = 0 v1 = 0

y2 = - 10.0 m v2 < 0

Recorder: __________________ Manager: __________________ Speaker: _________________

0 1 2 3 4 5

Adapted from: Laws, P. Workshop Physics Activity Guide. Module II. John Wiley, 2004

Flow

94

SPH4U: Transfers of Energy Homework Name:

1. Represent. Each row in the chart below gives multiple representations of one situation. Complete the missing parts for each row – come up with new and interesting situations where appropriate.

Word Description Sketch of Initial and Final States Work-Energy Bar Chart

and Equation (i) A stunt car has a spring-powered ejector seat. When the spring is released, the seat with its passenger is launched out of the car and reaches a maximum height y2 above its starting position. (The elastic energy stored in the compression x of a spring is Ee)

System: Equation:

(ii) An elevator is initially moving downwards at speed v1. It approaches the ground floor and slows to a stop in a distance h.

System: Equation:

(iii)

System: Equation:

(iv)

System: Equation:

Wext = Eg2 + Eth2

+ 0 -

Ek1+ Wext = Ek2

+ 0 -

+ 0 -

y

0

y1 > 0 v1 < 0

y2 = 0 v2 = 0

+ 0 -

y

0 y1 = 0 v1 = 0 x > 0

y2 > 0 v2 = 0 x = 0

95

SPH4U: Transfers of Energy Nobody really knows what energy is, but we do know how it behaves. Energy is a quantity that can be transferred between systems and which can be used to predict whether some event may occur. By carefully keeping track of where the energy is located, or stored, we can construct equations to help with our predictions. A: An Energetic Example Your teacher has a ramp set up at the front of the class. A block is on top of a cart that will be released and glide down an inclined track. The cart will collide with a barrier and stop. The block slides across another track until it comes to a stop. Your challenge is to predict how far your block will slide along the bottom track surface. 1. Reason. Which objects interact with the block? Which interactions involve a transfer of energy? 2. Reason. There are three important events in this situation.

Describe them here and label them on the diagram.

To understand the role of energy in a problem, we need to choose a system, or a collection of objects, whose energies we will track. We have a freedom to include any objects in our system and, if we do our work properly, everyone should always agree on the final answer. If an interaction can easily be described using energy (e.g. gravity), include those objects in your system. If an interaction is more easily described using work (e.g. applied, tension), make one object external. Friction effects involving transfers to thermal energy are best described as internal energy that is shared between system objects (they each get warmer). If this is the case, we will keep objects like the ground surface, roads, and tracks inside the system.

3. Reason. Based on the advice above, chose your system. Draw an energy-flow

diagram for the system between moments 1 and 2. Do the same for moments 2 and 3.

System =

The law of the conservation of energy states that the total energy of a system remains constant unless energy flows into or out of the system. This flow of energy can be due to the work done by an external force. The total energy of the system and its changes may be expressed by a work-energy equation: ET1 + Wext = ET2. 4. Reason. What is a convenient position for the vertical origin in the situation? Explain how you chose it and label it in the

diagram above. 5. Represent. Complete a work-energy bar chart for

moments 1 and 2. Complete a work-energy bar chart for moments 2 and 3.

Recorder: __________________ Manager: __________________ Speaker: _________________

0 1 2 3 4 5

+y

Ek1 + Eg1 + Wext = Ek2 + Eg2

+ 0 -

© C. Meyer 2014

Ek2 Eg2 Wext Ek3 Eg3 Eint

+ 0 -

ET2 + Wext = ET3

1-2 2-3

ET1 + Wext = ET2

96

6. Represent. The bar charts help you to write the work-energy equations for the system comparing the two moments in time. Do this using the same symbols as in the chart.

7. Explain. When you practice problems like this in your homework, part C asks for a word explanation. Let’s practice:

describe the motion of the system and use energy transfers and work to explain why it moves the way it does. To find the amount of thermal energy (which we are calling Eint) due to the sliding block, we need to use the definition of work to create a new equation. W = Fdcos. When an object is sliding, the angle between Ff and d is 180o. This gives W=-Ff d. The amount of thermal energy should be a positive number so we create the equation, Eth = Ff d. 8. Measure. Choose a block for your group. Make the measurements you think will be necessary to find all the energies.

Record these in your diagram above. 9. Predict. Use your work-energy equations to predict the sliding distance of the block. Write this up as you would part D

of your homework solutions. (Hint: only substitute in your final step!)

D: Mathematical Representation Describe steps, complete equations, algebraically isolate, substitutions with units, final statement

10. Evaluate. Is your answer reasonable? Explain why you think so before you test it! 11. Test. Ask your teacher to watch you test your prediction. Note: there are very large uncertainties in this test! Try it a few

times to see how it compares with your prediction. If you are within about 20 cm it is a good result!

97

SPH4U: The Ballistic Pendulum Here’s a problem for you: how can you determine the speed of a bullet using only measurements of mass and distance? The answer was found in 1742 by the English mathematician Benjamin Robbins using his invention, the ballistic pendulum. Now it’s your turn to repeat his clever work – but since guns are frowned upon in schools, you will launch a blob of play-doh at our ballistic pendulum. A: Pictorial Representation Sketch with “before” and “after”, coordinate system, label givens & unknowns, conversions, events B: Physics Representation Energy / momentum bar chart, flow diagram, events

C: Word Representation Describe motion, energy / momentum transfers, assumptions Moments 1-2: Moments 2-3:

Recorder: __________________ Manager: __________________ Speaker: _________________

0 1 2 3 4 5

pA1 + pB1 + J = pA2 + pB2

+ 0 -

Ek2 +Eg2 + Wext = Ek3 + Eg3

+ 0 -

© C. Meyer, 2014

Measurements: Do these later. Hint: There are three important events in this problem (not including the throw). You may assume that its velocity does not change much while traveling through the air.

1-2 2-3

Ek1+ Eg1 + Wext = Ek2 + Eg2 +Eint

+ 0 -

98

D: Mathematical Representation Describe steps, complete equations, algebraically isolate, substitutions with units, final statement

E: Evaluation Answer has reasonable size, direction and units?

Now try out the pendulum and make your measurements. Record these values in your sketch for part A. Hint: Make sure you consider the geometry of the swing carefully!

99

SPH4U: Spring Force and Energy How does the force exerted by a spring change as we stretch it? How much energy is stored in the spring? In this investigation we will find out. A: Force and Springs A few things about springs: Don’t overstretch springs – they can be permanently deformed and damaged. The position of the end of a spring experiencing no other forces, whether oriented horizontally or vertically, is called the equilibrium position. Physics springs have no mass. 1. Explain. There is an important different between the length of a spring and the amount of stretch. Explain this

difference. Which one are you being asked to use? 2. Explain. Describe an experiment to determine how the size of the spring force is affected by the amount of stretch in a

spring. A good experiment will feature many data points to reduce uncertainties and use consistent intervals to help find patterns.

3. Represent. Draw a sketch of your experiment showing the measurements involved. 4. Observe. Conduct your experiment and make your

measurements.

5. Represent. Plot a graph that shows the relationship between your variables. 6. Analyze. Use appropriate techniques to analyze the relationship in the graph. (Always show units!) 7. Represent. Construct an equation using the symbols Fs (the force the spring exerts), Δx (the stretch of the spring), and k

the spring constant (your slope) based on your graphical analysis.

Recorder: __________________ Manager: __________________ Speaker: _________________

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© C. Meyer, 2014

100

8. Predict and Test. You will hang a 1.0 kg object from your spring. Predict the amount of stretch that will result. Test your prediction.

9. Interpret. What does the value of the spring constant, k, tell us about the spring itself? What would a large value

indicate? What would a small value indicate? The relationship found above was first discovered by Robert Hooke around 1660. Within a certain range of stretch or compression most springs behave according to Hooke’s Law. Note that the direction of the force the spring exerts is opposite to the direction of the stretch it experiences. Official version: Magnitude only: B: Energy and Springs 1. Reason. Why do we believe there is a flow of energy when you stretched the spring during your

experiment? Represent this flow with an energy flow diagram. 2. Reason. Emmy is discussing how to measure the energy transferred while she stretched

the spring. “I measured a force of 2.1 N when the spring had 20 cm of stretch and a force of 3.3 N when it had 30 cm of stretch. I moved it by 10 cm so the work I did is W = (2.1 N)(0.10 m) = 0.21 J.” Marie says Emmy’s result is too low. Who do you agree with? Explain.

3. Reason. Albert joins in the discussion. “But the force gets bigger as you stretch it, so we

have to use the bigger value. The work will be (3.3 N)(0.10m) = 0.33 J.” Isaac says Albert’s result will be too high. Who do you agree with? Explain.

4. Reason. Both Albert and Emmy were wrong! The force is changing in a steady way when

the spring is stretched from 20 to 30 cm. What single value of force would be best to use in our work equation? How much work did Emmy actually do?

** check your answer with your teacher **

3.3

2.1

20 30

3.3

2.1

20 30

3.3

2.1

20 30

101

5. Calculate. Complete the chart below. Calculate the work done during each increment of stretch based on your force data from earlier. The “Increment of Stretch” is the extra distance the spring was stretched between one force measurement and the next. If you’re not sure, try two rows and call your teacher over. Displacement (stretch) from equilibrium (m)

Force (N) Increment of stretch (m)

Average force during increment (N)

Work done during increment (J)

0

6. Calculate. Now you are ready to calculate the total work done to stretch a spring from its equilibrium position to each

displacement. To do this, consider that the work to displace the spring to 0.20 m is equal the work to go from 0 to 0.10 m plus the work to go from 0.10 to 0.20 m. Complete the chart to the right and plot a graph of total work vs. displacement.

Work can be found by calculating the area under a force-displacement graph. This is especially helpful for forces that change magnitude.

7. Calculate. Find the work needed to stretch the spring to a displacement of 0.30 m by computing the area under the Fs vs. Δx graph you created earlier. Does this agree with the result in your chart above?

