About the ResourceWelcome to Perimeter Institute’s Revolutions in Science, a classroom resource based on three serious-but-fun Alice & Bob in Wonderland animations.

Challenge and inspire your students with the wonder and mystery of our universe—we really do live in an “Alice in Wonderland” world where things are not always what they seem to be:

• Gravity is not a force pulling down; it is the ground accelerating up in “curved spacetime.”

• Atoms cannot exist in a commonsense universe; they require a strange “quantum” reality.

• Energy has inertia; our energy to be alive comes from literally eating the mass of the Sun!

Engage your students in the powerful, but surprisingly accessible, creative and critical thinking processes that led to three of the most profound revolutions in science.

The resource focuses not only on basic scientific literacy—what these enduring understanding are—but more importantlyhow they were discovered. Using these discoveries asexemplars of the power of inquiry, students can experience for themselves how new scientific knowledge is created.

About the ANimatioNsThe three 60-second animations serve to hook your students’ interest—to show them that the everyday world is far more fascinating than they may have realized.

Along with the two characters, Alice & Bob, students will discover that the simplest questions can lead to the most profound shifts in our understanding of reality.

Alice is a delightfully precocious little girl, brimming with curiosity. Each episode opens with Alice wondering about something that seems so obvious it sounds silly, such as, “What keeps us stuck to the Earth?”

Bob is Alice’s older brother who feels it is his duty to ‘educate’ his sister. Without thinking, he blurts out the commonsense answer to her ‘foolish’ questions.

Alice gives us reason to question the commonsense answer. Together, our characters use their imaginations and simple reasoning to arrive at amazing insights into the universe.

Your students are sure to enjoy their mind-warping adventures with Alice & Bob in Wonderland!

About the DVD: The accompanying menu-driven DVD contains the plug-and-play Alice & Bob in Wonderland animations, as well as the following files, which can be accessed by closing the menu software and using your computer’s file browser: this Teachers’ Guide in PDF format and the animations in various file formats. View the animations now! The Student Worksheets and Assessments in editable DOC format can be found at www.perimeterinstitute.ca

Curriculum CoNNectioNs 2

What KeepS Us Stuck to the Earth? 3 Introduction 4 Teacher Demonstrations 5-6 Student Worksheets: SW1: Scientific Models: Gravity 7- 8 SW2: Scientific Revolution: General Relativity 9-11 Student Assessments: SA1: Scientific Models: Gravity 12 SA2: Scientific Revolution: General Relativity 13 Answers 14-16

How CaN Atoms Exist? 17 Introduction 18-19 Student Worksheets: SW1: Scientific Models: The Atom 20-21 SW2: Scientific Revolution: Quantum Mechanics 22-23 Student Assessments: SA1: Scientific Models: The Atom 24 SA2: Scientific Revolution: Quantum Mechanics 25 SA3: Applications of Quantum Mechanics 26 Answers 27-28

Where Does ENergy Come From? 29 Introduction 30-31 Student Worksheets: SW1: Scientific Models: Time 32- 34 SW2: Scientific Revolution: Special Relativity 35-36 Student Assessment: SA: Scientific Revolution: Special Relativity 37 Answers 38- 40

CredIts 41

2

Curriculum coNNectioNs

Topic Connection to Resource Module

Nature of Science Science involves both creative and critical thinking, leading to new and sometimes revolutionary ways of understanding nature. Educated guesswork and intuitive leaps can lead the scientific imagination to very strange ideas, but as long as these ideas fit the experimental evidence they must be taken seriously. The ultimate judge of a theory is how well it matches the observations, not how well it matches our commonsense.

Process of Scientific Modeling

We build scientific models to explain complex phenomena. Good models must be logically self-consistent, explain the observations accurately, make testable predictions of new observations, and give new insights into the phenomena.

1, 2, 3

1, 2, 3

Force and Acceleration

Newton s second law of motion dictates that acceleration is the result of a net force. In Newton s model, gravity is a force causing acceleration; in Einstein s model, gravity is not a force so objects in freefall are not accelerating.

1

Weight For Newton, weight is the force of gravity pulling down on you. For Einstein, there is no force of gravity; weight is the magnitude of the force needed to accelerate you up along with the accelerating ground.

1

Gravity Students challenge the underlying assumption of Newton s mysterious “force of gravity,” which has no known cause, and replace it with an alternative explanation for gravity using Einstein s curved spacetime.

1

Frames of Reference All observations and measurements are made relative to a frame of reference. If that frame is moving with constant velocity, there is no experiment that can be done to show that it is moving. If the frame is accelerating, the law of inertia seems to be violated so we invent forces to reconcile our experiences.

1, 3

Bohr-Rutherford Model of the Atom

The Bohr-Rutherford model of the atom is an obsolete scientific model. The idea of electrons orbiting around the nucleus is examined and shown to fail due to simple, classical concepts that are within the students grasp.

2

Quantum Mechanical Model of the Atom

The quantum mechanical model of the atom uses waves to describe the behaviour of particles. Electrons can behave as if they are in many places at the same time, solving the problems encountered by the classical (and Bohr-Rutherford) models.

2

Wave-Particle Duality The electron is a point-like particle that behaves like a wave. This allows the electron to act as if it is in many places, or traveling in many directions, at the same time.

2

Electromagnetic Fields

The electron is charged so it is surrounded by an electric field. Accelerating electrons have changing electric fields so they emit electromagnetic waves.

2

Relative Motion Two observers watching the same event might have very different descriptions of the event if they are moving relative to each other. There is no preferred frame of reference in the universe so all motion is relative.

3

Time Dilation The Newtonian concept of absolute time is wrong. Two observers moving relative to each other will measure the other s time passing at a different rate—moving clocks run slow..

3

Length Contraction Two observers moving relative to each other will measure the other s space to be contracted in the direction of motion—moving objects occupy less space. 3

Energy Energy is not just “the ability to do work.” Closer inspection of energy leads to the surprising result that all forms of energy have inertia—heating a cup of coffee increases its resistance to acceleration.

3

Inertia Inertia is not just “the ability to resist acceleration.” The inertia of even an object at rest represents the presence of energy, as described by E=mc2. 3

3

This module contains two single-period lessons based on the Alice & Bob in Wonderland animation: What Keeps Us Stuck to the Earth? In this episode Alice and Bob ask questions about the nature of gravity and realize that there is a deep connection between gravity and acceleration. Lesson 1 is an introductory level lesson (no prior knowledge of physics is required) that guides students through a critical thinking activity to connect acceleration and gravity. Lesson 2 is a more advanced lesson (prior knowledge of dynamics is an asset) that builds on concepts developed in Lesson 1 to show that the effects of gravity are actually caused by curved spacetime.

LessoN 1: SCIENTIFIC MODELS: GRAVITY

Use D1: Black Box to engage the students in the creative process of building and evaluating models.

Follow with D2: Sagging Rod to explore the force modelof gravity and introduce the acceleration model.

Distribute SW1: Scientific Models: Gravity after D2.This worksheet walks the students through an exercise in critical thinking about gravity and acceleration.

>> Show the Alice & Bob in Wonderland animation: What Keeps Us Stuck to the Earth?

SA1: Scientific Models: Gravity. This worksheet includes additional questions to be done in class or for homework.

LessoN 2: SCIENTIFIC REVOLUTION: GENERAL RELATIVITY

Use D3: Toy and Bungee Cord to highlight the differences between the two models.

>> Show the Alice & Bob in Wonderland animation: What Keeps Us Stuck to the Earth?

Distribute SW2: Scientific Revolution: General Relativity. This worksheet guides the students into a discovery of curved spacetime. In Part B, they will use masking tape and beach balls to model curved spacetime.

SA2: Scientific Revolution: General Relativity. This worksheet includes additional questions to be done in class or for homework.

Introduction 4

Teacher Demonstrations 5-6

Student Worksheets:

SW1: Scientific Models: 7-8 Gravity

SW2: Scientific Revolution: 9-11 General Relativity

Student Assessments:

SA1: Scientific Models: 12 Gravity

SA2: Scientific Revolution: 13 General Relativity

Answers 14-16

4

WHAT KEEPS US STUCK TO THE EARTH?

“It doesn’t matter how beautiful your theory is,

it doesn’t matter how smart you are. If it doesn’t

agree with experiment, it’s wrong.”

– RICHARD FEYNMAN

“…that one body may act upon another, at a

distance through vacuum, without the mediation

of anything else, by and through their action and

force may be conveyed from one to another, is to

me so great an absurdity, that I believe no man

who has in philosophical matters a competent

faculty of thinking, can ever fall into it.”

– ISAAC NEWTON

“I was sitting in a chair at the patent office in Bern,

when all of a sudden a thought occurred to me:

If a person falls freely, he will not feel his own

weight. I was startled. This simple thought made

a deep impression on me. It impelled me toward a

theory of gravitation.”

– ALBERT EINSTEIN (Happiest Thought)

Science is a process of building models to explain observations and then refining those models through careful thought and experimentation. Good models explain existing observations and make testable predictions. This Perimeter Institute classroom resource engages students in this process by exploring models of a common real world phenomenon—gravity. Students will exercise their critical and creative thinking skills to demonstrate why Einstein’s model of gravity is better than Newton’s.

Our everyday experiences of gravity suggest that the Earth exerts an attractive force on nearby objects. Newton successfully extended this force model of gravity to the Moon, Sun and planets. Nevertheless, the force model of gravity deeply troubled Newton because it did not explain the cause of the force. Moreover, in the 1850’s, a more careful look at existing observations suggested that something might be

wrong with Newton’s model—Mercury did not orbit the Sun quite as predicted. Scientists tried various ways to explain this discrepancy within the context of Newton’s model, but all attempts failed. Newton’s model of gravity had reached its limit.

Newton’s force model of gravity also troubled Albert Einstein. In his “happiest thought,” Einstein realized that when you are in freefall you do not feel your own weight, like an astronaut floating weightlessly in deep space. However, when an astronaut’s rocket accelerates, she feels as if there is a force pulling her down toward the floor, like weight. In reality, what the astronaut feels is the floor pushing up on her, accelerating her up. Could gravity be like this? Could it be that there is no force pulling us down, but instead the ground is accelerating up? Yes! Einstein showed how curving spacetime can make it possible for the ground to be forever accelerating up without the Earth expanding faster and faster! Students explore this idea through a simple, concrete activity involving just tape and a ball.

Einstein’s curved spacetime model of gravity makes several testable predictions that distinguish it from Newton’s force model. Einstein’s model predicts that time passes more slowly at the surface of a planet compared to farther away. This effect has been precisely measured and is evident daily in the Global Positioning System (GPS). Einstein’s model also correctly predicts the bending of light as it passes by a massive object, such as a star. Such gravitational lensing has become a powerful tool in astronomy. Einstein’s model also provides a very accurate description of the orbits of all the planets, including Mercury.

Einstein’s model of gravity has passed every experimental test to date. These same tests have conclusively ruled out Newton’s model. The old idea of gravity as a force may feel right but it is wrong. The “force of gravity” is an inference, not an observation. We observe the ground compressing under our feet. We infer that gravity is a force pulling us down. In reality, the ground is accelerating up in curved spacetime, pushing up on us, forcing us to accelerate along with it. Dropped objects don’t accelerate down: it is the ground that accelerates up in curved spacetime. These statements may strike us as odd, but they agree with experimental data. Gravity is not a force. Our everyday experiences of gravity are actually the effects of the ground accelerating up through curved spacetime. Gravity is curved spacetime.

5

Teacher DemoNstratioNsD1 - BLACK BOX: (see building instructions below)

1. Pull the top cords back and forth. Invite students to guess how they are connected inside. Now pull one of the bottom cords. Continue pulling different combinations of cords while drawing students into the mystery.

2. Ask students to draw a picture of what they imagine is inside the box. Encourage creative thinking!

3. Have students share their ideas on the board. Engage the class in a discussion about the various models that are on the board. Verify that the models correctly explain the observations. Highlight the following points:

• The same set of observations can generate different models.

• All models that explain the observations are equally valid.

• Models that fail to explain one or more observations are wrong, or need revision.

4. Ask the students if the models on the board predict any new observations that may help distinguish between them. For example, shake the black box to see if it rattles. Return to the models on the board and re-evaluate them, emphasizing the role of testable predictions in the process of developing robust scientific models.

Note: Never divulge what is inside the Black Box. In science, we only ever have access to indirect observations—we never “see reality” directly!

BUILDING YOUR BLACK BOX

Materials: (all dimensions are approximate)

• 2 pieces of 8 mm nylon rope, each 70 cm long

• 1 harness ring with a 4 cm diameter

• 35 cm long piece of drainage pipe (7.5 cm diameter)

• 2 drainage pipe end caps (7.5 cm diameter)

Tools:

• power drill with 3/8” drill bit

Procedure:

1. Drill the top holes directly across from one another, each 5 cm from the top. Repeat for the bottom holes, each 5 cm from the bottom (see top Figure).