8. Speculate. Create an equation that relates W and Δx. Hint: Consider how you computed the area to get the work.

Official version: ** check your result with your teacher **

Displacement from equilibrium (m)

Total Work (J)

102

C: The Test Launch Your teacher has a dynamics cart set up on a track at the front of the class. It has a spring launching mechanism. Your challenge is to make a few measurements and predict the launch speed of the cart. When thinking about this process, there are three important events: (1) the cart is at rest and the spring is uncompressed, (2) the spring is now fully compressed due to your hand, and (3) the cart has been launched and the spring is uncompressed.

1. Represent. Draw an energy flow diagram for the system of the cart between moments 1 and 2 and between moments 2

and 3. Draw an energy bar chart for each moment in time.

2. Plan. What steps will you take in your solution in order to find the launch speed?

3. Measure. Make any appropriate measurements to determine the quantities you will need to find the launch speed. (Hint:

to find k, simply place a weight vertically on top of the cart’s spring.)

4. Calculate and Predict. Use your measurements to predict the launch speed of the dynamics cart. D: Mathematical Representation Describe steps, complete equations, algebraically isolate, substitutions with units, final statement with uncertainty

5. Test. Use the motion detector. You probably noticed that the speed was a bit slower than you predicted. Explain why.

1-2 2-3

Ek1 +Ee1 + Wext = Ek2 + Ee2

+ 0 -

Ek2 +Ee2 + Wext = Ek3 + Ee3

+ 0 -

103

SPH4U: Velocity and Frames of Reference All physics quantities that we measure depend on the frame of reference of the observer. Alice is standing on the Earth and watches a train go by with a velocity of 150 km/h [E]. Inside the train stands Bob. Both Alice and Bob are physicists and make observations about each other’s motion.

1. Complete the chart showing the measured velocity of each object from each reference frame.

Object Frame A (Alice) Frame B (Bob) Alice Bob Earth Train

2. Bob has a ball and throws it. He measures the velocity of the ball to be 40 km/h [E]. The train keeps going at its usual

speed. Complete the chart showing the measured velocity of the ball from each reference frame. Explain how you found the velocity of the ball relative to frame A.

Object Frame A (Alice) Frame B (Bob) Ball

3. Bob throws a second ball and measures the velocity to be 30 km/h [W]. Complete the chart showing the measured

velocity of the ball from each reference frame. Explain how you found the velocity of the ball relative to frame A.

Object Frame A (Alice) Frame B (Bob) Ball

4. Bob pulls out a flashlight, points it east and turns it on. Using a fancy apparatus he measures the velocity of a particle of

light from his flashlight to be 300 000 000 m/s [E]. Using the previous logic, what is the velocity of the light relative to Frame A in m/s?

5. Imagine Bob was on an “express” train that travelled at 2 x 108 m/s [E] and turned on his flashlight just as in question 4.

What is the velocity of the light relative to Frame A? 6. Alice now has her flashlight turned on and points it east. Bob’s same express train passes by. What is the velocity of the

light from Alice’s flashlight relative to Bob?

B

A

B

A

Recorder: __________________ Manager: __________________ Speaker: _________________

0 1 2 3 4 5

© C. Meyer, 2012

104

SPH4U: Relativity Problems 1. A cosmic ray travels 60 km through the earth’s atmosphere in 400 µs (10-6 s), as measured by experimenters on the

ground. How long does the journey take according to the cosmic ray?

2. At what speed, as a fraction of c, does a moving clock tick at half the rate of an identical clock at rest.

3. An astronaut travels to a star system 4.5 ly away at a speed of 0.9c. Assume the times needed to speed up and slow down are negligible.

(a) How long does the journey take according to Mission Control on Earth? (b) How long does the journey take according to the astronaut?

4. How fast must an astronaut travel on a journey to a distant star so that the astronaut ages 10 years while the Mission

Control workers age 120 years?

5. Jill claims that her new rocket is 100 m long. As she flies past your house, you measure the rocket’s length and find that it is only 80 m/ Should Jill be cited for exceeding the 0.5c speed limit?

6. A muon travels 60 km through the atmosphere at a speed of 0.9997c. According to the muon, how thick is the

atmosphere? 7. Our Milky Way galaxy is 100,000 ly in diameter. A spaceship crossing the galaxy measures the galaxy’s diameter to be a

mere 100ly. (a) What is the speed of the spaceship relative to the galaxy? (b) How long is the crossing time as measured in the galaxy’s frame of reference?

8. What are the kinetic energy, the rest energy and the total energy of a 1.0 g particle with a speed of 0.8c?

9. At what speed is a particle’s kinetic energy twice its rest energy?

10. In an attempt to reduce the extraordinarily long travel times for voyaging to distant stars, some people have suggested

travelling close to the speed of light. Suppose you wish to visit the red giant start Betelgeuse, which is 430 ly away, and that you want your 20 000 kg rocket to move so fast that you age only 20 years during the journey. (a) How fast must the rocket travel relative to the earth? (Hint: roughly how long will the journey take according to an

observer on the earth?) (b) How much energy is needed to accelerate the rocket to this speed? (c) Compare this amount of energy to the total used by the United States in the year 2010, which was roughly 1.0x1020J.

11. The nuclear reaction that powers the sun is the fusion of four protons (1.673x10-27 kg) into a helium nucleus (6.645x10-27

kg). The process involves several steps, but the net reaction is simply: 4p He + energy. How much energy is released overall in each fusion process? Give your answer in J and MeV.

12. Consider the inelastic collision e- + e- e- + e+ + e- + e+ in which an electron-positron pair is produced in a head-on collision between two electrons moving in opposite directions at the same speed.

(a) What is the minimum kinetic energy each electron must have to allow this process to occur? (b) What is the speed of an electron with this energy?

1. 347 µs 2. 0.866c 3. (a) 5 yr, (b) 2.18 yr 4. 0.965c 5. Yes 6. 1.47 km 7. (a) 0.9999995c, (b)100 000.05 yr 8. 5.99x1013 J, 8.99x1013 J, 1.50x1014 J 9. 0.943c 10. (a) 0.999c, (b) 3.69x1022J, (c) 370 times greater! 11. 4.22x1012J, 26.4 MeV 12. (a) 8.19x10-14 J, (b) 0.866c

From R. Knight, Physics for Scientists and Engineers. A Strategic Approach.Pearson Education, 2013

105

SPH4U: The Light Clock Einstein thought deeply about the train scenario we studied last class and suggested two ideas that change our understanding of what happens when Bob turns on his flashlight. Einstein postulated that: (1) All observers, whether they are moving fast or slow in an inertial frame will observe light to travel at c = 2.998 x 108 m/s

in vacuum, and (2) the laws of physics are the same for all inertial frames – there are no special rules if you are moving fast or slow. These suggestions are known as the two postulates of special relativity. A: The World According to Bob

Bob is now travelling in a spacecraft at a velocity v relative to the earth. He is carrying with him a light clock – a special kind of clock imagined by Einstein. The clock consists of two perfect, smooth mirrors that face each other and are separated by a small distance ∆d. A particle of light (a photon) reflects back and forth between the mirrors which are lined up carefully so that the photon always reflects off the same two points on the mirror’s surface. We note two events that take place: event 1where the photon leaves the bottom mirror and event 2 where the photon reaches the top mirror. The time it takes for the photon to travel between the mirrors represents the ‘tick’ of the clock. 1. Reason. Use the symbols from the description of this situation to help complete the chart of

measurements from Bob’s frame of reference.

2. Represent. Construct an equation that relates the speed of the photon to the distance and time it

travels as measured in Bob’s frame. B: The World According to Alice Alice is standing on Earth watching Bob and his light clock travel by in the rocket ship. She is able to make careful measurements of the light clock and its photon. 1. Represent. From Alice’s frame, we see the light clock at three moments in time corresponding to three events: the

photon at 1, the photon at 2 and then the photon returning to the bottom mirror. You may assume the rocket is travelling quite fast! Draw the path of the photon through space. Label the distancebetween events 1 and 2 as ∆D.

2. Reason. Complete the chart of

measurements from Alice’s frame of reference. No calculations are required!

Time interval between events1and2 (one tick) ∆to

Velocity of the light clock Distance between events1and2 Speed of the photon

Time interval between events 1 and 2 (one tick) ∆t

Velocity of the light clock Distance between events 1 and 2 Speed of the photon

B

A

+x

2

1

∆d

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3. Represent. Construct an equation that relates the speed of the photon to the distance and time it travels as measured in Alice’s frame.

4. Reason. Compare the size of the results from the two frames of reference.

Note that both observers must agree on the speed of light according to the first postulate of special relativity.

5. Speculate. What does your comparison imply about the flow of time on the spacecraft according to Alice? C: Time Dilation A direct consequence of Einstein’s two postulates is that the time interval between two ticks of a clock is shortest when the clock is at rest relative to an observer.This is not an optical illusion, a delay effect, or a mechanical defect of the clock. The flow of time actually depends on who is observing and their speed. This idea is called time dilation. To help carefully describe time intervals we introduce two definitions: Proper time (∆to): The time interval between two events that occur at the same position in space. Relativistic time (∆t): The time interval between two events that occur at two different positions in space. The relativistic time interval will always be greater than the proper time interval (∆t > ∆to) as long as there is relative motion of the two observers. Each observer (and you too!) needs to decide every time, whether the time interval you are studying is proper or relativistic according to their own frame of reference. Gone are your days of innocence when time intervals were just time intervals! 1. Explain. Use the new definitions to help explain which type of time interval Alice and Bob measured. Note: we will

assume that ∆d is very small and can be ignored. 2. Interpret. Alice has a toasterwhich she starts.Bob measures a time interval of 72 s between the two events of starting the

toaster and the ‘pop’ at the end. Alice measures a time interval of 60 s between the same two events on her light clock. Explain what type of time interval each observer measured.

The two different time intervals (as measured in different frames) between the same pair of events do not lead to any kind of logical contradiction. A contradiction in physics only occurs when two observers in the same frame of reference get different results. Measurements of the same events by different observers in a single reference frame shouldagree with one another. This is what we have been assuming all along in our study of physics! To figure out how much larger the relativistic time interval is, we can do a mathematical analysis of the light clock and carefully compare ∆D with ∆d. If we do this (try it – it’s fun!) we get the time dilation equation: ∆t = ∆to, where

21

2

2

1

c

v . We need to learn how to use the expression for (gamma), but that will come later.