2. Thread one rope through the top holes and the harness ring (see middle Figure).

3. Tie a knot 15 cm from each end of the rope.

4. Thread the other rope through the bottom holes. Again, ensure that the rope passes through the harness ring as indicated (see bottom Figure). Tie a knot 15 cm from each end of the rope.

5. Secure the end caps.

Note: Variations on the design (without a ring for example) will enrich the discussion and work equally well. You may also wish to encourage students to build their own versions of the device with bathroom tissue tubes and string.

6

D2 - Sagging Rod: (a very flexible 2 m long rod with two small masses on each end)

1. Hold the rod horizontally with your hand in the middle so the rod sags. Ask students to explain why the rod is sagging–typically students will say “force of gravity!”

2. Place the rod on a table. Have two students apply horizontal forces on the ends while you hold the middle in place by applying an opposing horizontal force. The class observes the same shape as in step #1. Reinforce the concept that when opposing forces are applied to the rod it will bend.

3. Emphasize the distinction between Observation (when opposing forces are applied to the rod it bends) and Inference (the sagging rod is bent so there must be opposing forces; a “force of gravity” opposes your hand).

4. Have students suggest ways to make the rod bend without using opposing forces. Hold the rod vertically and accelerate it to the side. The ends of the rod will lag

behind the middle because of inertia. Emphasize that your hand is applying a force but there is no opposing “force of gravity.”

5. Distribute SW1: Scientific Models. Show the animation: What Keeps Us Stuck to the Earth? after students have worked in small groups to complete the table and discussion sections of the worksheet.

D3 - Toy, Bungee Cord and Board:

1. Show the animation: What Keeps Us Stuck to the Earth?

2. Demonstrate Newton’s model of gravity by stretching the bungee cord over the toy (see Figure). “According to Newton gravity is a force, like an invisible bungee cord, that pulls objects to the ground.” Pull the toy away from the board and let it ‘snap’ back down. The bungee cord exerts a force on the toy making it accelerate.

3. Demonstrate Einstein’s model of gravity by removing the bungee cord, holding the toy in the air and accelerating the board up to hit it. “According to Einstein, gravity is not a force. The toy does not accelerate down; rather, the ground accelerates up!” Place the toy on the board and accelerate it up. Ask students to imagine that

they are in deep space (no gravity); what would it feel like to stand on an accelerating board?

4. Distribute SW2: Scientific Revolution. Students work in small groups to complete the worksheet.

D4 - Curved Spacetime Exemplar:

1. In SW2, the students will use masking tape and a beach ball to model curved spacetime. Read through the activity and make an exemplar on a large exercise ball, if possible (see Figure).

2. The tape describing Alice’s path through spacetime must lie flat. She is experiencing no “force of gravity” and no acceleration so she must follow a straight path.

3. The tape describing Bob’s path must be crinkled. He is experiencing the ground pushing up on him, accelerating him up, and so he must follow a curved path.

4. Time dilation is demonstrated by comparing a length of tape connecting the tops of the ladders with the length of tape connecting the bottoms (Bob’s path).

Note: The time dilation demonstrated by this beach ball analogy is actually reversed to the real time dilation– analogies have limits.

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SW1: Scientific Models: Gravity Scientists use models to try to explain the observations they make. In this activity you are going to use two different models to explain the same observations of an everyday phenomenon—gravity.

Force Model: You are standing in a room that is on the Earth; the Earth exerts a downward force on objects inside the room. Explain the following phenomena using this downward force. Follow the sagging rod example.

Explain the Sagging Rod

- the Earth pulls down on the rod and your hand pushes up

- the rod bends because your hand is only in the middle

- the rod does not accelerate because the two opposing forces are balanced

Explain Weight (use words and arrows)

Explain Freefall (use words and arrows)

Acceleration Model: You are standing in a room that is inside a rocket; the rocket is accelerating “upwards” in deep space. Explain the following phenomena using this upward acceleration. Follow the sagging rod example.

Explain the Sagging Rod

- the room is accel-erating up; so are you and the rod

- the rod accelerates up because there is now only one force—your hand pushing up

- the rod bends because the ends have mass, which resist acceleration (inertia)

Explain Weight (use words and arrows)

Explain Freefall (use words and arrows)

SUMMARIZE: Force Model Acceleration Model

What is the “big idea” behind each model? How does each explain effects we call “gravity”?

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Discussion:

1. Examine both of your explanations for freefall. (a) What do you actually observe about an object in freefall? (b) What can you infer about the nature of gravity from your observations of freefall?

2. A flexible rod bends when opposing forces act on it. The same rod bends when suspended horizontally from the middle. Does this prove that gravity is a force? Explain.

3. A friend shows you a video on the Internet of a guy who can make objects “float” in the air. You know this is impossible—how might you explain the video?

4. You wake up in a closed room with no windows, with no idea how you got there. Describe an experiment you could do to determine if the room is on the Earth or inside a rocket accelerating in deep space.

>> Watch the animation: What Keeps Us Stuck to the Earth?

Thinking Deeper:

1. Both the force model and the acceleration model make claims that are hard to accept. What are they?

2. Both models of gravity explain everyday observations equally well. However, Newton’s force model fails to correctly describe the orbit of Mercury, so it ultimately fails the test for a valid scientific model. Inspired by the acceleration model, Einstein developed an alternative model of gravity. His curved spacetime model made several successful predictions that have conclusively ruled out Newton’s model. Does this mean we should throw out Newton’s model? Does a model have to be correct in order to be useful?

9

SW2: Scientific Revolution: General Relativity Scientific models must make predictions that match our observations, or they must be revised or replaced. New scientific models can be revolutionary. In this activity you are going to examine two models of gravity: Newton’s classical force model, and Einstein’s revolutionary curved spacetime model.

Part A: Modeling Gravity

Complete this table after watching >>Alice & Bob in Wonderland: What Keeps Us Stuck to the Earth?

Force Model Acceleration Model

Gravity: How does it work?

What’s hard to accept?

Alice steps off the top of a tall ladder

Bob stands at the bottom of the ladder

In the boxes, sketch snapshots of Alice as she falls to the ground and Bob as he stands at the bottom of the ladder, showing their progression in time. [Hint: Alice moves faster and faster as she falls.]

Connect-the-dots of Alice’s position in SPACE(height above the ground) as TIME goes on. Is her path through spacetime straight or curved?

Connect-the-dots of Bob’s position in SPACE(height above the ground) as TIME goes on. Is Bob’s path through spacetime straight or curved?

According to Newton...

Alice’s path through spacetime is ______________ because she is accelerating. She is accelerating

because gravity is a force pulling on her. Bob’s path through spacetime is ______________ because he is

not accelerating —the force of gravity is balanced by the ground pushing up.

There is no “force of gravity” pulling down on Alice so she _________ accelerating. Her path through

spacetime should be ______________ . The ground pushes up on Bob and since there is no opposing

“force of gravity” to balance this force, he should accelerate up and follow a _______________ path

through spacetime.

Discussion:

1. Alice has a video camera in her hands as she falls. If she takes a video of herself as she falls, could she tell that she was accelerating by viewing the video? (Ignore the background.)

2. Alice takes a video of Bob as she falls. Could she tell who was accelerating by viewing the video? (Ignore the background.)

3. Alice closes her eyes as she falls. What does she feel? Can she tell that she is accelerating?

4. Bob closes his eyes. What does he feel? Can he interpret this feeling as accelerating up?

(straight/curved)

(straight/curved)

(straight/curved)

(straight/curved)

According to Einstein... (is/is not)

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Einstein knew that Newton’s model of gravity is wrong. For one thing, it fails to correctly predict the orbit of Mercury; for another, it fails to obey the speed limit of the universe—the speed of light. In his search for a better model, the simple fact that acceleration up mimics force down was too strong of a coincidence to ignore. Einstein needed to find a way to make sense of the ground accelerating up without moving up. How can the ground be accelerating up when the Earth is not expanding? He found the answer in the geometry of spacetime.

Part B: Bending Spacetime

In Part A, we used the fact that accelerating objects trace out curved paths in spacetime and non-accelerating objects trace out straight paths. We also saw that Newton and Einstein would disagree on who is accelerating and who is not. In this part of the activity you will use tape to transfer the spacetime diagram from Part A onto the surface of a large ball to reveal how curving spacetime resolves the problem of who is accelerating.

1. Use a strip of tape to connect two points on your desk with a straight line. Use another strip of tape to make a curved line. Compare the two pieces of tape. Which strip of tape lies flat on the desk and which is crinkled?

2. Build your spacetime diagram on the surface of a large ball. Start with the space and time axes.

• The space axis is a strip of tape that runs vertically along a line of longitude. • The time axis runs horizontally along a circle of latitude (about 15˚ above the equator).

3. Add three identical strips of tape to represent the ladder in three consecutive snapshots. The ladders must follow lines of longitude on the surface, starting about 2 cm above the time axis and ending about 10 cm from the top.

4. Alice’s path is a strip of tape that connects the top of the first ladder with the bottom of the last ladder. Can you make it a straight line? Why would you want to?

5. Bob’s path runs parallel to the time axis along a circle of latitude. It will connect the bottoms of the three ladders. Does the tape lie flat or is it crinkled? What does this indicate?

6. The time elapsed for Bob at the bottom of the ladder is the length of his path (i.e. distance in the time direction). If Alice stayed at the top of the ladder, would her elapsed time be the same? Einstein’s model predicts time dilation: time passes at different rates depending on height about the ground, which has been verified by atomic clocks. Newton’s model makes no such claim. Models cannot be proven right—but they can be proven wrong and time dilation proves that Newton’s model of gravity is wrong!

Curved Spacetime:

When we transfer the spacetime diagram to the ball we find that the tape for Alice’s path can be

______________ , which means the line is ________________ so Alice is _____________________

through curved spacetime. The tape describing Bob’s path is ______________________, which

means the line is _________________ so Bob is ____________________ through curved spacetime.

Drawing the spacetime diagram on a curved surface reverses who is accelerating and who is not—just what Einstein needed to make the acceleration model make sense. The ground can be forever accelerating up without moving up! Gravity is not a force—it is curved spacetime.

(flat/crinkled)

(flat/crinkled)

(straight/curved)

(straight/curved)

(accelerating/not accelerating)

(accelerating/not accelerating)

Evaluating Models:

Newton’s model fails to predict the orbit of Mercury accurately. Einstein’s model does and it also

accurately predicts time dilation and the bending of light. We must conclude that the best model of

gravity is __________________ ______________________ model.(Newton’s/Einstein’s) (force/curved spacetime)

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By curving spacetime, Alice’s path changes from curved to straight—she experiences no “force of gravity” and no acceleration. By curving spacetime, Bob’s path changes from straight to curved—he experiences the ground pushing up on him, continually accelerating him up, but without him moving up. Einstein was able to show that gravity is not a mysterious, invisible force—it is the curvature of spacetime. This curved spacetime model asserts that you feel heavy because the surface of the Earth is forever accelerating up without actually moving up.

Part C: Accelerating Up without Moving Up

Consider the type of motion (accelerating or not) in each of the following scenarios:

In Deep Space Near the Ground

Rocket 1: Floating in deep space, engines off

Rocket 2: Accelerating “up” in deep space, engines on

Rocket 3: In freefall near the ground, engines off

Rocket 4: Hovering near the ground, engines on

1. In Rocket 1, the astronaut knows she is not accelerating; the rod is straight and she is floating. In which other rocket does she make these observations?

2. In Rocket 2, the astronaut knows he is accelerating; the rod is bent and he feels the force of the floor pushing up on him. In which other rocket does he make these observations?

3. The astronaut in Rocket 3 uncovers the window and looks out. She can see the ground and Rocket 4. (a) What was her type of motion before looking out the window? (Accelerating or not accelerating) (b) How would she describe her motion when she looks out the window? (c) Combine your answers from (a) and (b) into a statement.

4. The astronaut in Rocket 4 uncovers the window and looks out. He can see the ground and Rocket 3. (a) What was his type of motion before looking out the window? (Accelerating or not accelerating) (b) How would he describe his motion when he looks out the window? (c) Combine your answers from (a) and (b) into a statement.

We have discovered that astronauts in very different scenarios can experience the same type of motion. This insight is called Einstein’s Equivalence Principle: Freefalling in a uniform gravitational field (Rocket 3) is physically identical to floating in deep space (Rocket 1). Hovering in a uniform gravitational field (Rocket 4) is physically identical to constant acceleration in deep space (Rocket 2). The mass of the Earth curves spacetime so that objects in freefall appear to accelerate down, but there is no force causing this “acceleration”. It is the same kind of “acceleration” you feel when a car accelerates towards you. You are not accelerating—the car is!

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SA1: Scientific Models: Gravity

1. Observation: using your senses to gather information from your environment. Inference: using logic to interpret the information gathered from your environment. Identify the observations and inferences in the following narrative. Bob wakes up and looks out the window. There are drops of water on the window. “It must have rained last night,” he thinks. He goes downstairs and notices that the ladder is leaning against the house, so he goes outside to help his dad with the roof repair work. “Hey Alice, what are you doing up there?” shouts Bob. Alice is so startled that she loses her grip on the ladder. As she falls to the ground, she sees Bob getting closer and closer. “The force of gravity is making me accelerate down at 9.8 m/s2,” yells Alice. Bob reaches out and catches her just before she hits the ground. “Good thing I was accelerating up at 9.8 m/s2 so I could rescue you,” says Bob. Alice gives Bob a quizzical look and then she tells him about how she was washing the windows when he made her fall. OBSERVATIONS INFERENCES

2. True or False? Rewrite any false statements to make them true. (a) There can only be one model that explains a set of observations. (b) We prove a model is right when we observe the predictions it makes. (c) Models that do not make new predictions are wrong. (d) A model is valid if it can explain the observations. (e) Any model that cannot explain the observations is useless and should be discarded. (f) We design experiments to prove that a given model is correct.

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SA2: Scientific Revolution: General Relativity

1. Alice and Bob are arguing over whether gravity is a force or curved spacetime. Bob says, “You honestly believe the ground is accelerating up? That’s weird!” Alice replies, “Mysterious invisible force? Who’s weird now, Bob?” Which side of the argument do you hold? How would you convince someone to agree with you?

2. According to Newton, gravity is an invisible, attractive force that acts between massive objects. If his model of gravity is wrong does that mean his equation for universal gravitation is also wrong?

3. According to Einstein, gravity is the curvature of spacetime. If Einstein’s model of gravity is better, why do we still use Newton’s model? When do we have to use Einstein’s model?

4. How is the “force of gravity” similar to centrifugal force? Explain.

5. Newton and Einstein are looking at a book sitting on a table. How would each of them describe the forces acting on that book and how would they justify their description?

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fere

nce.

3. T

here

are

sev

eral

way

s to

do

this

: eith

er a

forc

e is

app

lied

that

you

are

no

t abl

e to

see

(mag

nets

or w

ires)

, or t

he ro

om is

in fr

eefa

ll w

ith a

ll ob

ject

s fa

lling

at th

e sa

me

rate

as

the

room

. Ast

rona

uts

train

for w

eigh

tless

ness

by

fallin

g in

side

a p

lane

that

is d

ivin

g.

4. T

here

is n

o si

mpl

e ex

perim

ent t

hat y

ou c

ould

do

to d

istin

guis

h be

twee

n th

e tw

o sc

enar

ios.

Thin

king

Dee

per:

1. T

he fo

rce

mod

el c

laim

s th

at th

ere

is a

mys

terio

us, i

nvis

ible

forc

e th

at

reac

hes

out t

hrou

gh s

pace

to in

fluen

ce m

ass

but c

anno

t exp

lain

the

phys

ical

nat

ure

or c

ause

of t

his

forc

e. T

he a

ccel

erat

ion

mod

el c

laim

s th

at

the

grou

nd is

fore

ver a

ccel

erat

ing

up w

ithou

t mov

ing

up.

2. N

ewto

n’s

mod

el is

stil

l ver

y us

eful

. It g

ives

a s

impl

e in

tuiti

ve p

ictu

re o

f gr

avity

that

wor

ks fo

r alm

ost a

ll si

tuat

ions

. A m

odel

doe

s no

t hav

e to

be

cor

rect

in o

rder

to b

e us

eful

—th

ere

are

man

y m

odel

s th

at a

re u

sefu

l in

lim

ited

cont

exts

that

ulti

mat

ely

fail.

We

don’

t nee

d to

use

Ein

stei

n’s

curv

ed s

pace

time

mod

el to

cal

cula

te th

e tra

ject

ory

of a

bas

ebal

l; N

ewto

n’s

mod

el is

ade

quat

e fo

r thi

s ta

sk. E

inst

ein’

s m

odel

is n

eces

sary

onl

y to

un

ders

tand

wha

t is

real

ly h

appe

ning

to th

e ba

seba

ll. U

sing

New

ton’

s m

odel

is

ana

logo

us to

say

ing

that

the

Sun

revo

lves

aro

und

the

Earth

—it

is s

till a

co

nven

ient

way

of t

hink

ing,

eve

n if

it is

gro

ssly

inco

rrect

.

Expl

ain

Wei

ght (

use

wor

ds a

nd a

rrow

s)

- the

Ear

th p

ulls

dow

n on

the

obje

ct

and

your

han

d pu

shes

up

- the

obj

ect d

oes

not a

ccel

erat

e be

caus

e th

e tw

o op

posi

ng fo

rces

are

ba

lanc

ed

Expl

ain

Free

fall (

use

wor

ds a

nd a

rrow

s)

- the

Ear

th p

ulls

dow

n on

the

obje

ct

- the

obj

ect a

ccel

erat

es b

ecau

se

ther

e is

no

oppo

sing

forc

e

Acc

eler

atio

n M

odel

Expl

ain

Wei

ght (

use

wor

ds a

nd a

rrow

s)

- the

room

is a

ccel

erat

ing

up; s

o ar

e yo

u an

d th

e ob

ject

- the

obj

ect a

ccel

erat

es u

p be

caus

e th

ere

is n

ow o

nly

one

forc

e—yo

ur

hand

pus

hing

up

- wei

ght i

s th

e se

nsat

ion

of

push

ing

up o

n an

obj

ect t

o fo

rce

it

to a

ccel

erat

e up

, alo

ng w

ith th

e

acce

lera

ting

room

Expl

ain

Free

fall

(use

wor

ds a

nd a

rrow

s)

- the

room

is a

ccel

erat

ing

up

- an

obje

ct in

free

fall

has

no fo

rces

ac

ting

on it

so

it do

es n

ot a

ccel

erat

e

- the

floo

r con

tinue

s to

acc

eler

ate

up

and

mee

ts th

e ob

ject

- the

obj

ect a

ppea

rs to

acc

eler

ate

dow

n bu

t it i

s ac

tual

ly th

e ro

om (a

nd

you)

acc

eler

atin

g up

SW

1: A

nsw

ers

For

ce M

odel

Sum

mar

ize:

Acce

lera

tion

Mod

el

Our

fram

e of

refe

renc

e is

som

ehow

ac

cele

ratin

g. W

eigh

t and

free

fall

are

effe

cts

of th

is a

ccel

erat

ion.

Forc

e M

odel

The

Earth

som

ehow

exe

rts a

n

attra

ctiv

e fo

rce

on n

earb

y ob

ject

s,

like

a m

yste

rious

invi

sibl

e ha

nd.

15

SW

2: A

nsw

ers

Part

A: M

odel

ing

Gra

vity

Forc

e M

odel

Acc

eler

atio

n M

odel

Gra

vity

: How

doe

s it

wor

k?G

ravi

ty is

a m

yste

rious

in

visi

ble

forc

e th

at

eman

ates

from

obj

ects

th

at h

ave

mas

s.

Our

fram

e of

refe

renc

e is

ac

cele

ratin

g. W

eigh

t and

fre

efal

l are

effe

cts

of th

is

acce

lera

tion.

Wha

t’s h

ard

to

acce

pt?

Gra

vity

is a

mys

terio

us

invi

sibl

e fo

rce

The

grou

nd is

acc

eler

atin

g up

with

out m

ovin

g up

Alic

e st

eps

off t

he

top

of a

tall

ladd

er

Bob

stan

ds a

t th

e bo

ttom

of t

he

ladd

er

Acco

rdin

g to

N

ewto

n...

Alic

e’s

path

thro

ugh

spac

etim

e is

CU

RVED

bec

ause

she

is

acce

lera

ting.

She

is a

ccel

erat

ing

beca

use

grav

ity is

a fo

rce

pullin

g on

her

. Bob

’s p

ath

thro

ugh

spac

etim

e is

STR

AIG

HT

beca

use

he is

not

acc

eler

atin

g—th

e fo

rce

of g

ravi

ty is

ba

lanc

ed b

y th

e gr

ound

pus

hing

up.

Ther

e is

no

“forc

e of

gra

vity

” pul

ling

dow

n on

Alic

e so

she

IS

NO

T ac

cele

ratin

g. H

er p

ath

thro

ugh

spac

etim

e sh

ould

be

STR

AIG

HT.

The

gro

und

push

es u

p on

Bob

and

sin

ce th

ere

is n

o op

posi

ng “f

orce

of g

ravi

ty” t

o ba

lanc

e th

is fo

rce,

he

shou

ld a

ccel

erat

e up

and

follo

w a

CU

RVED

pat

h th

roug

h sp

acet

ime.

Dis

cuss

ion:

1. If

Alic

e ig

nore

s th

e ba

ckgr

ound

she

can

not t

ell t

hat s

he w

as a

ccel

erat

ing.

2. A

lice

will

be a

ble

to te

ll th

at o

ne o

f the

m is

acc

eler

atin

g bu

t she

can

’t te

ll w

hich

one

.

3. A

lice

will

not f

eel a

ccel

erat

ion.

She

cou

ld ju

st a

s w

ell b

e flo

atin

g w

eigh

tless

ly in

spa

ce, w

ith a

bre

eze

blow

ing

over

her

face

. She

can

not t

ell

that

she

is a

ccel

erat

ing

until

she

refe

rs to

som

ethi

ng in

a d

iffer

ent f

ram

e of

re

fere

nce

(e.g

. the

gro

und)

and

eve

n th

en s

he c

an o

nly

tell

that

som

ethi

ng

is a

ccel

erat

ing—

not n

eces

saril

y he

r.

4. B

ob w

ill fe

el a

ccel

erat

ion.

He

coul

d ju

st a

s w

ell b

e in

side

a ro

cket

ac

cele

ratin

g “u

p” in

dee

p sp

ace.

Part

B: B

endi

ng S

pace

time

1. T

he S

TRAI

GH

T TA

PE is

FLA

T an

d th

e C

URV

ED T

APE

is C

RIN

KLED

.

Cur

ved

Spac

etim

e:W

hen

we

trans

fer t

he s

pace

time

diag

ram

to th

e ba

ll w

e fin

d th

at th

e ta

pe fo

r Alic

e’s

path

can

be

FLAT

, w

hich

mea

ns th

e lin

e is

STR

AIG

HT

so A

lice

is

NO

T AC

CEL

ERAT

ING

thro

ugh

curv

ed s

pace

time.

Th

e ta

pe d

escr

ibin

g Bo

b’s

path

is C

RIN

KLED

, w

hich

mea

ns th

e lin

e is

CU

RVED

so

Bob

is

ACC

ELER

ATIN

G th

roug

h cu

rved

spa

cetim

e.

Eval

uatin

g M

odel

s:W

e m

ust c

oncl

ude

that

the

best

mod

el o

f gra

vity

is

EIN

STEI

N’S

CU

RVED

SPA

CET

IME

mod

el.

Part

C: A

ccel

erat

ing

Up

with

out M

ovin

g U

p

1. T

he a

stro

naut

is a

lso

float

ing

and

the

rod

is s

traig

ht in

Roc

ket 3

.

2. T

he a

stro

naut

als

o fe

els

a fo

rce

and

the

rod

is b

ent i

n R

ocke

t 4.

3. (a

) The

ast

rona

ut in

Roc

ket 3

is N

OT

acce

lera

ting.

(b

) She

see

s th

at s

he is

in F

REE

FALL

. (c

) Obj

ects

in F

REE

FALL

are

NO

T ac

cele

ratin

g.

4. (a

) The

ast

rona

ut in

Roc

ket 4

is a

ccel

erat

ing.

(b

) He

sees

that

he

is n

ot m

ovin

g re

lativ

e to

the

grou

nd.

(c) O

bjec

ts N

OT

MO

VIN

G re

lativ

e to

the

grou

nd a

re A

CC

ELER

ATIN

G.

Acco

rdin

g to

Ei

nste

in...

16

SA1:

Ans

wer

s

1.

OBS

ERVA

TIO

NS

I

NFE

REN

CES

2. T

rue

or F

alse

? R

ewrit

e an

y fa

lse

stat

emen

ts to

mak

e th

em tr

ue.

(a) T

here

can

onl

y be

one

mod

el th

at e

xpla

ins

a se

t of o

bser

vatio

ns.