Measurement Comparison Speed of photon Distance between 1 and 2 (∆d vs. ∆D) Time interval between 1 and 2 (∆t vs. ∆to)

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SPH4U: The Moving Ruler Consider a subatomic particle called the muon () which is moving rapidly at a speed v relative to a ruler in the direction of the ruler’s length. We note two events: event 1 when the muon is at the 0 centimeter mark, and event 2 when the muon is at 30 centimeter mark. You need a ruler and an eraser (muon) for this investigation. A: The Ruler’s Frame 1. Represent. Act out this situation from the frame of reference of the ruler. Pretend to use a stopwatch. Say out loud, “1”

and “2” when events 1 and 2 occur. Demonstrate this for your teacher. Sketch this situation and label the two events. Indicate the velocity of the muon v and the distance between the two events ∆xo.

2. Reason. An observer in the ruler’s frame of reference measures the time interval between the two events. Carefully

explain what type of time interval this is. Act this out by showing when you start and stop your pretend stopwatch. What symbol should you use to represent it?

3. Represent. Write an equation that relates speed, distance and time of the muon as measured by an observer in the ruler’s

frame. B: The Muon’s Frame 1. Represent. Act out this situation from the frame of reference of the muon. Say out loud, “1” and “2” when events 1 and

2 occur. Demonstrate this for your teacher. Sketch this situation and label the two events – clearly show that the ruler has moved! Indicate the velocity of the ruler v and the distance the ruler travels between events 1 and 2.

2. Reason. An observer in the muon’s frame of reference measures the time interval between the two events 1 and 2. Act

this out by showing when you start and stop your pretend stopwatch. Carefully explain what type of time interval this is. What symbol should you use to represent it?

3. Represent. What distance does the metre stick travel in this frame of reference? Label this as x on the diagram. 4. Represent. Write an equation that relates speed, distance and time for the metre stick as measured by an observer in the

muon’s frame.

5. Reason. Compare the size of the results from each frame.

Speed of muon / metre stick Time between events 1 and 2 Distance between events 1 and 2

6. Reason. What can we conclude about the distance measurement of each observer?

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C: Length Contraction The second consequence of Einstein’s two postulates is that the spatial interval between two events (the distance) also depends on the observer! Moving objects (or intervals of space) become smaller along their direction of motion. This is called length contraction. This is not an optical illusion – space itself (even if it’s empty) contracts. So a ruler moving towards us contracts. If we travel past Earth, the space between Earth and the moon will contract. We define two different types of distances or lengths. Proper length (xo): The distance between two points (ends of an object, positions in space) that are at rest relative to an

observer. Relativistic length (x): The distance between two points (ends of an object, positions in space) that are moving relative to

an observer. The relativistic length is always smaller than the proper length (x < xo). A quick calculation gives the expression: x = xo/

109

SPH4U: Relativity Problem Solving A: Why Don’t We Notice? The consequences of Einstein’s two postulates seem really crazy to us largely because we have never noticed the changes to time and length intervals. We must now address this: why have we never noticed time slowing down or lengths contracting for drivers on the 401? The expression for gamma: = (1 – v2/c2)-1/2 will help us to answer this question. In special relativity, express your velocity values as a fraction of c. For example, v = 1.5 x 108 m/s = 0.5 c. When you substitute the velocity written this way into , the c’s divide out nicely and the math is much friendlier. 1. Calculate and Represent. Complete the chart below. Rewrite the first five speeds in terms of c. Calculate for each

speed. Sketch a graph of vs. v.

In relativity we often encounter extreme numbers. We need to judge significant digits by the digits which are not zero for (1.00007 has one useful digit), or which are not 9 for velocities in terms of c (0.9994c has one useful digit). 2. Explain. Should the first five values you calculate be exactly the same? 3. Explain. Based on the chart, offer a simple explanation for why relativistic effects are not noticed in daily life. 4. Describe. What happens to the size of as v approaches the value c?

5. Reason. What does this tell us about the flow of time for a highly relativistic object (speeds close to c)?

Speed (m/s) Speed (c)

Fast Runners, 10 m/s Fast Cars, 40 m/s Fast Jets, 600 m/s The Space Shuttle, 7 860 m/s Voyager Space Probe, 17 000 m/s 0.1 c TV screen electrons 0.3 c 0.5 c 0.7 c 0.9 c 0.99 c X-Ray Machine Electrons 0.999 c LHC protons, 0.999 999 999 95 c

Recorder: __________________ Manager: __________________ Speaker: _________________

0 1 2 3 4 5

| 0.2 c

| 0.4 c

| 0.6 c

| 0.8 c

| 1.0 c

5.0 —

10.0 —

15.0 —

20.0 —

0

110

6. Apply. Relativistic effects are important for GPS satellites which orbit at a similar speed to the space shuttle relative to the ground. Precision timing is absolutely essential for determining an object’s location on the earth. For a GPS satellite observed from the earth, = 1.000 000 000 3.

a) Over the course of one day, how much time in seconds does the GPS clock gain or lose compared to a ground clock? Watch your math!

b) How far does light travel during that time discrepancy? What would be the implication for the GPS system?

B: Relativity Problem Solving A: Pictorial Representations Draw a reference frame (x-y axis) for each observer in the problem. Attach the given information to the frame in which that value was measured. Carefully label the intervals as proper or

relativistic. Clearly show the objects and events used to define the time intervals. C: Word Representations Describe the motion and what types of time or length intervals are involved. E: Evaluate Based on your understanding of relativistic or proper intervals, you should know if the results should be larger or

smaller. Tips: (1) A convenient unit of distance is the light year: 1 ly = c a (speed of light year). (2)When possible, solve for and calculate first. 1. Solve. An astronaut travels rapidly past Earth. She notices the lights in Toronto turn on at night and then off in the

morning and measures the time interval to be 14.3 hours. In this time she measures that Earth moves 5x109 m. Complete the rest of part A below and then solve the problems: What is the speed of the earth? What is the time interval between lights on and off as measured by a person in Toronto?

A: Pictorial Representation Sketch, reference frames, label givens & unknowns with symbols, conversions, describe events

C: Word Representation Describe motion (no numbers), explain why, assumptions

D: Mathematical Representation Describe steps, complete equations, algebraically isolate, substitutions with units, final statement

E: Evaluation Answer has reasonable size, direction and units? Why?

Astronaut’s Frame

1 = lights on 2 = lights off

111

SPH4U: Relativity and Energy The consequences of Einstein’s bold suggestion, that the speed of light is constant for all inertial reference frames, go far beyond just space and time – they also extend to our notions of energy. Using a clever argument, Einstein created the world’s most famous equation:

E = mc2 where = (1-v2/c2)-½

This is usually written, for the general public, as Eo = mc2, where the “o” is carelessly left out! Sometimes physics ideas stretch beyond our common sense and we begin to rely on equations to help us understand how our universe works. Let’s explore this equation and try to figure out what it tells us about energy. A: The Mass-Energy Relationship 1. Reason. Describe carefully how this energy depends on the speed of an object.

2. Reason. What other type of energy depends on an object’s speed? What does this tell us about the type of energy

Einstein’s equation describes? 3. Reason. According to the equation, how much energy does an object have when it is at rest?Explain how the equation

for E becomes the equation for Eo. Is Einstein’s equation still describing kinetic energy? 4. Reason. When at rest, what is the only characteristic of the object that could be changed and affect the amount of energy

Eo? What does this suggest about where this energy might be stored? An object at rest possesses a form of energy called its rest energy, Eo, Einstein’s complete expression (E = mc2) gives the total energy of the object, which always includes the rest energy and possibly some kinetic energy depending on the object’s velocity. To the best of our knowledge, this equation is correct under all circumstances and replaces the ones we have previously learned. 5. Represent. Write an expression that shows the relationship between E, Eo and Ek. ** check this with your teacher before moving on **

Recorder: __________________ Manager: __________________ Speaker: _________________

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B: Relativistic Energy 1. Calculate. Consider a 1.0 kg block initially at rest. It experiences a force that eventually causes it to reach an impressive

speed of 0.6 c. Imagine we had learned nothing about relativity - determine the energies for the “Before Einstein” column in the chart below. Use Einstein’s equation to help find the energies for the “After Einstein” column.

Before Einstein (B. E.) After Einstein (A. E.) Rest Energy zero

Kinetic Energy

Total Energy

2. Explain. How do you use Einstein’s equation to find the kinetic energy? 3. Reason. Under what condition is the expression ½mv2 valid? What should we conclude about the limitations of the

traditional kinetic energy equation?

Accelerating an object to speeds near that of light is extremely challenging and with our current technology, we can only accomplish this for atoms and sub-atomic particles. According to Newton, all we need to do is exert a steady force on something for long enough and the uniform acceleration will eventually cause the object to reach 3.0 x 108 m/s and our science fiction dreams will come true. According to Einstein, things are different. 4. Reason. How much energy is required to bring the 1.0 kg block to the speed of light? Explain the mathematical

difficulty with performing this calculation. Explain to your kid sister how much energy would you “need”. 5. Reason. What does the difficulty of the previous calculation imply about the possibility of ever reaching or exceeding

the speed of light?

This is the main reason why the latest and greatest particle accelerator, the Large Hadron Collider ($ 9 000 000 000), is such a colossal engineering feat. A tremendous amount of energy is required to accelerate the collider’s protons to 0.999 999 991 c.

113

C: Particle Physics 1. Calculate. A proton is a very small particle with a mass of 1.673 x 10-27 kg. How much energy is stored in the mass of

the particle? Subatomic particles usually possess very small quantities of energy. A new unit is needed to conveniently notate these small values. One electron volt (eV) is a unit of energy equivalent to 1.602 x 10-19 J. 2. Calculate. Find the proton’s rest mass energy in terms of MeV (106eV).

3. Explain. Physicists often write the mass of the proton as 938.3 MeV/c2. Use the rest-energy equation to help explain why this is a valid unit for mass. Explain why these units make it easy to calculate the rest energy.

Is it possible to release the energy stored in a particle’s mass? You may have already studied the process of nuclear fusion or fission in another course and have learned that, yes, this is possible. In a typical fusion reaction (like in the sun), a deuterium particle (1876 MeV/c2) fuse with a tritium particle (2809 MeV/c2)producing a helium nuclei (3729 MeV/c2) and a neutron (937 MeV/c2).

D + T He + n 4. Reason. How does the mass of the reactants compare with the mass of the products? What happened to the mass? What

does this imply about the conservation of mass? Speculate on a new, better conservation law.

5. Calculate. How much energy is released during the fusion process? Give your answer in joules and electron volts.

The conversion of matter to energy can be total if a matter particle collides with a corresponding anti-matter particle. This is the raison d’etre of the Large Hadron Collider: to collide protons and anti-protons, which releases a tremendous amount of energy. This is also the physics behind the medical imaging technique positron imaging tomography (PET scans), where an electron collides with a positron (the anti-electron). In the case of the PET scan, radioactive materials are injected into the blood stream of a patient. The decay process releases a positron (an anti-electron) which collides with an electron of a nearby atom. In the process, the two particles annihilate and produce two gamma-ray photons ().

e- + e+ energy (two photons) 6. Calculate. How much energy is released when an electron (0.511 MeV/c2) collides with a positron (same mass) and the

two annihilate (leave no mass behind)? You may assume they are both essentially at rest.

114

7. Calculate. In Star Trek, the main power source for the starshipEnterprise is a matter-antimatter engine. How much energy would be produced by annihilating 1.0 L of gasoline (0.720 kg) with 1.0 L of anti-gasoline (0.720 kg)? What form of energy is the annihilation energy transformed into? What speed would that accelerate a typical car (1200 kg) to (use ½mv2)?

8. Calculate. The previous result is quite fast! We should confirm this with a more reliable calculation using Einstein’s equation to solve for and then v. Use the result for explain why the result from #7 was reasonably accurate.

9. Calculate. The reverse process can also take place! Energy can be converted into a particle – antiparticle pair.

e- + e+ + → p + p In this case, the extra kinetic energy of the electron-positron pair is converted into the mass of the proton and anti-proton. This is exactly what used to happen at the LEP (Large Electron Positron) collider at CERN in Switzerland. What should the speed be of an electron and positron in the LEP to allow this to happen?

Eo car Eo gasWextEo car Ek car

+ 0 -

+ 0 -

115

SPH4U: A New Kind of Interaction A: Observe and Find a Pattern You will need: 2 ebonite rods, a sheet of acetate and a watch glass. Carefully balance one ebonite rod (rod A) on the watch glass. Keep it in place with a small ball of tape – see the diagram to the right. Now you are ready to investigate how the ebonite rod interacts with other materials. 1. Observe. Vigorously rub the objects as described below. Bring them close and look for an interaction. Sometimes the

watch glass gets stuck and needs a little tap. Record your observations. (Hint: all three are different!) Rod A unrubbed Rod B unrubbed A:

Rod A rubbed with acetate Rod B rubbed with acetate B:

Rod A rubbed with acetate The acetate that rubbed rod A C:

2. Interpret. According to your observations, how do similarly charged objects interact? How do differently charged

objects interact? How to uncharged objects interact? We model ordinary, uncharged matter as a substance consisting of very large, but equal, quantities of positively and negatively charged particles. Careful physics experiments indicate that only the negatively charged particles are able to move in solid materials. Therefore, when an object is rubbed, negatively charged particles (the valence electrons) can either be added to or removed from the material. When a valence electron leaves, the remaining atom (the ionic core) is positively charged. We can represent these two parts (the electron and the ionic core) with charge-pairsrepresented by a positive and negative charge. A charge diagram shows a selection of these many charge pairs in the material and illustrates movement of the negative charges. Note that the pairs are drawn with the “+” and “-”symbols overlapping. This indicates a neutral atom in the material.

Ordinary Matter

3. Reason. A student draws a charge diagram of the object shown above after is was rubbed on the left-hand side and

gained some negative charge. Explain the errors in the diagram and draw a correct one.

4. Explain. Use the charge-pair model to explain the patterns observed in the previous experiment. Draw a charge diagram

to help illustrate the two objects in each type of interaction.Draw a vector for the force each object experiences. The ebonite rod exerts a stronger pull on electrons than the acetate.

Recorder: __________________ Manager: __________________ Speaker: __________________

0 1 2 3 4 5

A: neutral rod

B C

© C. Meyer, 2014

116

B: The Unrubbed Rod 1. Observe. Place a fresh, unrubbed rod (rod A) on the watch glass.Rub one end of another rod (rod B) with acetate. Bring

the rubbed end of rod B near rod A. Describe the interaction you observe. The atoms and molecules in an insulator (ebonite, a plastic, is an insulator) hold their electrons close to the atoms. They do not allow them to freely move. When there is an electric interaction with another object, the valence electrons of the insulator can only shift around the atoms and don’t travel far away from the atomic nuclei (even any added negative charges!). To

show this we draw a small circle around the charge pair and show the shifted charges inside . We say that the material is now polarized (still neutral, but with slightly shifted charges). 2. Explain. Why do the rods interact as you observed?Complete a charge diagram for each. 3. Predict and Test. You are curious to know what will happen if you rub rod B with acetate and bring the acetate near the

unrubbed rod. (If this doesn’t work, try rubbing the acetate on the table.) Prediction

Explain your prediction + Draw charge diagram Observations

C: The Two Spheres Your teacher has set up two Styrofoam spheres hanging sidebyside from the edge of a table. One sphere is covered with aluminum foil. 1. Observe. Note: This experiment work best the first time – once the spheres touch the rod a new

effect takes place. Rub a rod with acetate. Veryslowly bring the rod closer to both spheres. Try to keep the rod parallel to the table edge. Describe the interaction between the spheres and the rod.

The electrons of conductors (the foil, a metal, is a conductor) are not bound close to the atomic nuclei. The conduction electrons in a metal are free to move any distance throughout the material. The conduction electrons do repel one another, so they don’t all bunch up in one location. Due to the mobility of the electrons, a metal will become polarized when it interacts with a charged object. 2. Explain. Why are the strengths of the interactions different?Draw charge diagrams showing these differences. 3. Reason. Explain how the difference in the mobility of the electrons in a metal compared with an insulator produce the

different results you observed.

+ −

spherebefore

Styrofoam sphere after

foil sphere after

117

D: Party Time - Confetti You are throwing a party for a friend and happen to have two kinds of confetti – the traditional paper kind and confetti made from aluminum foil (you though it might be nice and sparkly). Since your friend is late, you decide to try an experiment. You rub a rod with acetate and hold it close above a little pile of each type of confetti. Don’t do it yet! Complete the chart below. The piles of confetti are set up at the front of the room. Predict what you

will observe Explain your prediction Do the experiment and

record your observations

Revise your explanations, if necessary

Pape

r Con

fetti

Alu

min

um

Con

fetti

5. Explain. Explain in three steps what happens to the charges of an aluminum confetti particle during this experiment.

Draw a charge diagram and show the electrostatic forcesfor the rod and confetti particle in each step. The confetti on the table before the rod is charged.

The confetti travels upwards towards the charged rod.

The confetti has touched the rod and is travelling downwards

E: Apply Your Understanding! 1. Predict. What will happen if you place a charged rod near a falling stream of water from a tap? Explain your prediction

using a charge diagram for a droplet of water. Draw a second illustration the path of the entire stream of water. 2. Test. Try it out and describe any changes that might be necessary to your explanation above.

3. Apply. Your clothing tends to cling together after going through the dryer. How might this occur inthe dryer? Is your

answer consistent with what you observed in class? Explain and representyour answer with a diagram.

118

F: Just ‘Scoping Things Out An electroscope consists of a vertical metal rod that sticks out of a glassenclosure; a piece of plastic prevents electric charge from going from themetal rod to the enclosure. A pair of thin metal leaves hang down from thebottom of the rod inside the enclosure. Complete the chart below. Experiment Record your observations. Explain your observations.

1. Bring a rubbed rod close to the electroscope’s top but do not touch. Then remove the rod.

2. Rub a rod with acetate and then rub the top of the electroscope with the rod.

3. Touch the top of the electroscope with your hand.

4. Bring the rubbed rod near the electroscope; while it is being deflected, place your hand on the electroscope then remove your hand; after remove the rod.

1. Explain. What is “grounding” and why does grounding a charged object discharge it?

119Adapted from Van Heuvelen, Physics Union Mathematics, Rutgers University, 2010

SPH4U: The Strength of Electrostatic Interactions In 1785 Charles Coulomb used a torsion balance to measure the force that one charged sphere exerts on another charged sphere to find how the force between two electrically charged objects depends on the magnitudes of the two charged objects and on their separation. Coulomb could not measure the absolute magnitude of the electric charge on the metal spheres. However, he could divide the charge in half by touching a charged metal sphere with a third identical uncharged metal sphere. A: Looking for Patterns 1. Explain the mechanism behind his method of dividing the charge and provide an illustration.

Why did he have to use metal spheres? Would plastic spheres work? 2. Look at the variables in the chart. Which quantity is the dependant variable? Which are the independent variables? 3. The table to the right provides data that resembles what

Coulomb might have collected. Let’s say you want to find the data that demonstrate how the size of the force is affected by the charge q2. Explain how you choose this data from the table. Note that all the quantities are measured in generic “units”.

4. Find patterns in the data and construct three graphs showing how size of the force is affected by the other quantities in

this experiment.

5. Describe how the strength of the electrostatic force depends on your chosen variables.

Experiment Charge q1

Charge q2

Distance r

Force Fq1 on q2

1 1 1 1 1 2 ½ 1 1 1/2 3 ¼ 1 1 ¼ 4 1 ½ 1 ½ 5 1 ¼ 1 ¼ 6 ½ ½ 1 ¼ 7 ¼ ¼ 1 1/16 8 1 1 2 ¼ 9 1 1 3 1/9

10 1 1 4 1/16

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B: Coulomb’s Law Two point-like objects with a net charge exert a force upon one another along a line from the centre of one object to the centre of the other. The magnitude of the electrostatic force that object 1, with electric charge q1, exerts on object 2, with electric charge q2, when they are separated by a center-to-center distance r is given by the expression below. Note that this is also the equal magnitude electric force that object 2, with electric charge q2, exerts on object 1, with electric charge q1:

221

1221 r

qqkFF qonqqonq

where k = 9.0 x 109 Nm2 /C2. We assume that the objects are much smaller than their separation r (i.e. point-like objects). The number of electric charge, q, is measured in units of coulombs (C) where 1 C = 6.24x1018 electrons or protons. 1. The diagrams show two charged objects and their separation. Rank the size of the force that the left object exerts on the

right object from the strongest to the weakest force. Explain how you made the ranking.