FALS

E: T

here

can

be

seve

ral m

odel

s th

at e

xpla

in a

set

of o

bser

vatio

ns.

(b) W

e pr

ove

a m

odel

is ri

ght w

hen

we

obse

rve

the

pred

ictio

ns it

mak

es.

FALS

E: W

e pr

ove

a m

odel

is w

rong

whe

n w

e do

n’t o

bser

ve th

e pr

edic

tions

it m

akes

.

(c) M

odel

s th

at d

o no

t mak

e ne

w p

redi

ctio

ns a

re w

rong

. FA

LSE:

Goo

d m

odel

s m

ake

new

pre

dict

ions

, but

a m

odel

onl

y ne

eds

to

expl

ain

the

exis

ting

data

to b

e va

lid.

(d) A

mod

el is

val

id if

it c

an e

xpla

in th

e ob

serv

atio

ns. T

RU

E

(e) A

ny m

odel

that

can

not e

xpla

in th

e ob

serv

atio

ns is

use

less

and

sho

uld

be

disc

arde

d. F

ALS

E: M

odel

s th

at d

o no

t exp

lain

all

the

obse

rvat

ions

can

st

ill b

e us

eful

in a

lim

ited

cont

ext.

(f) W

e de

sign

exp

erim

ents

to p

rove

that

a g

iven

mod

el is

cor

rect

. FA

LSE:

We

desi

gn e

xper

imen

ts to

pro

ve th

at a

giv

en m

odel

is w

rong

.

SA2:

Ans

wer

s

1. F

orce

is in

tuiti

vely

obv

ious

. A fa

lling

obje

ct a

ccel

erat

es d

own,

so

ther

e m

ust b

e a

forc

e pu

lling

it do

wn.

The

idea

that

the

grou

nd is

acc

eler

atin

g up

whe

n th

e Ea

rth is

not

exp

andi

ng ju

st s

ound

s ab

surd

!

Acce

lera

tion

is s

impl

er. O

bjec

ts “f

all”

beca

use

they

hav

e in

ertia

. The

fram

e of

re

fere

nce

is a

ccel

erat

ing

so it

look

s lik

e ob

ject

s fa

ll bu

t the

y do

n’t.

They

look

lik

e th

ey a

re a

ccel

erat

ing

in o

ur fr

ame

but i

n sp

acet

ime

they

are

act

ually

not

ac

cele

ratin

g. E

xper

imen

ts h

ave

confi

rmed

the

pred

icte

d cu

rvat

ure

of s

pace

time,

w

hich

con

clus

ivel

y ru

les

out t

he fo

rce

mod

el.

2. N

ewto

n’s

equa

tion

for u

nive

rsal

gra

vita

tion

mak

es re

ason

ably

acc

urat

e pr

edic

tions

for t

he e

ffect

s of

wea

k gr

avity

(e.g

. the

effe

cts

of th

e Su

n on

the

orbi

ts o

f the

pla

nets

), bu

t giv

es g

ross

ly w

rong

pre

dict

ions

for t

he e

ffect

s of

ver

y st

rong

gra

vity

(e.g

. nea

r a b

lack

hol

e). T

he e

quat

ion

is a

lso

wro

ng in

the

sens

e th

at it

refe

rs to

a fo

rce,

and

gra

vity

is n

ot a

forc

e.

3. W

e st

ill us

e N

ewto

n’s

mod

el b

ecau

se it

is in

tuiti

vely

sim

ple

and

the

mat

h is

st

raig

htfo

rwar

d. W

e m

ust u

se E

inst

ein’

s m

odel

whe

n ac

cura

cy is

ver

y im

porta

nt

(e.g

. spa

ce p

robe

s an

d G

PS),

whe

re N

ewto

n’s

mod

el b

reak

s do

wn

com

plet

ely

(e.g

. bla

ck h

oles

and

neu

tron

star

s), o

r whe

n w

e ar

e try

ing

to g

et a

cle

arer

pi

ctur

e fo

r how

the

univ

erse

wor

ks.

4. C

entri

fuga

l for

ce is

a fi

ctiti

ous

forc

e in

voke

d w

hen

obje

cts

in a

non

-iner

tial

fram

e of

refe

renc

e ex

perie

nce

iner

tia. F

or e

xam

ple,

whe

n a

car t

urns

a c

orne

r it

acce

lera

tes

but t

he o

bjec

ts in

the

car w

ant t

o ke

ep g

oing

stra

ight

ahe

ad s

o th

ey

feel

a “f

orce

” pus

hing

them

aga

inst

the

mot

ion

of th

e ca

r. Si

mila

rly, t

he “f

orce

of

grav

ity” i

s a

fictit

ious

forc

e cr

eate

d to

exp

lain

iner

tial b

ehav

iour

in a

non

-iner

tial

fram

e of

refe

renc

e. T

he E

arth

cur

ves

spac

etim

e in

suc

h a

way

that

the

grou

nd

is a

non

-iner

tial f

ram

e of

refe

renc

e. F

allin

g ob

ject

s se

em to

acc

eler

ate

tow

ards

th

e gr

ound

, but

ther

e is

no

forc

e ca

usin

g th

is “a

ccel

erat

ion”

; so

we

inve

nt o

ne—

the

“forc

e of

gra

vity.

” Bot

h fo

rces

des

crib

e re

al e

ffect

s ca

used

by

acce

lera

tion;

ne

ither

one

des

crib

es a

n ac

tual

forc

e.

5. N

ewto

n w

ould

say

that

the

book

is “a

t res

t” an

d th

eref

ore

not a

ccel

erat

ing

so

the

forc

es a

ctin

g on

it a

re b

alan

ced.

The

com

pres

sion

of t

he ta

ble

give

s cl

ear

evid

ence

that

the

tabl

e is

pus

hing

up

on th

e bo

ok s

o a

dow

nwar

d fo

rce

of

grav

ity is

nee

ded

to b

alan

ce th

e fo

rces

.

Eins

tein

wou

ld a

gree

that

the

book

is “a

t res

t”, b

ut “a

t res

t” re

lativ

e to

wha

t?

The

grou

nd, w

hich

is a

ccel

erat

ing

up in

cur

ved

spac

etim

e. T

he o

nly

forc

e ac

ting

on th

e bo

ok is

the

tabl

e pu

shin

g up

and

sin

ce th

ere

is n

o “fo

rce

of g

ravi

ty”

oppo

sing

the

forc

e of

the

tabl

e, th

e bo

ok m

ust a

ccel

erat

e up

, al

ong

with

the

acce

lera

ting

grou

nd. U

ltim

atel

y, Ei

nste

in w

ould

just

ify h

is

desc

riptio

n by

app

eal t

o ex

perim

ents

—tim

e di

latio

n ha

s be

en o

bser

ved

us

ing

atom

ic c

lock

s.

- dro

ps o

f wat

er o

n w

indo

w

- not

ices

the

ladd

er

- dis

tanc

e be

twee

n Al

ice

and

B

ob d

ecre

ases

at 9

.8 m

/s2

- “It

mus

t hav

e ra

ined

”

- dad

is fi

xing

the

roof

- the

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e of

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pul

ls h

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own

- Bob

is a

ccel

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ing

up

17

This module contains two single-period lessons based on the Alice & Bob in Wonderland animation: How Can Atoms Exist? In this episode Alice and Bob ask questions about the structure of the atom and discover that the commonly accepted planetary model of the atom (including the Bohr-Rutherford model) cannot possibly exist. Lesson 1 is an introductory level lesson (no prior knowledge of physics is required) that explores why the planetary model fails. Lesson 2 is a more advanced lesson (prior knowledge of waves is an asset) that extends the concepts developed in Lesson 1 to build the quantum mechanical model of the atom—a model that explains how atoms can exist. An additional student activity sheet (SA3) is included that could be combined with either lesson to address applications and implications of scientific discoveries.

LessoN 1: SCIENTIFIC MODELS: THE ATOM

>> Show the Alice & Bob animation: How Can Atoms Exist?

Distribute SW1: Scientific Models: The Atom. This worksheet walks students through a critical examination of atomic models using existing knowledge and a computer simulation to reveal the problems with classical models of the atom.

SA1: Scientific Models: The Atom. This worksheet includes additional questions to be done in class or for homework.

LessoN 2: SCIENTIFIC REVOLUTION: QUANTUM MECHANICS

>> Show the Alice & Bob animation: How Can Atoms Exist?

Distribute SW2: Scientific Revolution: Quantum Mechanics. This worksheet engages the students in the creative process of building a quantum model of the atom. Students will use a computer simulation to assist in visualizing the atom.

SA2: Scientific Revolution: Quantum Mechanics. This worksheet includes additional questions to be done in class or for homework.

Introduction 18-19

Student Worksheets:

SW1: Scientific Models: 20-21 The Atom

SW2: Scientific Revolution: 22-23 Quantum Mechanics

Student Assessments:

SA1: Scientific Models: 24 The Atom

SA2: Scientific Revolution: 25 Quantum Mechanics

SA3: Applications of 26 Quantum Mechanics

Answers 27-28

18

How CaN Atoms Exist?Science is a process of building models to explain observations and then refining those models through careful thought and experimentation. This Perimeter Institute classroom resource engages students in this process as they explore models of the atom. Atoms are the building blocks of matter. They are central to our existence; and yet, there is no “commonsense” way to understand how they can exist. The best commonsense atom we can imagine—the one with electrons orbiting the nucleus, like planets orbiting the Sun—would almost instantly self-destruct. Students will exercise their critical and creative thinking skills as they examine how various commonsense models fail and how the very non-commonsensical quantum nature of our universe makes atoms possible.

By the early 1900s, experiments had revealed that atoms consist of particles much smaller than the atom itself: one tiny, positively charged nucleus comprising almost all of the atom’s mass, plus a number of even tinier, negatively charged electrons. The challenge was to construct a working model of the atom based on these particles and the forces between them.

Electrons are attracted to the nucleus (since opposite charges attract) and repelled from each other (since like charges repel). Any configuration of the atom in which the electrons don’t move is not stable; the attractive force always wins and the electrons collapse into the nucleus. One way to prevent this collapse would be to add “struts” that hold the electrons in place, but we have never seen evidence of any kind of support structure when we strip electrons off an atom. And besides, what type of matter would the struts be made

from? Another possibility would be to invent a new force that acts inside the atom, but such a force has never been observed. In science, we exhaust all existing possibilities before introducing a new type of matter or force.

If the atom cannot exist with static electrons, then the only remaining possibility is a dynamic model where the electrons are moving. In order for a moving electron to stay near the nucleus, its trajectory must bend. The net attractive force towards the nucleus—which defeated the static model—is exactly the sort of force needed to bend an electron’s trajectory into an orbit around the nucleus. But even in the simplest case of a circular orbit, where the electron’s speed is not changing, only its direction is continually changing—the electron is accelerating. This is a problem. When a charged object accelerates (changing its speed or direction), it emits energy in the form of electromagnetic waves. For instance, this is exactly how a cell phone works: electrons in the antenna are accelerated, emitting radio waves. In the atom, the accelerating electrons would emit electromagnetic waves in the form of light. This light would carry energy away from the atom, causing the electron to drop to lower energy orbits, quickly spiraling into the nucleus.

So electrons can’t stand still (the static model fails); nor can they move (the dynamic model fails). Both models would result in all the atoms in your body collapsing in a blast of light energy on par with an atomic bomb. There is no way to escape the catastrophic collapse of any commonsense atom. This raises the question: if the electrons in an atom can’t stand still, and can’t move, what could they possibly be doing?

As a first step towards a working model, imagine spreading an orbiting electron into a rotating ring. A perfectly smooth rotating ring is moving but you cannot see any motion—it appears to be static; this is what physicists call a stationary state. A charged rotating ring is stationary so it does not emit electromagnetic waves and would be a simple solution to the energy loss problem. However, such a spreading out of a particle is fraught with severe problems of its own. Each part of the ring would be repelled from all the other parts (since like charges repel), and there would be very strong electrostatic forces tending to make the ring fly apart. We would have to invent a new kind of matter or force to hold it together. Also, whenever we “look” at an electron we always

“If, in some cataclysm, all scientific knowledge

were to be destroyed, and only one sentence

passed on to the next generation of creatures,

what statement would contain the most

information in the fewest words?

I believe it is the atomic hypothesis... that all things

are made of atoms...”

– RICHARD FEYNMAN

19

“see” a point-like particle, with the full mass and full charge of one electron. We never see evidence of a spread out electron in the shape of a ring, or any other shape.

Nature’s solution to the unstable atom problem is very strange. An atomic electron does something very much in the spirit of spreading itself out into a rotating ring (avoiding the energy loss problem), without literally spreading out its matter (avoiding the other severe problems mentioned above).