A B C

Ranking: Why:

A B C

Ranking: Why:

A B C

Ranking: Why:

A B C

Ranking: Why:

2. For each of the cases in the previous activity, compare the force that the left object exerts on the right object to the force

that the right object exerts on the left object. Explain how you know. 3. In each diagram below small charged particles are fixed on a grid. All particles have a charge which is a multiple of q.

Draw a force diagram particles A, B and C. Rank the magnitudes of the net electrostatic force experienced by particles A, B and C. Provide a brief explanation for your ranking.

+q

-3q +3q C

+q

+9q

B

+3q

+q A

2d

+3q +q

2d

+2q +q

d

+q +q

½ d

+q +q

3d

+q +q

d

+q +q

d

+q -q

d

+q +q

d

-q -q

d

+4q +q

d

+3q +q

d

+2q +2q

121Adapted from Van Heuvelen, Physics Union Mathematics, Rutgers University, 2010

SPH4U: Analyzing Electrostatic Forces A: Gravity vs. Electricity – Fight! 1. Represent and Reason. Imagine two point-like charged objects of mass m1 and m2 that have electric charges q1 and q2,

respectively. Complete the table below that compares their electric and gravitational interactions.

What property of the objects determines

whether they participate in the

interaction?

What is the direction of the force between

the interacting objects?

Write an expression for the magnitude of

the force that one object exerts on the

other.

How does the magnitude of the force depend on

properties of the objects?

How does the magnitude of the

force depend on the distance between the

objects?

Gra

vita

tiona

l It is an attractive

force.

It is directly

proportional to the masses m1 and m2.

Elec

tric

The electric charge determines whether they will interact.

B: Combinations of Interactions 1. Represent and Reason. Two equal-mass stationary spheres are attached to the end of two

strings, as shown at the right. The sphere on the left has an electric charge of +5Cand the sphere on the right has an electric charge of +1 C. Each string makes an angle less than 45o with respect to the vertical. Fill in the tablesbelow.

ID

FD – left sphere FD – right sphere

Which angle is bigger? Explain. Write down the 2nd law for the

right sphere. Rank the FT, Fe, and Fg, from largest to smallest. Explain.

Recorder: __________________ Manager: __________________ Speaker: __________________

0 1 2 3 4 5

+ +

θ1

122

2. Es

D: Ma(comp

3. R

ca

(

(

(

4. A

ath

Equation Jeopshown in the eq

100.9( 9

athematical Repplete algebraic eq

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SPH4U: Picturing Electric Forces One of the reasons why electrostatic forces seem so mysterious is that they have an effect at a distance, without any objects being in contact. We are familiar with one other force that also has this property. One way of explaining how gravity can have an effect on objects far away is with the idea of a field. We say that the earth has a gravitational field that extends throughout space and that the earth is the source of the field.The earth acts on distant objects throughits gravitational field. A: In Search of the Electric Field For this investigation you will need an electroscope, an ebonite rod, an acetate strip, and a piece of aluminum foil. An electroscope can be used as an electric field detector. If there is a fairly strong electric field at the location of the metal sphere atop the electroscope, the metal leaves inside will repel each other. 1. Observe. Charge the rod and move the electroscope in the region of space around the rod. Always be careful with the

glass electroscope. Don’t let the rod and sphere touch! Describe what you observe. 2. Reason. Based on your observations, where is the electric field in that region of space strongest? What happens to its

strength further away? What is the source of this field? The sphere of the electroscope acts as a test particle in the rod’s electric field. A test particle is a small, point particle whose charge is much less than that of the source. As a result, the presence of the test article does not affect the field of the source. 3. Observe. Place the electroscope on a table and bring the charged rod nearby. Place the piece of paper between the rod

and sphere – don’t let them touch!Describe your observations. Next, place the aluminum foil between the rod and sphere – don’t let them touch! Observe. What effect do different materials have on an electric field?

4. Apply. How would you protect sensitive electronics from unwanted electric fields? 5. Predict and Test. What would happen if there is aluminum foil around your cell phone? How many bars do you think

you will get? Try phoning the foiled phone. Describe your observations. B: Picturing the Electric Field 1. Represent and Reason. For each situation pictured below, represent with arrows the gravitational force or theelectric

force that a test particle would experience due to the sources at the points shown. In the study of electricity we will always choose all test particles (test charges) to have a positive charge.

The planet Earth

An object with a large positive charge

An object with a large negative charge

-

● ●

● ● +

● ●

● ● Earth

● ●

● ●

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2. Reason. In the diagrams on the previous page, explain how you decided on the lengths of the vectors you drew. A force field (a field representing forces) is a way of representing the force vectors from an interaction at every point in space. This would require many, many vectors so instead we draw field lines. To do this we use a set of rules for electrostatic (constant) fields: (1) Any field line follows the “path” of the vectors. The vector is always tangent to the field line. (2) Electrostatic field lines start at positively charged objects and end at negatively charged objects. (3) The strength of the field at a point is represented by the density or concentration of thelines near that point. (4) A corollary to this idea is that the number of lines leaving or terminating on a chargedobject is proportional to the

magnitude of its electric charge. 1. Interpret. The diagrams to the right show the field

vectors and field lines for a particle with charge +2q. Use the dashed circles to explain how the field line diagram shows regions of space with different strengths. How does this agree with the field vector representation?

2. Represent. A new particle is presented with a charge of

–q. Explain how the field vector diagram and the field line diagram will be different from the two in the previous question. Draw these diagrams.

3. Apply. A region of space has a strange looking electric field as shown to the right.

Note that we do not know the location of the source of this field. Three particlesare placed in the field. The particles have the same size of charge but different signs. (a) Draw the force vectors experienced by each particle. (b) Rank the size of the electric force each particle experiences. Explain your ranking

in terms of the field line density. 4. Reason. We replace particle C with a similar particle that has twice the charge. How would the size of the force acting

on particle C change? Explain.

Electric Field Vectors Electric Field Lines

Electric Field Vectors Electric Field Lines

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SPH4U: The Field Concept The idea of the field is fundamental to modern physics. An object or a system of objects is the source of the field. An additional object, much smaller than the sources, can be placed in the field without affecting the field. Depending on the properties of the object, it may experience a force from the field due to the sources. A: It Was a Windy Day To help us get used to the ideas behind a field, we will think about an analogy with wind. A very large, industrial strength fan creates a wind pattern as shown in the diagram to the right. Someone holds a small kite at point A. Then the person holds a larger kite at that same point. In both cases, the kite directly faces the fan and therefore catches the wind. 1. Reason.In what sense is the wind stronger on the large kite than it is on the small

kite? 2. Reason. In what sense is the wind equally strong at both kites? What could you

measure about the wind to illustrate this point? As a convenient catchphrase, let’s define the wind field as the strength and direction of the wind itself at a given point (whether or not an object is held there). So, the wind field at point A stays the same whichever kite you put there; but that same wind field produces a different wind force on kites of different sizes.Now you’ll figure out a way to define the wind field more precisely. 3. Reason. The smaller kite has cross-sectional area 0.50 m2. When held at point A, it feels a wind force of 3.0 N. The

larger kite has exactly twice the cross-sectional area (1.0 m2) of the smaller kite. What wind force would you expect the larger kite feel at point A? Explain.

4. Reason. Now a kite of cross-sectional area 2.0 m2 is held at point A. What wind force do you expect it to feel? Why?

Complete the chart showing all your results and describe the pattern. 5. Evaluate. Below are two possible mathematical definitions of a wind field. Which of these, if either, better captures the

pattern you found above? Comment on the validity of each one.

wind field = wind force cross-sectional area wind field = wind force cross-sectional area

6. Reason. Based on the better definition you found above, what are the units of the wind field?

Surface Area (m2) Force (N) 0.50 m2 3.0 N

1.0 m2

2.0 m2

Adapted from Tutorials in Physics Sense-Making,University of Maryland Physics Education Research Group, Spring 2008.

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7. Summarize. Does the wind force depend on the fan, the kite, or both? What about the wind field? Explain. B: The Electric Field We just saw that the wind field is the strength and direction of the wind, independent of whether the wind acts on anything. In general, a field is the strength and direction of something, independent of whether that something acts on an object. Let’s apply these ideas to electric fields. A glass rod is given a positive charge by rubbing it with silk.You hold a bead with a small charge at point A. Then you pull it away and hold a bead with a larger charge at that same point. 1. Reason. When we place a bead with a greater charge at the same position near the rod it

experiences a greater “effect”. Does that “effect” correspond to a larger electric force, a larger electric field, or both? What law explains this greater effect?

2. Reason.There is another sense in which both beads feel the same “electric effect” at point A. Does that sense correspond

to the same electric force, the same electric field, or both?Explain. 3. Reason. As we saw with kites, the property of an object that determines how much it responds to a given wind field is

its cross-sectional area.What property of an object determines how much it responds (i.e., how much force it feels)? In a given electric field?

4. Summarize. Does the electric field experienced by a bead at point A depend on the charge of the rod, the charge of the

bead or both? Explain. C: Defining the Electric Field When held at point B, a bead of charge 2.0 nC (nanocoulombs, n = 10-9) is repelled by the rod with a force of 10 N. The charge on that bead is now doubled, to 4.0 nC, and the bead is again held at point B. 1. Reason. What force do you expect the bead now feels from the rod? Explain. 2. Reason. Now a bead of charge 5.0 nC is held at B. What electric force do you expect it feels from the rod?

A

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3. Reason. So, the 2.0, 4.0, and 5.0 nC beads feel different electric forces at point B. But the electric field at B is the same no matter which bead you hold there, just as the wind field at a given point in front of the fan doesn’t depend on which kite you hold there. Is there some number having to do with electric force and bead charge that’s the same for all three beads at point B and could therefore formalize the concept of electric field?

4. Reason. Let’s explore the meaning of the number and equation you just chose.What are the units of that number?

Explain what that number means, and why it corresponds to the notion of electric field. 5. Reason. You’ll now “translate” these concepts into an equation. Let E denote the electric field created by a rod or other

collection of charges; let q denote the charge of a bead or other particle placed in the field; and let F denote the electric force felt by that particle. Write an equation relating E, q, and F.

** call your teacher to check your result ** 5. Summarize. Now let’s compare three similar examples of fields: wind, gravitational and electric. Summarize the

comparisons in the chart below.

Wind Gravitational Electric Property of an object that

determines how much force it experiences.

Units of field

D: Problem Solving In the diagram below, beads 1 and 2 carry charges 1.0 nC and 2.0 nC, respectively. P is just a point in space, not a charge. The electric force exerted on bead 2 by bead 1 is 12 N. First let’s think about the effect of bead 2 on bead 1. 1. Reason. Find the electric force exerted by bead 2 on bead 1. Yes, you have enough information: use a basic law of

physics! 2. Calculate. Find the electric field due to bead 2 at the location occupied by bead 1. 3. Reason. Is your D#2 answer greater than, less than, or equal to the electric field due to bead 1 at the location occupied

by bead 2? Explain.

P 1 2

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4. Reason. Given that beads 1 and 2 feel different fields, it’s reasonable to expect that they also feel different forces. But they don’t! To reconcile this apparent conflict, explain in intuitive terms how beads 1 and 2 can end up experiencing the same force even though they feel different fields.

5. Reason. The charge on bead 1 is now tripled. How does that affect:

(a) the electric field due to bead 2 at the location occupied by bead 1? Explain.

(b) the electric field due to bead 1 at the location occupied by bead 2? Explain. (c) the force felt by bead 2? Explain.

(d) the force felt by bead 1? Explain. Definition of the Electric Field. An electrostatic field describes the electrical effects throughout space of a charged source object B on another charged particle A. The strength and direction of the electric field depend on only the charges of the source and the position in space. To measure the electric field we need to use a charged particle A. The value of the electrostatic field due to a source B at a point in space (for example, 10N/C) means that if we place particle A at that position, it will experience 10 N of force for each coloumb of its charge, qA.

Gr. 12 version: A

ABeB q

FE

Full Version: A

ABeB q

FE

Use the equation to find the magnitude of the electric field. Use diagrams and your understanding of fields to reason about the directions. Use a sign convention to show the directions, just like you do with forces, when doing the math.

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SPH4U: The Superposition of Fields A: Two Sources and Only One Field, Ja! This is a class demonstration that your teacher will lead. 1. Observe. (as a class) Rub the acetate against paper and hold the negatively charged acetate near the electroscope.

Remove the acetate and hold the paper near the electroscope. Describe your observations. 2. Reason.(as a group) Is there anything different about what’s happening to the charges inside the electroscope when the

acetate or paper is brought near? Explain. 3. Predict. (as a group) Your teacher will hold the freshly rubbed papervery close to the electroscope. Then, starting from

far away, slowly bring the acetate close. What will initially happen to the electroscope leaves as the paper gets closer? 4. Observe and Reason. (as a class) Observe as the paper and then acetate are brought close to the electroscope. What can

we say about the combined effect of the two objects on the electroscope? What is the strength of the electric field at that position in space? Explain.

Electric field vectors are related to electric force vectors (they are the force per unit charge). When there are multiple sources, we can construct a type of force diagram for electric fields which shows the individual electric field vectors at a point in space due to each source. Just like with forces, we can then find the total or net electric field: ...21 EEEnet

In

fact, we can work with field vectors using all the same techniques as force vectors (force diagrams, sign conventions, etc.) 5. Represent. The acetate and paper are positioned such that the electroscope leaves are vertical (not deflected). We

represent the acetate, paper and electroscope as point-like particles (or small spheres). (a) Complete the charge diagram below for each object. (b) Draw a force diagram for a positive test particle in the electroscope. (c) Draw an electric field vector diagram showing the electric field vectors of the two source objects at the position of the electroscope. (d) Write a complete equation for the net force experienced by the test particle. (e) Write a complete expression for the net field vector at the position of the test particle. What are these equations equal to?

Diagram Charge Diagram Force Diagram E-Field

Vector Diagram Net Force Equation

Net Field Equation

** check with your teacher before continuing **

acetate

‘scope

paper paper acetate

‘scope

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B: Let’s See Opposites Attract Imagine that we position two equal but opposite charges along a line. This configuration is called an electric dipole. 1. Represent and Reason. Imagine we place a small test particle

with positive charge at each of the points shown to the right. Draw a force diagram at each point for the electric field due to the individual sources– simply estimate their magnitudes. With a different colour, draw a vector to represent the net field at each point (if you are familiar, use the parallelogram trick).

2. Predict. Review the rules for electric field lines. Use the net field vectors you found to help construct the field lines for the two charges on the grid below.

3. Observe. Use the applet: http://www.falstad.com/vector3de/

Select “dipole” for the field selection and select “Display: Field Lines”. Make any changes to your prediction if necessary.

4. Apply. What are some realistic examples of systems with the

field of a dipole (think of chemistry examples!) 5. Predict. What will happen to the net electric field if the distance between the positive and negative charge becomes very

small (small enough we can’t notice it)? Explain your prediction. 6. Test. Test your prediction using the simulation. Gradually decrease the charge separation and describe what happens to

the total field. What is a realistic example of a system like this? C: Charged Plates Another important electrostatic field example is the field that exists between two parallel metal plates that have opposite electric charges. Use the applet and set the field to “charged plate pair”. Make the sheet size as large as possible and increase the field line density. 1. Observe. Draw the field lines. How do the directions of the field lines near the middle

of the plate compare with one another? 2. Interpret. An idealized field between two oppositely charged parallel plates

is uniform, as shown in the diagram to the right. This is the field we will consider when working with parallel plates. Rank the strength of the field in each indicated region. Explain your ranking.

The strength of the electric field between two oppositely charged parallel plates depends on the voltage applied across the plates and the distance between them. |E| = |V|/d

+ -●

●

●

● ●

●

● ●

+ -

top plate

bottomplate

A

B C

+

-

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SPH4U: Moving Charges in an Electrostatic Field Aside from particles accelerating due to gravity, all other particle motion that we experience can ultimately be explained (usually at a microscopic level) by electric fields. This is especially important in the areas of molecular biology and chemistry where electric fields rule the day and explain everything that happens. A: The Simplest Field The simplest field to explore is a uniform electrostatic field like to one we came across last class. This type of field is a good model for the fields present in old cathode ray tube TVs, in the massive particle accelerator at CERN, in simple wires connected to a battery, or in ion channels through a cell membrane. Below is illustrated a pair of oppositely charged metal plates. 1. Interpret.Use the field lines to determine the sign of the charge on the

metal plates. Explain.

2. Reason. Emmy says, “I think the force on a particle at point 1 is stronger than the force on the same particle at point 3. It’s closer to the top plate.” Marie says, “I think the forces at point 1 and 3 are the same.” Who do you agree with? Explain.

3. Reason. A positively-charged particle is placed at point 1. A negatively charged particle is placed at point 3. Explain

how you find the direction of the electric force experienced by each particle. 4. Represent.A negatively charged particle is at rest at point 2 and is released. Draw a motion diagram for the particle after

its release. Explain how it moves and why. 5. Represent. A positively charged particle is launched downwards from point 1. It doesn’t hit the bottom. Draw a motion

diagram for the particle after its release. Describe the motion of the particle. 6. Represent.An electron is launched horizontally from point 3. Draw a motion diagram for the particle after its release.

Describe the motion of the particle.

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2 3

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7. Calculate. The separation between the two metal plates on the previous page is 7.3 cm. A high voltage power supply is connected to the plates producing a voltage of 1.7x103 V between the two plates. An electron, a proton and a neutron are separately released at point 3 in the electric field. Determine the net force acting on each particle and its acceleration. |qp|=|qe|=1.60x10-19 C, me = 6.67x10-31 kg, mp = mn = 1.67x10-27 kg D: Mathematical Representation Complete equations, describe steps, algebraic work, substitutions with units, final statement

B: Electrostatic Field Summary 1. Describe.Where do electrostatic field lines start and end? Draw an illustration.

2. Describe.Whatparticles do electric fields affect? Do they affect all particles? 3. Explain. You know the charge of a particle and the direction of the electric field. How do you find the direction of the

electric force acting on the particle? Draw an illustration.

4. Explain. When examining the picture of an electric field, how can we find the regions where the field is strong? Where

it is weak? Draw an illustration.

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SPH4U: Magnetostatic Fields Fields can describe the force of any interaction that has an effect throughout space. Another force you have no doubt played with is magnetism. We need to develop some new rules and ideas to describe magnetic effects. For example, with magnetism we won’t use test particles we will use test-compasses! A: Searching for a Field You need a bar magnet and a compass. A compass is a special device whose north end points in the direction of the magnetic field at that position in space. 1. Observe. Check that your compass works – it should jiggle at the slightest touch. Place the compass at each point in the

region around the bar magnet as shown below. Draw an arrow showing the magnetic field vector ( B

) at that position in space.

2. Observe. Use the applet: http://falstad.com/vector3dm/. Set the field selection to “solenoid”. Set the display to “Field

Vectors” and then “Field Lines”. It may be easier to see if you choose “Show Y Slice”. We can’t examine the field inside the bar magnet with a compass, but we can study it using a solenoid. A solenoid is a coil of wire whose field is very similar to that of a bar magnet. Sketch the magnetic field lines around and inside the bar magnet. Add arrows to the field lines to match your field vectors above.

3. Explain. Point out two regions where the field strength is particularly strong and weak. How can you tell?

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B: The Birds and the Bees of Magnetism You need an alligator wire, a compass and a 9V battery. WARNING: Never clip both ends of the wire to the battery – it will overheat! 1. Observe. Clip only one end of the wire to the battery. Move the middle of the wire above the compass. Describe your

observations. Is there any evidence for a magnetic interaction between the compass and the wire? Reminder: The conventional current flows from the positive terminal to the negative terminal of the battery. We can imagine positive charges flowing through the wire in a direction opposite to that of the actual electron flow. We understand that it is only electrons that flow in most electrical circuits, but we are stuck with using the conventional current for historical reasons. 2. Observe. Place the wire over the top of the compass such that it is parallel to the needle.