How can an electron spread out, and not spread out? In an atom, an orbiting electron can be thought of as a particle, like a very tiny baseball, but unlike a baseball, one that doesn’t move along a definite trajectory. It exists in a profoundly weird state in which, at any instant of time, it is not definitely at any location in its orbit. Instead, it is only potentially at each location in its orbit (all at the same time), with an equal potential of being found at any particular location if we were to “look” at the atom (e.g. shine light on it). The very act of light hitting an electron somehow forces the electron to “take a stand”—to assume a definite location. (How this happens is still a mystery today.) This potential, or indefinite, location is described by a fuzzy donut-shaped wave that circulates around the nucleus. It’s not a physical wave, like a sound wave or a water wave; nor is it the electron’s matter physically spread out; instead, it’s a mathematical wave that describes the probability of finding the electron (as a whole point-particle) here or there if we were to “look.”

In short, the electron is a particle that behaves like a wave. This weird blending of “particle” and “wave” properties into a single entity is called quantum mechanics. At the foundations of everything we currently know about matter and forces is the discovery of the quantum nature of our universe. This breakthrough was a 20th century equivalent to the

Copernican revolution, with equally vast and far-reaching consequences that go well beyond the atom. Quantum ideas have allowed us to not only understand how atoms can exist, and how they work; they also underlie a huge array of technologies from cell phones and computers, to laser surgery and the Internet, representing millions of jobs and trillions of dollars of the world’s economy.

The quantum nature of the atom is non-commonsensical. An orbiting electron behaves like a wave, effectively allowing it to be in many places and moving in different directions at the same time! If you wiggle both ends of a Slinky simultaneously, you will create two waves moving along the Slinky in both directions at the same time, resulting in a standing wave. In exactly the same way, we can have two quantum waves circulating in opposite directions around the nucleus. The resulting quantum standing wave describes a single electron behaving as if it is orbiting both clockwise and counter-clockwise at the same time! The mathematics of these waves is well understood. What is not well understood, and still the subject of much debate, is what this mathematics implies about the ultimate nature of reality. The quantum model results in a stable atom and has been experimentally verified to unprecedented precision—it’s decidedly strange, but it works.

“I think I can safely say that nobody understands

quantum mechanics.”

– RICHARD FEYNMAN

20

SW1: Scientific Models: The Atom By the early 1900s, experiments had revealed that atoms consist of particles much smaller than the atom itself: one tiny, positively charged nucleus comprising almost all of the atom’s mass, plus a number of even tinier, negatively charged electrons, such that the total electric charge is zero. In this activity you will build and evaluate possible configurations of these particles to try to produce a stable model of the atom.

Part A: Static Model

The Law of Static Electricity states that OPPOSITE charges ATTRACT and LIKE charges REPEL.

1. Hydrogen is the simplest atom. It has one negatively charged electron and a positively charged nucleus. What would happen if you put the electron near the nucleus and “let go”?

2. How can Hydrogen exist as a stable atom if its electron and nucleus are attracted to each other? Can you think of a fix for this problem?

Part B: The Planetary Model

If electrons in the atom cannot be standing still, then they must be moving. Maybe the atom looks like a tiny solar system, with electrons orbiting around the nucleus, like planets around the Sun. As you consider this model, recall that objects that are moving will continue moving on a straight path unless pushed or pulled to the side.

1. What has to happen to a moving electron to change its direction of motion?

2. How might the positively charged nucleus of an atom bend the path of a moving electron?

3. A circular path, or orbit, is the simplest trajectory that an electron could follow. What would happen to the electron’s orbit if we gradually removed energy from the atom?

21

Part C: The Failure of the Planetary Model

Any charged object is surrounded by an electric field. It is this field of the nucleus that exerts an attractive force on an electron inside the atom. The electron, too, is surrounded by an electric field. Let’s use the PhET simulation (http://phet.colorado.edu/en/simulation/radio-waves) to investigate what happens to that field when the electron accelerates (wiggles around).

1. Begin with the following settings: Manual, Full Field, Electric Field, Static Field. What happens to the electric field when you wiggle the electron in the transmitting antenna?

2. Change the settings to: Manual, Full Field, Electric Field, Radiated Field. What happens when you wiggle the electron in the transmitting antenna?

3. Change the settings to: Oscillate, Full Field, Electric Field, Radiated Field. Watch the electron in the receiving antenna. Where does it get the energy to move?

4. An electron orbiting around the nucleus is accelerating just like the electron you wiggled in the antenna. (Imagine looking at the atom from the side. As the electron orbits, it will appear to move up and down.) What would be emitted by the electron as it orbits around the nucleus?

5. Whenever a charged object accelerates (changes its speed or direction of motion), it emits electromagnetic (EM) waves. It takes energy to create these waves, and the waves carry this energy away. Why would this be a problem for the Planetary Model of the atom?

Summary:

1. Electrons can’t stand still because:

2. Electrons can’t move because:

There is no way to escape the catastrophic failure of any commonsense atom. This raises the question: if the electrons in an atom can’t stand still, and can’t move, what could they possibly be doing? The answer lies in Quantum Mechanics—a completely new set of laws that describe how nature behaves at a deeper level.

“How wonderful that we have met a paradox. Now we have some hope of making progress.”– NIELS BOHR

22

SW2: Scientific Revolution: Quantum Mechanics Any commonsense model of the atom is destined to fail. In static models, the atom collapses due to the electrostatic force of attraction the nucleus exerts on the electrons. Dynamic models, like the planetary model, also fail because the atom loses energy as the accelerating electrons emit EM waves, again collapsing the atom. We need a model in which the electron is somehow dynamic (orbiting) but at the same time static (not emitting EM waves)—something physicists call a stationary model. For example, a perfectly smooth spinning top is dynamic (rotating), but appears to be static—you can’t tell that it’s spinning because nothing is changing; it always looks the same.

Part A: The Rotating Ring

The electron cannot orbit around the nucleus as a point-like particle. What if we spread the mass and charge of the electron out into a rotating ring?

1. A rotating ring of charge behaves like a current-carrying wire. Would the rotating ring emit EM waves? Why or why not?

2. Consider the electrostatic forces acting inside the ring. Would such a structure be stable? Why or why not? Would we be able to observe it?

Part B: Standing Waves

The rotating ring idea is on the right track, but we have never observed such rings. We always “see” electrons as point-like particles. In preparation for Part C we will need to review some facts about waves: (1) A wave can be in many places at the same time, and (2) Two waves can exist simultaneously in the same place.

1. Stretch a coiled spring (e.g. a Slinky) between two people, on a smooth, horizontal surface (hard floor or table). Wiggle one end of the spring at a constant rate. Where is the wave? What is the direction of the wave?

2. Wiggle both ends of the spring at the same rate. This creates two waves travelling in opposite directions along the spring, existing simultaneously in the same place. Adjust the rate until you get a stable pattern. Notice that the combined wave is not travelling in either direction. It is a wave—it oscillates side to side—but it is not travelling. This is called a standing wave. What happens to the standing wave as you gradually increase the frequency of vibration? Can you create standing waves at higher frequencies of vibration?

23

Part C: The Quantum Model

In the quantum model of the atom, the electron is a point-like particle whose behaviour is described by a wave. If the wave is moving, the electron is moving. Wherever the wave exists, the electron can potentially exist. The weird thing is that the electron does not exist at any definite location until its location is measured. Left undisturbed, the electron behaves as if it is spread out like a wave, and stationary states similar to the rotating ring become possible. Use this simulation (http://www.falstad.com/qmatom/) to visualize these waves. Note that these waves are mathematical—the electron’s mass and charge are not physically spread out.

1. Start the simulation. In the top-right drop down menu select “Complex Combos (n=1-4)”. Click on “Clear” then move your mouse over the little circles in the bottom-left panel, noting the yellow text that appears just above the panel. Click on the “n=2, l=1, m= –1” circle, which is the top circle in the second column. Finally, rotate the view by clicking on the z-axis in the top right corner of the main panel and dragging it down until the z disappears at the origin and the y-axis points straight up. This is a “top down” view of a single electron “orbiting” the nucleus of a Hydrogen atom. (The nucleus is at the centre, but not shown.) What do you see?

2. The colours represent the “phase” of a donut-shaped wave circulating around the nucleus, showing that the wave “crests” and “troughs” are moving. Observe that the moving electron is behaving as if it is in two places at once—actually everywhere at once, wherever the wave is non-zero! Select the “View” drop down menu from the top menu bar and deselect “Phase as Color.” You will now see a probability pattern: the probability, at any instant of time, of finding the electron at various locations around the nucleus. In what way is the electron static? In what way is it dynamic? Do you think the electron is emitting EM waves? Draw comparisons with the rotating ring in Part A.

3. Reselect “Phase as Color,” click on “Clear,” and then choose the “n=2, l=1, m=+1” circle. Note the direction of rotation of this wave. Now click on the “n=2, l=1, m=–1” circle. You have just combined two waves circulating in opposite directions around the nucleus to produce a standing wave. This standing wave describes an electron behaving as if it is moving both clockwise and counterclockwise at the same time! Is the electron “moving”? Click on the x-y-z coordinate system and rotate it to view this standing wave from different angles. Deselect “Phase as Color” to reveal the corresponding probability pattern. In what way is the electron static? In what way is it dynamic? Do you think the electron is emitting EM waves? Why or why not?

By describing the behaviour of a particle using a wave, anything a wave can do a particle can do. A wave can be in many places at once, or be moving in different directions at once—so can a particle! This leads to very non-commonsensical behaviour of electrons inside atoms, and yet these are the lengths scientists have gone to in order to construct a working model of the atom—one that allows us to understand how atoms can exist in our universe.

24

SA1: Scientific Models: The Atom

1. Why does the Hydrogen atom collapse if the electron isn’t moving?

2. Lithium has 3 electrons and a nucleus with a +3 charge. Show that there is no way to put electrons near the nucleus in a stable, static arrangement.

3. Explain how having the electrons move improves the model.

4. The PhET simulation shows a radio station transmitting EM waves. The energy it takes to create these waves is carried off by the waves. Describe some other examples of EM waves and identify the sources of energy.

5. What is the major problem with the planetary model of the atom? Why do we need new “quantum” rules?

Thinking Deeper:

1. If the planetary model doesn’t work, why is it included in almost every introductory Chemistry textbook?

2. What problem does the Law of Electrostatics have for the nucleus of the atom? Suggest a possible solution.

3. All matter is made out of atoms, but there is no way to build a commonsense model of the atom. Summarize the problems and identify properties that a new model must have.

25

SA2: Scientific Revolution: Quantum Mechanics

1. Explain how the rotating ring solves the dilemma of orbiting electrons emitting EM waves.

2. Why does the rotating ring idea fail?

3. A traveling wave is a wave pattern that moves. How does describing the “orbiting” electron by a traveling wave circulating around the nucleus solve the problem of the electron emitting EM waves?

4. A standing wave is composed of two oppositely-directed traveling waves. How is the behaviour of the “orbiting” electron described by a standing wave similar to its behaviour in the static model of the atom? How is it different?

Thinking Deeper:

1. Both standing (and traveling) waves can only exist around the nucleus when an integer number of wavelengths fit around the “orbit”. How can this property be used to explain discrete energy levels in an atom?

2. Start the simulation from SW2 (http://www.falstad.com/qmatom/) and use similar settings (and deselect “Phase as Color”). The state produced by selecting “n=2, l=1, m=–1” and “n=2, l=1, m=+1” at the same time is an example of an excited state of the atom. The ground state of the atom (the state of minimum energy) is given by the “n=1, l=0, m=0” circle (top circle in the first column). Both of these are stationary states—they do not emit EM waves. Click on the “n=1, l=0, m=0” circle to put the electron in both states at once—an excited state and the ground state. Increase the Simulation Speed using the slider in the right panel. Do you think the electron is emitting EM waves? Draw comparisons with the antenna simulation in Part C of SW1. What is the quantum atom in the process of doing?

3. Quantum mechanics is often referred to as weird or strange. What is so strange about it?

26

SA3: Applications of Quantum Mechanics Quantum Mechanics is one of the most successful scientific models ever created. Not only has it passed every experimental test to date, but it has become the basis for a huge number of applications, resulting in trillions of dollars of economic activity every year.

1. The quantum mechanical model of the atom says that light is emitted when electrons go from a higher energy state to a lower energy state. A light emitting diode (LED) is a device that uses this property to produce light very efficiently. LEDs do not get hot, do not burn out and do not contain any harmful materials. (a) Where do you find LEDs being used? (b) For a typical home, about 15% of its electricity bill is for lighting. How much money would you be willing to invest in new lighting technology in order to reduce your energy consumption? (c) Research the LED bulb technology that is currently available for residential use. How much would it cost to convert your house over to LED bulbs? How many years would it take for this investment to pay off? (d) What are the factors that you would consider when choosing which technology is the best for you?