Briefly touch the second alligator clip to the other battery terminal – DO NOT CLIP IT ON! Hold the wire in place only long enough for the needle to stabilize. Record your observations. Illustrate the direction of the conventional current flow and the compass needle. Is there any evidence for the existence of a new magnetic field?

3. Observe. Now hold the compass just above the wire. Repeat your observations. Magnetic fields ( B

) are created by movingelectric charges. Charges at rest (static charges) do not create magnetic fields. To

illustrate the motion of the charges and the magnetic fields, we need a new set of symbols. For a vector pointing out of a page draw: For a vector pointing into a page draw: 4. Observe and Explain. Use the applet: http://falstad.com/vector3dm/. Set the field selection

to “current line”. Set the display to “Field Vectors”. Select “Show Z Slice”. Sketch a sample of the field vectors around the wire. Is the current flowing into or out of the page? Does this representation agree with your compass observations? Explain.

5. Observe. Now set the display to “field lines”. Sketch the field lines and the current in the

box to the right. Which way should the arrows on the field lines point? Now draw them! The Right Hand Rule for the Magnetic Field of an Electric Current relates the motion of positive charges (conventional current) and the magnetic field they create. When the thumb of your right hand points in the direction of the conventional current, your fingers curl in the direction of the magnetic field lines. 6. Predict Imagine the wire is oriented across our page as shown below. It is difficult to draw

the field lines from this point of view. We can simply show the direction of the field vectors as they pass through the page surface. Explain whether the field lines will point into or out of the page in the region around the wire.

7. Test.Switch back to field vectors and choose “Show X Slice”. You may need to

rotate the image to understand what you are seeing. Explain any differences with your prediction. Draw field vectors illustrating your observationsusing and .

Field Vectors

Field Lines

wire current

Field Lines

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SPH4U: Magnetic Forces on Charges A: Charges at Rest This is a class demonstration. We want to study the interaction between a magnetic field and a charged object that is stationary. An electromagnet will create the magnetic field and rubbing a balloon will create a static electric charge. 1. Observe. For each situation, record your observations in the chart below. Charged balloon, magnet turned on Charged balloon, magnet current reversed Charged balloon, magnet turned off

2. Reason. Is there any evidence from your observations for the existence of a magnetic interaction between a charged

object at rest and a magnetic field? Justify your answer using your observations. B: Charges on the Move This is a class demonstration. We want to study the interaction between a magnetic field and moving charges. An oscilloscope tube and high tension power supply will produce moving charges. A permanent magnet will provide the magnetic field. 1. Observe. The bar magnet is held perpendicular to the beam with the north pole beside

the tube. To a good approximation the magnetic field points straight out from the end of the bar magnet. Record your observations. Complete the diagram showing the velocity of the current, the magnetic field and the force acting on the charges.

2. Observe. The bar magnet is held perpendicular to the beam with the north pole above the tube.

Record your observations and complete the diagram. 3. Observe. The bar magnet is held parallel to the tube with the north pole pointing towards

you. Record your observations and complete the diagram. Why do the field lines look so different in this example?

4. Reason. (as a group) Two conditions must be met in order for magnetic fields to exert a force on a charged object. From

your observations in parts A and B, what are these two conditions? The force that results from the movement of a charged object in a magnetic field is given by: sinBvqFm , where q and v are the charge and speed of the object, B is the strength of the magnetic field the object is moving through and θ is the angle between the magnetic field lines and the velocity. The direction of the force is determined by the Right Hand Rule for the Magnetic Force on Positive Charges: The thumb points in the direction of the velocity of the charged object, the fingers in the direction of the magnetic field lines. The palm of your hand points in the direction of the force on a positive charge. If the object has a negative charge, just reverse you answer from the right-hand rule.

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B

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B v

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5. Reason. Use the equation for the magnetic force to verify your conditions from the previous question. Explain. 6. Predict. Use the right hand rule to predict how the beam will deflect when the bar magnet is held

perpendicular to the beam with the south pole on the left side of the tube. Draw a picture to illustrate your prediction. We will try this out as a class. Move on for now.

C: The Motion of Charges in a Magnetic Field Consider an electron traveling with an initial velocity v moving through a magnetic field that is oriented into the page as shown to the right. 1. Reason. What is the direction of the force acting on the electron? Explain and

draw the force on the diagram.

2. Reason. A short moment later, where might the electron be located? Draw it and

its new velocity vector. At this later moment, what will the direction of the force be? Draw this on the diagram. Explain.

3. Predict. Assume the electron never leaves the magnetic field. What will the complete path of the electron look like?

How will the directions of the velocity and force vectors compare? What type of motion results? Explain. 4. Reason. How much work does the magnetic field do on the electron? How does the electron’s speed and kinetic energy

change? Explain. C: Bubbles, Tiny Bubbles The image to the right shows the paths of particles travelling through a bubble chamber, a chamber of superheated fluid. A charged particle travelling through the fluid causes a bubble to form. This leaves a bubble trail behind the particle. A strong magnetic field passes through the bubble chamber and is oriented out of the page. The trails of five particles are labeled. 1. Apply. Indicate the charge of the particle that

produced each trail. Explain how you can tell. 2. Apply. Rank the momentum of the particles that

produced each trail. Explain. 3. Apply. Why do you think trails C and D spiral inwards?

A

B

CD

E

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B

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SPH4U: Electromagnetic Disturbances We have studied both electro- and magneto-static fields up to this point. Now we will combine these fields and begin to change them so they are no longer static. Something truly remarkable happens. A: The Physics of a Wire Consider a wire connected between the negative and positive terminals of a power supply. A steady current travels through the wire. The diagram to the right illustrates this situation and shows two sets of field lines. 1. Represent. Choose a colour to represent the conventional

current. Draw an arrow that represents the flow of conventional current.

2. Represent and Explain.Choose a colour to represent the electric

field. Explain how you decide which lines are the electric field lines. Label and colour code these lines. Add arrows showing the direction of the field lines.

3. Represent and Explain.Choose a colour to represent the magnetic field. Explain how you decide (as if you haven’t

done question #2) which lines are the magnetic field lines.

B: A Closer Look Now imagine that we zoom in to a small region of space located above the middle of the wire. The diagram below illustrates this along with the electric and magnetic field lines. Note that one field points in or out of the page and is represented with the circles. 1. Represent. Complete the diagram by: (a) labeling and colour

coding the electric and magnetic field lines and (b) showing the direction of all the field lines and current

2. Reason. How do the directions of the electric and magnetic field

lines compare in this region of space?

3. Explain. A positive test charge is placed at rest in this region of space. In this investigation, we will ignore the effects of the magnetic field since they are relatively small. What will happen to the positive charge which is initially is at rest? Explain and draw vectors representing any forces.

Below is a third diagram showing the same region of space as above. But for this example, we have switched the polarity of the power supply, reversing the positive and negative terminals shown in the first diagram.

- +

wire

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4. Represent and Explain. Complete thediagram as you did in question #4. Describe the how the fields have changed due to the reversed terminals.

5. Explain. What will happen to the positive test charge which is

initially at rest?

C: Shaking Things Up Now imagine that we continuously vary the polarity of each terminal of the wire from positive to negative. 6. Reason.Describe the motion of the charges in the wire.

7. Reason.Describe the motion of the positive test charge as the polarity varies.

8. Observe. When the polarity of the wire changes, the fields don’t change everywhere immediately. They begin to change closest to the wire and then spread outwards through space. Use the simulation: http://falstad.com/emwave1. Describe what this changing pattern looks like.

9. Observe. Freeze the simulation.Static electric or magnetic fields drop off in a steady way at positions farther from the

sources. Is this happening in this situation? Explain. An alternating current produces a changing pattern of electric and magnetic fields that spreads outwards from the wire through space. In 1864, James Clerk Maxwell studied these phenomena in detail and mathematically described the wave-like motion of these oscillating fields. From his calculations he determined the outwards speed of the rippling fields to be given by:

Where k is Coulomb’s constant (k = 8.99 x 109 Nm2/C2) and o is a similar constant for magnetism (o = 1.257 x 10-6 Tm/A).

10. Predict. Determine the velocity of this wave including units (1A = 1C/s) (1T = 1 kg/Cs). What are the implications of this result?

Rules for Electromagnetic Waves. What we have discovered is an electromagnetic wave which works according to these ideas: 1) An EM wave is created by the acceleration of electrical charges (our varying current in this case). 2) An EM wave might cause a distant charged particle to acceleration if the properties of the wave match properties of the

particle. 3) When an EM wave accelerates a charged particle, some of the wave’s energy is absorbed which might stop the wave.

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SPH4U: Understanding EM Waves A genuine, real-life EM wave is very tricky to visualize – it has much more detail than a simple water wave. To help us picture what takes place when an EM wave travels through space we will use the line model of an EM wave. A: The Line Model of an Electromagnetic Wave To the rightis a screen capture from the simulation we used in our previous investigation. It represents a two-dimensional slice in space and the field values along that slice. To help us talk about this image we will define the vertical direction as the y-direction and the x- and z-directions as the horizontal plane. The wire is positioned along the y-axis with its middle at the origin. This image represents the field values in the xy-plane. 1. Represent. Imagine we draw a line along the x-axis starting at the surface

of the wire. Remember that the simulation does not show the field strength using the arrow lengths. Sketch a graph of the electric field vs. position along the +x-axis (the white line). Do the same for the magnetic field. Upwards is positive for the E-field and outwards is positive for the B-field.

We can put these two graphs together and illustrate the field vectors to give us the line model of an electromagnetic wave. This is a picture of the electric and magnetic field vectors along one line through space at one moment in time. One extraordinary feature of an EM wave is the fact that the amplitude of the peaks decreases very, very little allowing EM waves to travel extremely far through space- even across the entire universe! 2. Reason. Examine the

line model of an EM wave shown to the right. Rank the magnitude of the electric field at points A, B, C and D in space. Explain.

3. Reason. In what direction is the EM-wave propagating (travelling)? Draw an arrow on the diagram above. 4. Reason. How do the directions of the electric and magnetic fields compare with the direction of propagation?

5. Reason. An electron is located at point C in the diagram above and is at rest. In what direction will it experience a force?

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In this example, we noticed that along the line through space that we chose, the electric field vectors point along one axis (the y-axis in this case). An electromagnetic wave with its electric field vectors pointing along one axis in space is a linearly polarized EM wave. We call the direction the electric field points in the direction of polarization of the EM wave. B: Wave, Meet Conductor. Conductor, Meet Wave. Imagine that the EM wave depicted above in question #2 is a radio wave. It is polarized in the y-direction. As it travels outwards, it passes by a long, thin conducting wire that is oriented along the y-axis. 1. Reason. As the wave propagates (travels) past the wire, would the electric field due

to the radio wave cause the charges in the wire to move? If so, would the charges move in a direction along the wire? Explain.

2. Reason. Imagine a very sensitive lightbulb is connected to two halves of the same wire in the middle. The

bulb and wire are again placed in the path of the previous radio wave and with the wire oriented parallel to the y-axis. Would the bulb glow? Explain.

3. Reason. Would it glow if the wire was oriented parallel to the z-axis? Explain. 4. Reason. Would it glow if the wire was oriented parallel to the x-axis? Explain. An antenna is a conductor designed to capture the energy from an electromagnetic wave. The electric field of the wave causes charges in the antenna to accelerate, transferring electromagnetic energy from the wave into kinetic energy of the moving charges. An electric current is created or induced and this current can be used in an electric circuit to do many things! We say the EM wave is the signal which is picked up by the antenna and used by the circuit. This is how a radio station signal, a Bluetooth signal, a Wi-Fi signal, a TV signal, and a cellphone signal all work! 5. Predict. In order to best detect the oncoming radio wave (that is, to maximize the induced current in the circuit), how

should the antenna be oriented relative to thedirection of polarization of the wave? Explain. 6. Test. Use the signal generator and oscilloscope set up at the front of the room to test your predictions. Describe your

observations and the results.

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Adapted from McDermott and Shaffer, Tutorials in Introductory Physics. Prentice Hall, 2002

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SPH4U: Polarization When a light wave is received by our eyes, we cannot tell which direction the electric field vectors point in – but some creatures can! Certain materials react differently to electric or magnetic field vectors in different directions. A: Through the Polarizing Glass You will need a pair of polarizing filters. Be careful since some of them are glass! 1. Observe. Look at the room lights through one of the polarizing filters. Describe how the filter affects what you see. Does

rotating the filter have any effect? 2. Observe. Hold a second polarizing filter in front of the first, and look at the room lights again. Describe how the filter

affects the light you see. How does rotating one of the filters with respect to the other affect what you see? A light beam is linearly polarizedwhen the electric field vectors throughout the beam point along a single axis. An unpolarized beam is made up of many rays of light, each of which has the electric field pointing in different, random directions. The EM wave that is transmitted from a polarizing filter (or polarizer) has its electric field vectors pointing parallel to the polarizer’s axis of polarization. Apolarizerabsorbs the parts of an EM wave that have electric field vectors perpendicular to the filter’s axis of polarization. The axis of polarization is a property of the material the filter is made from (usually long chains of plastic polymers). The axis is often marked by a line on the filter. A polarization diagram represents the direction of the E-field of the wave and the filter’s axis.

3. Interpret. Describe what is happening according to the polarization

diagram to the right. Compare the brightness of the light at moments 1 and 2.

4. Reason. Do the room lights produce polarized light? Explain how you can tell from your observations. 5. Represent. Draw a polarization diagram for the situation of minimum intensity you found using the room lights and two

filters. Describe the intensity of the light at each moment.

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When two polarizers are oriented with respect to one another such that the light is at a minimum intensity, the polarizers are said to be crossed and all the light has been absorbed.

6. Observe. What are other sources of polarized light from around the classroom or amongst your belongings?

7. Reason. Suppose that you had a polarizer with its direction of polarization marked. How could you use this polarizer to determine the direction of polarization of another unmarked polarizer? Explain your reasoning.

8. Reason. A beam of polarized light is incident on a polarizer, as shown in the side view diagram. The direction of E-field of the light makes an angle with respect to the polarizer’s direction of polarization (see front view diagram). The amplitude of the electric field of the incident light is E1. Write an expression for the amplitude of the transmitted component of the electric field, E2. Represent this in a polarization diagram.

9. Predict. A beam of unpolarized lighttravels through a pair of crossed filters – no light is transmitted from the last filter.

What will happen when a third filter, whose axis of polarization makes a 45o angle with both filters, is placed between two crossed filters? Draw a polarization diagram. Describe the light intensity. Test your prediction – we will also try this as a class.

Adapted from McDermott and Shaffer, Tutorials in Introductory Physics. Prentice Hall, 2002

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SPH4U: Exam Preparation Careful preparation for an exam helps you demonstrate the understanding you have gained through the hard work you have done during an entire course. Effective preparation depends on knowing your own strengths and weaknesses, using careful time management and employing the best studying strategies. A: Know Thyself Let’s begin by thinking about what you currently do to prepare for a typical test. 1. Reflect. (individually) How much time in total do you typically spend perparing for a physics test? When do you do your

preparations? 2. Reflect. (individually) When preparing for a test, what proportion of time do you spend on each of the following:

Summarization Reviewing homework solutions

Highlighting/ underlining Solving problems for practice

Rereading textbook or handbook sections Other:

3. Reason. (as a group) How effective have these strategies been for you? Rank the strategies listed above from most

effective (1) to least effective (6). Record your ranking in front of each description above. Provide a rationale explaing why you think your top ranked and lowest ranked strategies are the best and worst for you.

B: Performance Anxiety There are many occasions in life when we are faced with one shot at accomplishing a particular goal. Most people feel some level of stress or anxiety when performing under pressure. 1. Reflect. Describe a few situations that don’t involve a test or exam in which you or other people you know have faced a

high pressure situation. 2. Reason. High performance athletes and professional musicians are examples of people who routhinely perform under

high pressure situations. How would such a person prepare in the days leading up to a crucial performance?

3. Reason. What strategies do you think a high-pressure performer uses during the performace itself to deal with stress and anxiety?

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4. Reason. Describe some strategies you could realistically make use of for the upcoming exams.

C: Evaluating Study Strategies Research has been done to measure the effectiveness of different strategies in student leaning and studying. Here are some results from:Psychological Science in the Public Interest 14(1) 4–58 Technique Effectiveness Rationale Highlighting

Low

Summarizing

Low

Rereading

Low

Self-explanation (explain aspects of the process to yourself while studying)

Medium

Practice testing (practicing your understanding in a testing situation)

High

Distributed Practice (spreadingpractice over a number of days)

High

1. Reason. In the chart above, provide a rationale for the rating of each technique.

2. Plan. Which techniques would you like to include in your upcoming exam preparations? Explain how you might

incorporate them.

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Sleep's memory role discovered By James Gallagher, 5 June 2014, Health and science reporter, BBC News

The team in China and the US used advanced microscopy to witness new connections between brain cells - synapses - forming during sleep. Their study, published in the journal Science, showed even intense training could not make up for lost sleep. Experts said it was an elegant and significant study, which uncovered the mechanisms of memory. It is well known that sleep plays an important role in memory and learning. But what actually happens inside the brain has been a source of considerable debate. … And by disrupting specific phases of sleep, the research group showed deep or slow-wave sleep was necessary for memory formation. During this stage, the brain was "replaying" the activity from earlier in the day. Prof Wen-Biao Gan, from New York University, told the BBC: "Finding out sleep promotes new connections between neurons is new, nobody knew this before.” We thought sleep helped, but it could have been other causes, and we show it really helps to make connections and that in sleep the brain is not quiet, it is replaying what happened during the day and it seems quite important for making the connections." This is just the latest piece of science to highlight the importance of sleep. A new reason for sleep was

discovered last year when experiments showed the brain used sleep to wash away waste toxins built up during a hard day's thinking. … Commenting on the findings, Dr. Raphaelle Winsky-Sommerer, from the University of Surrey, told the BBC: "This is very impressive, carefully crafted and using a combination of exquisite techniques to identify the underlying mechanisms of memory."They provide the cellular mechanism of how sleep contributes to dealing with experiences during the day. "Basically it tells you sleep promotes new synaptic connections, so preserve your sleep."

1. Reason. Use the scientific evidence presented in this article to help give advice to a stressed out friend who has three

heavy exams next week. Your friend may object to some of your suggestions. Try to anticipate his or her concerns and address them in your advice.

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How to Prepare for an Exam

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SPH4U: Modeling 2-D Waves A: Picturing Two-Dimensional Waves So far when we have described light waves, we have pictured waves travelling along a single line in space. But let’s look at them another way now. We can draw pictures resembling the surface of water. Consider the illustration from the simulator www.falstad.com/ripple/. We can easily sketch this wave by drawing lines of constant phase such as crests or troughs. These represent the wave fronts and in this example, the wave fronts are fairly straight lines and are called planar waves. To understand how a 2-D wave works, we use an idea proposed by Christian Huygens in 1690. The fundamental wave motion is circular – imagine a pebble dropped in water and a circular wave rippling outwards. Huygens proposed that each point along a wave front can be considered to be a point source of circular waves or wavelets. When these circular waves travel outwards their wave fronts join together and form the future wave front of the present wave. Use a penny and draw a dot every penny radius along the present wavefront. Be sure to centre the penny at the two ends as well! Next, trace the outline of the penny when centred on each dot. Form the future wavefront by drawing a new surface tangent to the foreword edges of the wavelet circles. You can smooth this out, understanding that there are many more wavelets that we did not draw in between the ones we did. 1. Reason. What does the radius of the penny represent about the wave?

2. Represent. We will use this

technique to model what happens when a wave passes through the wide opening. Find the next three future wavefronts. Be sure to use points at the two ends of the previous wavefront!

3. Represent. Now try a really narrow opening.

Construct the next three wave fronts. 4. Describe. What happens to the shape of the

original wavefront after it passes through an opening?

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Recorder: __________________ Manager: __________________ Speaker: _________________

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© C. Meyer, 2013

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