2. The quantum mechanical model of the atom says that electrons can only occupy certain energy levels and that the atom will absorb or emit light as the electron changes energy levels. In 1917 Albert Einstein used the laws of quantum mechanics to predict that excited atoms could be stimulated with light to emit their extra energy as more of the same kind of light, thereby amplifying the light. Forty-three years later the first functioning laser was made. (a) Lasers produce very intense, coherent, monochromatic light. List all the applications of lasers that you know of and describe how the properties of laser light are well suited for that application. (b) The scientist who coined the term laser (Light Amplification by Stimulated Emission of Radiation) spent 27 years fighting with the patent office. What would he gain by winning the patent for this technology?

3. The quantum mechanical model of the atom says that electrons are particles that behave like waves. Waves can reflect and produce standing waves. This wave behaviour of electrons is essential for the functioning of transistors, which are the basis for all electronics. (a) Consider your bedroom. List all the devices that contain electronic components. (b) Research the electronics industry. How much money was generated last year by the production of transistors alone? How much money was generated by the production of devices that use transistors?(c) Look up Moore’s Law on the Internet. What does Moore’s Law say and why is it important to the electronics industry?

4. The quantum mechanical model of the atom says that electrons can behave as if they are in more than one place or state of motion at the same time. This strange behaviour of electrons is being explored to design a new type of computer called a quantum computer. Quantum computers will be able to do certain complicated tasks extremely quickly and will allow scientists to make very sophisticated models of quantum systems. As scientists gain the ability to model quantum systems, they will be able to design new and more powerful quantum technologies. Think back over the last century and reflect on how discoveries in basic science and their applications have worked together to produce the world we live in. Where do you think these new applications of quantum mechanics will take us in this next century? Use historical examples to support your insights.

27

SW

1: A

nsw

ers

Part

A: S

tatic

Mod

el

1. T

he e

lect

ron

wou

ld b

e at

tract

ed to

the

nucl

eus

and

acce

lera

te to

war

ds it

, em

ittin

g a

flash

of l

ight

.

2. T

he H

ydro

gen

atom

can

not b

e st

able

if th

e el

ectro

n is

sta

tic. T

he o

nly

othe

r op

tion

is to

mak

e it

dyna

mic

—w

e ne

ed to

mak

e th

e el

ectro

n m

ove.

Part

B: T

he P

lane

tary

Mod

el

1. W

e m

ust e

xert

a si

dew

ays

forc

e on

the

mov

ing

elec

tron

to c

hang

e its

di

rect

ion

of m

otio

n.

2. T

he n

ucle

us w

ill pu

ll si

dew

ays

on th

e m

ovin

g el

ectro

n, b

endi

ng it

s pa

th.

3. T

he e

lect

ron

wou

ld s

pira

l int

o th

e nu

cleu

s as

the

ener

gy is

gra

dual

ly re

mov

ed

from

the

atom

. Not

e fo

r tea

cher

s: T

he e

lect

ron

wou

ld a

ctua

lly s

peed

up

as it

spi

rals

in, i

ncre

asin

g its

kin

etic

ene

rgy;

but

the

elec

trost

atic

pot

entia

l en

ergy

of t

he e

lect

ron-

nucl

eus

syst

em w

ould

dec

reas

e by

a g

reat

er a

mou

nt,

resu

lting

in a

net

dec

reas

e in

the

atom

’s to

tal e

nerg

y.

Part

C: T

he F

ailu

re o

f the

Pla

neta

ry M

odel

Not

e fo

r tea

cher

s: A

n el

ectro

n at

rest

is s

urro

unde

d by

a s

tatic

, rad

ial e

lect

ric fi

eld

patte

rn. T

his

field

sto

res

ener

gy in

the

spac

e ar

ound

the

elec

tron

(ele

ctro

stat

ic e

nerg

y).

Whe

n th

e el

ectro

n m

oves

with

a c

onst

ant s

peed

and

dire

ctio

n, th

is fi

eld

patte

rn (a

nd

ener

gy) m

oves

with

the

elec

tron,

like

flie

s bu

zzin

g ar

ound

a m

ovin

g ga

rbag

e tru

ck.

But w

hen

the

elec

tron

acce

lera

tes

(cha

nges

its

spee

d or

dire

ctio

n of

mot

ion)

, som

e of

this

ene

rgy

is “s

hake

n of

f” (li

ke fl

ies

shak

en o

ff an

acc

eler

atin

g ga

rbag

e tru

ck) i

n th

e fo

rm o

f ele

ctro

mag

netic

wav

es. A

ccor

ding

to M

axw

ell’s

equ

atio

ns, a

cha

ngin

g el

ectri

c fie

ld c

reat

es a

mag

netic

fiel

d, a

nd v

ice-

vers

a, s

ettin

g up

a c

hain

reac

tion

that

is a

n el

ectro

mag

netic

wav

e. (T

he s

imul

atio

n sh

ows

only

the

elec

tric

part

of th

e el

ectro

mag

netic

wav

e.) I

n qu

estio

n #1

bel

ow, s

tude

nts

see

just

the

stat

ic p

art o

f the

fie

ld p

atte

rn th

at m

oves

alo

ng w

ith th

e el

ectro

n. In

que

stio

n #2

, stu

dent

s se

e th

e ra

diat

ed p

art o

f the

ele

ctric

fiel

d—th

e “fl

ies

that

are

sha

ken

off”.

(Not

e th

at th

e en

ergy

in

the

spac

e ar

ound

the

elec

tron

is im

med

iate

ly re

plac

ed w

ith e

nerg

y fro

m th

e “h

and”

th

at is

wig

glin

g th

e el

ectro

n, i.

e., i

t tak

es m

ore

effo

rt to

wig

gle

a ch

arge

d pa

rticl

e th

an

a ne

utra

l par

ticle

of e

qual

mas

s!) W

hen

the

acce

lera

tion

is a

sim

ple

up-a

nd-d

own

osci

llatio

n, th

e EM

wav

es fo

rm a

sim

ple

patte

rn th

at ra

diat

es o

utw

ards

from

the

elec

tron,

and

car

ry w

ith th

em th

e en

ergy

requ

ired

to m

ake

othe

r ele

ctro

ns m

ove.

Thi

s is

wha

t stu

dent

s se

e in

que

stio

n #3

.

1. T

he e

lect

ric fi

eld

chan

ges

as th

e el

ectri

c fie

ld p

atte

rn a

roun

d th

e el

ectro

n m

oves

with

the

elec

tron.

2. T

he e

lect

ric fi

eld

wig

gles

, cre

atin

g a

wav

e pa

ttern

in th

e fie

ld th

at m

oves

aw

ay fr

om th

e el

ectro

n.

3. T

he e

lect

ron

in th

e re

ceiv

ing

ante

nna

gets

its

ener

gy fr

om th

e w

ave

emitt

ed

by th

e tra

nsm

itter

.

4. T

he e

lect

ron

wou

ld e

mit

an E

M w

ave

as it

orb

its a

roun

d th

e nu

cleu

s be

caus

e th

e ac

cele

ratin

g el

ectro

n w

ould

cre

ate

a ch

angi

ng e

lect

ric fi

eld

in th

e re

fere

nce

fram

e of

the

atom

.

5. T

he E

M w

ave

wou

ld re

mov

e en

ergy

from

the

atom

cau

sing

the

elec

tron

to

spira

l int

o th

e nu

cleu

s; th

e at

om w

ould

col

laps

e in

a fl

ash

of li

ght.

SW

2:

Ans

wer

s

Part

A: T

he R

otat

ing

Rin

g

1. A

rota

ting

ring

will

not e

mit

EM w

aves

bec

ause

the

elec

tric

(and

mag

netic

) fie

lds

surro

undi

ng th

e rin

g ar

e no

t cha

ngin

g. It

is c

hang

ing

elec

tric

or

mag

netic

fiel

ds th

at p

rodu

ce E

M w

aves

.

2. T

he c

harg

e in

side

the

ring

wou

ld re

pel i

tsel

f, an

d th

e rin

g w

ould

tend

to fl

y ap

art.

Cla

ssic

ally,

at l

east

, we

wou

ld b

e ab

le to

obs

erve

suc

h a

stru

ctur

e by

us

ing

a m

icro

scop

e w

ith li

ght o

f suf

ficie

ntly

sho

rt w

avel

engt

h.

Part

B: S

tand

ing

Wav

es

1. T

he w

ave

is e

very

whe

re in

the

Slin

ky a

t the

sam

e tim

e. W

hile

eac

h pa

rt of

the

Slin

ky m

oves

sid

e-to

-sid

e on

ly, th

e w

ave

patte

rn tr

avel

s in

the

perp

endi

cula

r dire

ctio

n, fr

om o

ne e

nd o

f the

Slin

ky to

the

othe

r.

2. A

s yo

u gr

adua

lly in

crea

se th

e fre

quen

cy o

f vib

ratio

n, th

e st

andi

ng w

ave

will

disa

ppea

r and

the

Slin

ky w

ill ap

pear

‘cha

otic

,’ w

ith ra

ndom

vib

ratio

ns.

Even

tual

ly y

ou w

ill re

ach

a fre

quen

cy w

hich

pro

duce

s an

othe

r sta

ble

stan

ding

w

ave

patte

rn; t

his

patte

rn w

ill ha

ve o

ne m

ore

node

.

Part

C: T

he Q

uant

um M

odel

1. Y

ou s

ee a

col

ourfu

l fuz

zy ri

ng th

at s

low

ly ro

tate

s in

a c

lock

wis

e di

rect

ion.

The

ro

tatio

n sh

ows

the

mot

ion

of th

e w

ave

“cre

sts”

and

“tro

ughs

” as

the

wav

e ci

rcul

ates

aro

und

the

nucl

eus.

2. T

he e

lect

ron

is s

tatic

in th

at th

e pr

obab

ility

patte

rn (t

he “a

mpl

itude

” of t

he

wav

e) d

oes

not c

hang

e at

all.

The

ele

ctro

n is

dyn

amic

in th

at th

e “c

rest

s” a

nd

“trou

ghs”

(the

“pha

se” o

f the

wav

e) is

circ

ulat

ing.

The

pro

babi

lity

patte

rn te

lls

us th

at th

e “p

oten

tial l

ocat

ion”

of t

he e

lect

ron

is s

prea

d ou

t int

o a

perfe

ctly

sm

ooth

ring

. The

circ

ulat

ing

phas

e te

lls u

s th

at th

is ri

ng is

rota

ting.

So

the

elec

tron

is b

ehav

ing

exac

tly li

ke a

cla

ssic

al, s

tatio

nary

cha

rged

rota

ting

ring—

it w

ill no

t em

it EM

wav

es.

3. T

he e

lect

ron

is b

ehav

ing

as if

it is

mov

ing

in tw

o op

posi

te d

irect

ions

at o

nce.

In

this

sen

se it

is n

ot m

ovin

g (th

ere

is n

o an

gula

r mom

entu

m),

and

is li

ke th

e cl

assi

cal s

tatic

mod

el o

f the

ato

m. T

he d

iffer

ence

from

the

clas

sica

l sta

tic

mod

el is

that

the

elec

tron

does

not

get

pul

led

into

the

nucl

eus

beca

use

it is

ac

tual

ly m

ovin

g! A

s in

#2

abov

e, th

e el

ectro

n is

sta

tic in

that

the

prob

abilit

y pa

ttern

doe

s no

t cha

nge

at a

ll. It

is d

ynam

ic in

that

the

phas

e is

circ

ulat

ing,

al

beit

in tw

o op

posi

te d

irect

ions

at o

nce!

The

ele

ctro

n is

beh

avin

g lik

e tw

o cl

assi

cal,

stat

ic “b

lobs

” of c

harg

e—it

will

not e

mit

EM w

aves

.

28

SA1:

Ans

wer

s1.

The

nuc

leus

attr

acts

the

elec

tron,

pul

ling

it in

to th

e nu

cleu

s. (T

he n

ucle

us is

m

uch

heav

ier,

and

so h

ardl

y m

oves

.)

2. In

any

mul

ti-el

ectro

n at

om, e

ach

elec

tron

will

be re

pelle

d fro

m th

e ot

her e

lect

rons

, and

will

try to

mov

e as

far a

way

fro

m th

e ot

hers

as

poss

ible

, in

a sy

mm

etric

way

(see

Fig

ure

for L

ithiu

m).

But i

n al

l cas

es, e

ach

elec

tron

expe

rienc

es

a st

rong

er fo

rce

of a

ttrac

tion

tow

ards

the

nucl

eus

than

th

e ne

t for

ce o

f rep

ulsi

on fr

om th

e ot

her e

lect

rons

. The

se

unba

lanc

ed fo

rces

cau

se th

e at

om to

col

laps

e.

3. T

he p

revi

ous

prob

lem

poi

nts

out t

hat t

here

is a

net

forc

e pu

lling

elec

trons

to

war

ds th

e nu

cleu

s. In

stea

d of

allo

win

g th

is fo

rce

to s

impl

y pu

ll st

atic

ele

ctro

ns

into

the

nucl

eus,

we

use

this

forc

e to

ben

d th

e pa

th o

f the

mov

ing

elec

trons

, ca

usin

g th

em to

orb

it th

e nu

cleu

s.

4. L

ight

is e

mitt

ed w

hen

an e

lect

ron

in a

n at

om “d

rops

” to

a lo

wer e

nerg

y le

vel;

the

ener

gy c

omes

from

the

excit

ed a

tom

. Cel

l pho

nes

emit

EM w

aves

just

like

a ra

dio

stat

ion;

the

ener

gy c

omes

from

a b

atte

ry. X

-rays

are

em

itted

whe

n el

ectro

ns a

re

slowe

d do

wn b

y a

collis

ion;

the

ener

gy c

omes

from

the

mov

ing

elec

tron.

5. A

n or

bitin

g el

ectro

n is

con

tinua

lly a

ccel

erat

ing

(tow

ard

the

nucl

eus)

due

to it

s co

ntin

ually

cha

ngin

g di

rect

ion

of m

otio

n. T

his

caus

es th

e at

om to

radi

ate

ener

gy

in th

e fo

rm o

f EM

wav

es, a

nd th

e el

ectro

n to

spi

ral i

nto

the

nucl

eus.

New

ton’

s la

ws,

toge

ther

with

the

law

s of

ele

ctro

mag

netis

m, p

redi

ct th

e co

llaps

e of

the

atom

, so

atom

s ca

nnot

exi

st in

a c

lass

ical

uni

vers

e. Q

uant

um id

eas

are

need

ed

to e

xpla

in h

ow a

tom

s ca

n ex

ist.

Thin

king

Dee

per

1. T

he p

lane

tary

mod

el (i

nclu

ding

the

Bohr

-Rut

herfo

rd m

odel

) give

s a

simpl

e, in

tuitiv

e pi

ctur

e fo

r the

ato

m. I

t is

a go

od s

tarti

ng p

oint

for u

nder

stan

ding

sim

ple

chem

ical

reac

tions

. Mod

els

can

be u

sefu

l in a

limite

d co

ntex

t, ev

en if

they

are

wro

ng.

2. T

he p

roto

ns in

the

nucl

eus

elec

trost

atic

ally

repe

l one

ano

ther

ver

y st

rong

ly.

The

idea

of t

he s

trong

nuc

lear

forc

e, w

hich

hol

ds th

e nu

cleu

s to

geth

er, c

ould

be

intro

duce

d to

stu

dent

s. H

ere

natu

re d

oes

use

a ne

w fo

rce

to s

olve

a

stab

ility

prob

lem

!

3. T

he s

tatic

mod

el c

olla

pses

due

to e

lect

rost

atic

forc

es. T

he p

lane

tary

mod

el

colla

pses

due

to E

M w

aves

dra

inin

g en

ergy

from

the

atom

. We

need

a m

odel

in

whi

ch e

lect

rons

som

ehow

“orb

it” w

ithou

t em

ittin

g EM

wav

es.

SA2: A

nsw

ers

1. If

an

orbi

ting

poin

t-lik

e el

ectro

n is

spr

ead

out i

nto

a ro

tatin

g rin

g, n

othi

ng w

ould

be

“wav

ing”

bac

k an

d fo

rth, o

r sid

e to

sid

e, a

nd s

o it

wou

ld n

ot e

mit

EM w

aves

.

It w

ould

cre

ate

stat

ic e

lect

ric a

nd m

agne

tic fi

elds

, but

EM

wav

es a

re p

rodu

ced

only

whe

n th

ese

field

s ch

ange

.

2. D

iffer

ent p

arts

of a

spr

ead

out e

lect

ron

wou

ld re

pel e

ach

othe

r, te

ndin

g to

mak

e th

e rin

g fly

apa

rt. W

e w

ould

hav

e to

inve

nt a

new

type

of m

atte

r or f

orce

to h

old

the

ring

toge

ther

. Als

o, w

hene

ver w

e “lo

ok” a

t an

elec

tron,

we

alw

ays

see

a po

int-l

ike

parti

cle.

If a

n el

ectro

n to

ok th

e fo

rm o

f a ri

ng, i

t wou

ld h

ave

to tu

rn in

to

a po

int-l

ike

parti

cle

the

inst

ant w

e “lo

ok” a

t it.

This

wou

ld b

e ab

surd

.

3. T

he p

oint

-like

ele

ctro

n be

have

s as

if it

is in

man

y pl

aces

at o

nce

(whe

reve

r th

e w

ave

is n

on-z

ero)

, and

so

it is

effe

ctiv

ely

spre

ad o

ut e

xact

ly li

ke a

rota

ting

ring.

The

wav

e is

mov

ing

(the

elec

tron

has

angu

lar m

omen

tum

) but

the

corre

spon

ding

pro

babi

lity

patte

rn is

not

cha

ngin

g—a

stat

e ca

lled

a st

atio

nary

st

ate.

Ele

ctro

ns in

suc

h st

atio

nary

sta

tes

do n

ot e

mit

EM w

aves

.

4. It

is th

e sa

me

in th

at a

sta

ndin

g w

ave

does

not

mov

e, li

ke a

n el

ectro

n in

the

clas

sica

l sta

tic m

odel

. It i

s di

ffere

nt in

that

the

elec

tron

does

not

get

pul

led

stra

ight

into

the

nucl

eus.

Thi

s is

bec

ause

the

elec

tron

is a

ctua

lly “m

ovin

g”

(alb

eit i

n tw

o di

rect

ions

at t

he s

ame

time!

), an

d so

the

net f

orce

tow

ards

the

nucl

eus

just

ben

ds th

e pa

th o

f the

ele

ctro

n in

to tw

o si

mul

tane

ous,

cou

nter

-ro

tatin

g “o

rbits

”!

Thin

king

Dee

per

1. E

ach

ener

gy le

vel c

orre

spon

ds to

a d

iffer

ent “

harm

onic

,” lik

e th

e ha

rmon

ics

on a

vio

lin s

tring

. For

the

first

ene

rgy

leve

l, on

e w

avel

engt

h fit

s ar

ound

the

“orb

it”. F

or th

e se

cond

ene

rgy

leve

l, tw

o w

avel

engt

hs fi

t aro

und

the

“orb

it”,

and

so o

n. T

he e

lect

ron

is n

ever

foun

d be

twee

n th

ese

disc

rete

ene

rgy

leve

ls

beca

use

you

don’

t get

sta

ble

stan

ding

(or t

rave

ling)

wav

es th

ere.

2. T

he s

imul

atio

n sh

ows

a pr

obab

ility

patte

rn th

at is

cha

ngin

g—on

e th

at is

sl

oshi

ng b

ack

and

forth

, exa

ctly

like

an

elec

tron

mov

ing

up a

nd d

own

in

an a

nten

na w

ire. A

n el

ectro

n in

this

non

-sta

tiona

ry s

tate

is e

mitt

ing

(or

abso

rbin

g) E

M w

aves

, i.e

., a

phot

on. I

n th

e ca

se o

f em

issi

on, t

he a

tom

is

in th

e pr

oces

s of

“dro

ppin

g” fr

om th

e ex

cite

d st

ate

to th

e gr

ound

sta

te; a

nd

the

reve

rse

in th

e ca

se o

f abs

orpt

ion.

Not

e to

teac

hers

: Ele

ctro

ns d

o no

t m

yste

rious

ly “j

ump”

bet

wee

n at

omic

ene

rgy

leve

ls! T

here

is a

ver

y se

nsib

le

phys

ical

pro

cess

invo

lved

.

3. T

he w

eird

ness

of q

uant

um m

echa

nics

may

be

stat

ed a

s th

e w

ave-

parti

cle

dual

ity: t

he id

ea th

at a

ll qu

antu

m p

artic

les

(e.g

. ele

ctro

ns a

nd p

hoto

ns) e

xhib

it bo

th w

ave

and

parti

cle

prop

ertie

s. A

n el

ectro

n ca

n ex

hibi

t the

wav

e pr

oper

ties

of b

eing

in tw

o lo

catio

ns a

t onc

e, o

r mov

ing

in tw

o di

rect

ions

at o

nce,

whi

ch

is n

atur

al fo

r wav

es, b

ut n

ot fo

r cla

ssic

al p

artic

les.

Onc

e w

e ac

cept

this

w

eird

wav

elik

e be

havi

our o

f par

ticle

s (a

nd v

ice-

vers

a), o

ther

qui

ntes

sent

ially

qu

antu

m a

spec

ts o

f nat

ure

follo

w n

atur

ally.

For

exa

mpl

e, q

uant

izat

ion

of

atom

ic e

nerg

y le

vels

is n

ot w

eird

—it

is a

sim

ple

cons

eque

nce

of e

lect

rons

be

havi

ng li

ke w

aves

.

29

This module contains two single-period lessons based on the Alice & Bob in Wonderland animation: Where Does Energy Come From? In this episode Alice and Bob ask questions about energy and discover a deep connection between mass and energy. Lesson 1 (introductory level; some prior knowledge of physics is an asset) introduces the concept of relativity, and shows how it led to Einstein’s model of relative time and a universal speed limit. Lesson 2 is a more advanced lesson (prior knowledge is expected) that starts with relative time to show how energy has inertia.

LessoN 1: SCIENTIFIC MODELS: TIME

>> Show the Alice & Bob animation: Where Does Energy Come From?

Distribute SW1: Scientific Models: Time. This worksheet presents the students with several thought experiments that will help them develop Einstein’s Special Theory of Relativity.

LessoN 2: SCIENTIFIC REVOLUTION: SPECIAL RELATIVITY

>> Show the Alice & Bob animation: Where Does Energy Come From?

Distribute SW2: Scientific Revolution: Special Relativity. This worksheet presents the students with several thought experiments that will help guide them to develop E=mc2.

SA: Scientific Revolution: Special Relativity is a worksheet that includes additional questions to be done in class or assigned for homework.

Introduction 30-31

Student Worksheets:

SW1: Scientific Models: 32-34 Time

SW2: Scientific Revolution: 35-36 Special Relativity

Student Assessment:

SA: Scientific Revolution: 37 Special Relativity

Answers 38-40

30

Where Does ENergy Come From?

Science is a process of building models to explain observations and then refining those models through careful thought and experimentation. Good models explain existing observations, make testable predictions, and give deeper insights into the phenomena. This Perimeter Institute classroom resource engages students in this process by exploring models of two common real world phenomena—time and energy, with an emphasis on the role of thought experiments in science. Students will exercise critical and creative thinking to discover how Albert Einstein’s intuitive belief that an observer in a closed room cannot tell whether the room is moving leads to a radical new understanding of time and energy, known as the Special Theory of Relativity.

We have all experienced relativity. When you are inside a closed room, such as an airplane with the window shades drawn, you can’t tell that the room is in motion. You can’t feel inertial motion, (i.e. motion in a straight line at a constant speed with no rotation). Everything that happens inside the room (e.g. drinking coffee or juggling balls) happens the same way it does when the room is at rest, no matter how fast the room is moving. From this we learn that “at rest” and “in motion” are relative concepts—they make sense only when compared to objects outside the room, such as the ground moving beneath an airplane. There is no such thing as absolute rest.

But is relativity universal? Is it really true that you can’t detect the inertial motion of a closed room by any experiment done inside the room? Ever since Galileo first suggested the concept of relativity in 1632, it has been accepted as true for mechanical experiments (such as drinking coffee, or juggling balls). But in the early 1800s, strong experimental evidence emerged to show that light behaves like a wave, and this presented a problem for relativity. The problem was that, like sound waves, light waves presumably could not travel in empty space. So space must be filled with a light-wave medium they called the “ether”: an immobile substance whose vibrations constitute light, but through which matter could move freely. The wave nature of light created the possibility of detecting motion relative to the medium. Ether might represent a state of absolute rest.

Students will explore the challenge that waves pose for relativity using a thought experiment: Alice is inside a spaceship floating in deep space, at rest in the hypothetical ether. She sends a pulse of light (a single wave front) upwards from the floor, which moves at speed c relative to the ether. She measures the time it takes to reflect off the ceiling and return to the floor. Alice then repeats exactly the same experiment in her “closed room”, except now it is drifting through the ether (causing an “ether wind” to blow through the spaceship). She still sees the pulse of light travel vertically up and down, but Bob—floating at rest in the ether, sees the light pulse move diagonally up and down. It is an established fact that waves of any kind move at a fixed speed relative to the medium. Once created, a wave propagates on its own, independent of any motion of the source. Even though Alice’s light source is in motion, the wave front it creates will move with speed c relative to the ether. And since it travels a greater distance in the second experiment, Bob will measure a greater return time.

Did you know? Einstein showed that the very nature of

time and space prevents us from detecting motion relative

to the ether. We cannot know if it exists. It might and it

might not.

“More careful reflection teaches us, however, that

the special theory of relativity does not compel us to

deny ether.” – ALBERT EINSTEIN

Did you know? Einstein’s “Speed of Light Principle” only

asserts that the speed of light is independent of the motion

of the source of light, which is obvious for the wave-in-

ether model of light. It does not suggest that the speed of

light is independent of the motion of the observer of light.

This absurd-sounding statement is true, however, and

Einstein showed how it is an obvious consequence of his

new model of time and space.

“My solution was really for the very concept of time, that is,

that time is not absolutely defined...” – ALBERT EINSTEIN

31

The crucial question is, “What elapsed time will Alice measure?” The obvious answer is, “The same as Bob!” But, since this time is different from Alice’s “at rest” time, we would have to admit that it is possible to use an experiment with light inside a closed room to detect the inertial motion of that room. Universal relativity would not hold. For universal relativity to hold, Alice must measure the same time whether she is at rest or moving, but then Isaac Newton’s model of absolute time would be wrong. (Absolute time says when one second elapses for you, one second elapses for everyone in the universe, regardless of their location or state of motion; in this case, Alice must measure the same time as Bob.) Students see that if we adopt a wave-in-ether model of light (as most physicists did until after 1905), then absolute time and universal relativity are incompatible. One is wrong, and must be jettisoned from our thinking.

For many years scientists, including Einstein, struggled with the tension between relativity—which seemed so simple and natural that it ought to be universal—and the nature of light. The definitive breakthrough came in 1905 when Einstein realized that the problem had nothing to do with light, but rather the nature of space and time. In particular, he realized that absolute time was just an assumption that had never really been tested beyond everyday experience. He immediately jettisoned Newton’s model of absolute time and worked out the logical consequences of universal relativity. Students will work through the following rational and intuitive progression:

Time Dilation. According to the second part of our thought experiment, Alice’s moving clock runs slowly relative to Bob’s. Note that, according to universal relativity, from Alice’s perspective she is “at rest”, and it is Bob’s clock that is moving, and running slowly (not faster!). Time dilation is reciprocal.

Length Contraction. During the time measured on his clock, Bob sees Alice cover a certain horizontal distance as measured in his frame of reference. But for Alice, it takes less time. For both to be moving at the same relative speed, Alice must measure this distance to be less in her frame of reference. She must see Bob’s space—and everything in it, contracted. And again, by universal relativity, this effect must be reciprocal: Bob must see Alice’s space—and everything in it, equally contracted.

Universal Speed Limit. Now suppose Alice throws a ball forward inside her moving spaceship so that it is covering a distance of one metre every second according to her ruler

and clock. Bob will see the ball covering less than one metre (length contraction) in more than one second (time dilation). The additional speed of 1 m/s that Alice has given to the ball will be less than 1 m/s for Bob. As Alice’s rocket approaches the speed of light, this effect becomes more pronounced, so that it is never possible for Bob to see the ball reach (or exceed) the speed of light. By universal relativity, no matter how fast Alice is moving relative to Bob, she can consider herself to be “at rest”, and can throw the ball forward as close to the speed of light as she wants, relative to herself.

Unified Model of Energy: E=mc2. Using only time dilation, students engage in a thought experiment to conclude that a fast moving ball is more difficult to deflect sideways than one at rest. Its inertia relative to someone at rest increases along with its kinetic energy. By placing two balls connected to the ends of a spring inside a box, and letting the balls oscillate rapidly, students see that the balls’ increased inertia relative to the box increases the mass of the box—kinetic energy has inertia. As the system energy oscillates between kinetic energy and spring potential energy, students realize that potential energy has inertia too. Extending this to the molecules in a hot object, students learn that thermal energy has inertia. Since a block can be heated with light, students also discover that electromagnetic energy has inertia. All forms of energy possess inertia (resistance to changes in motion). Students also explore the converse: how Einstein correctly guessed that the inertia (i.e. mass) of even an object at rest is equivalent to an enormous amount of energy.

This resource introduces students to Einstein’s Special Theory of Relativity not as a sequence of counterintuitive facts, but rather as a logical argument based on the simple and natural Principle of Universal Relativity, supported by ample experimental evidence.

Did you know? What makes time dilation and length

contraction reciprocal is a subtle effect called relativity

of simultaneity. What’s simultaneous for Bob is not

simultaneous for Alice, and vice versa! Time dilation,

length contraction, and relativity of simultaneity work

together to enforce universal relativity in a logically self-

consistent manner.

“The phenomena of electrodynamics and mechanics

possess no properties corresponding to the idea of

absolute rest.” – ALBERT EINSTEIN

32

SW1: Scientific Models: Time

In this activity you will conduct several thought experiments and use logic to discover something fascinating about the nature of time and space. You will work in groups of three using chart paper or a large whiteboard. Each group will have a Reader (who keeps the group on task), a Recorder (who writes the answers on the chart paper), and a Reviewer (who asks questions to check for understanding).

Part A: Relativity—“Can you tell you are moving?”

Commonsense dictates and experience confirms that there is no mechanical experiment that can be done in a closed room to tell whether or not the room is moving at a constant speed in a straight line.

1. Imagine Alice sitting in a parked car, tossing an apple straight up. Later, she repeats this experiment while the car is moving at a constant speed. (a) Sketch the path of the apple, as seen by Alice, both when the car is parked and when it is moving. (b) Can Alice use this experiment to tell that she is moving? (c) Sketch the path of the apple, as seen by Bob standing on the sidewalk, when the car is moving.

2. Imagine Alice on a raft made from two pontoons joined by a couple of poles. She anchors the raft in a still pool of water and sends a wave from one pontoon across to the other. It reflects back and she measures the total time taken. Later, the raft is propelled through the water at a constant speed and she repeats her experiment (see Figure). An important fact to consider is that the speed of a wave produced by a moving source will be the same as the speed of a wave produced by a stationary source. (a) Sketch the path of the wave Alice observes when the boat is anchored. (b) Sketch the path of the wave, as seen by Bob floating at rest in the water, when the boat is moving through the water. (c) Can Alice use the times measured in this experiment to tell that she is moving?

3. Imagine Alice inside a rocket deep in space with a clock that measures the time taken for a pulse of light to go up to a mirror on the ceiling and back again. Alice has learned that light behaves like a wave, and assumes it travels through some medium in space that flows freely through the moving rocket. Suppose she starts at rest in that medium and measures the time taken for the pulse to go up and back again. Later, she repeats the experiment when she is moving at a constant speed. Bob is just floating in space, at rest in the medium. He also times the light pulse as Alice cruises by. (a) Sketch the path of the light pulse Alice observes when she is at rest. (b) Sketch the path of the light pulse Bob observes when Alice is moving. (c) Can Alice use this experiment to tell that she is moving?

Part B: Newton vs Einstein

Newton believed in absolute time: the rate at which time passes is the same for everyone regardless of their motion. Einstein believed in universal relativity: there is no way to tell, from inside a closed room, that the room is moving. The natural assumption that light waves travel through some medium in space means that the speed of light along both paths in the diagram is the same. Since the path observed by Bob is longer he must record a longer time than Alice did when she was at rest. This puts absolute time and universal relativity into direct conflict—only one of them can be right.

33

1. Absolute time dictates that in the moving experiment Alice should measure the same time as Bob does. What are the implications for universal relativity if this is true?

2. Universal relativity dictates that in the moving experiment Alice should measure the same time as she did when she was at rest. What are the implications for absolute time if this is true?

Absolute time and universal relativity are incompatible. One of them must be wrong. The only way to resolve the conflict is through experiment.

3. The Large Hadron Collider (LHC) is the largest scientific experiment in history. It is a 27 km long particle accelerator that smashes protons together with unprecedented energy. Neutral kaons are unstable particles produced during the collisions that decay with a half-life of 8.9x10-11s. Use this half-life as the decay time for the kaons. (a) How far would you expect the kaons to travel before decaying, if they are travelling at 0.995c? (b) Kaons are detected 27 cm from the centre of the collision. How does this data refute Newton? (c) How would Einstein interpret these results?

4. Consider again the kaons produced at the LHC. From the perspective of the kaons, they are at rest and they survive for 8.9x10-11s. The kaons “see” the detector rushing by them at 0.995c. What length of the detector rushes by during this time? How can you reconcile this with the 27 cm mentioned above?

A logical consequence of universal relativity is time dilation—“moving clocks run slow.” The flip-side of time dilation is length contraction—“moving objects occupy less space.”

Part C: Time Dilation and Length Contraction

Experimental evidence supports Einstein’s predictions of time dilation and length contraction. With this as motivation, let’s take a closer look at the diagram above for the rocket experiment.

1. According to universal relativity, Alice cannot tell that she is moving. The time taken by the light pulse must be the same for her, whether she is moving or not. Write the algebraic expression for the time taken, tAlice , when Alice is not moving.

2. Bob sees the light pulse travel up and down in the vertical direction, with a vertical speed of (which simplifies to ). Write the algebraic expression for the time Bob measures, tBob.

3. Compare tAlice and tBob. What is the “time dilation” factor that relates Alice’s time to Bob’s time?

4. In the moving case, Bob sees the vertical speed of the light pulse to be . Does this mean that Alice sees the vertical speed of light to be less than c? Explain, using time dilation.

5. When Alice is moving relative to Bob, Alice and Bob disagree on how much time elapses for the light pulse to return. How will this disagreement affect their understanding of how far Alice has travelled in her rocket?

Time and length are both changed by the same amount, the Lorentz factor, and it shows up in so many relativity calculations that it gets its own symbol, γ. Note that γ is always ≥ 1.

34

Part D: Speed Limit

One of the misunderstandings about relativity is that Einstein began with this statement that “nothing goes faster than the speed of light” and derived everything from that premise. While this statement is true, it is not fundamental—it is a logical consequence of time dilation and length contraction.

1. Alice is inside her rocket, moving relative to Bob, when she starts running forward at 1 m/s. She covers 1 m in 1 s according to her ruler and clock.(a) How far is Alice’s 1 m as measured by Bob? (Express your answer in terms of γ.)(b) How long is Alice’s 1 s as measured by Bob? (Express your answer in terms of γ.)(c) How fast is Alice running inside the rocket as measured by Bob? (Express your answer in terms of γ.)

2. As Alice begins to run, her speed inside the rocket changes from zero to 1 m/s. How will this speed change appear to Bob? How will this speed change be affected by the speed of the rocket as it gets closer to c?

Time dilation and length contraction “enforce” a universal speed limit, and allow it to make sense.

3. Suppose Alice needs to apply a force F, for one second, to get herself running. Now imagine, instead, that Bob “reaches” into her moving rocket and pushes her with the same force, F. (He stays at rest, but his hand moves very fast!) How long does he need to apply the force, according to his clock, to have the same effect on Alice?

It’s harder for Bob to accelerate Alice, as if her mass somehow increases. Actually, nothing happens to Alice’s mass. It is time dilation and length contraction that make Alice’s effective inertia greater, relative to Bob.

4. Plot the following historical data for particle accelerators with Energy of the proton (in GeV) on the x-axis and Speed of the proton (as a % of c) on the y-axis. What happens to the speed as more and more energy is given to the particle? Where is the energy going, if it’s not going into increasing the proton’s speed?

Suppose we give a particle some energy to accelerate it from rest up to speed v. By universal relativity we can catch up with the particle and see it as “at rest” again. We can then repeat this process—again and again, forever. While its speed is limited (by the nature of time and space), the amount of energy we can give a particle is unlimited. This is just one of the fascinating ideas contained in Einstein’s most famous equation, E=Mc2.

Bringing It All Together

1. Review the work that you have done as a group and discuss any points that need clarification. Summarize the concepts in your notebook. Be sure to address the following points: • What is universal relativity and absolute time, and how do they conflict with each other? • What is time dilation and length contraction, and how are they related? • Why is there a universal speed limit? • List some of the experimental evidence for Special Relativity. • What happens to an object’s effective inertia as its speed increases?

Proton accelerator Energy (GeV) Speed (%c)

CERN Linac 2 0.050 31.4

TRIUMF 0.48 75

CERN PS Booster 1.4 91.6

BNL Cosmotron 3.3 97.5

CERN PS 25 99.93

35

SW2: Scientific Revolution: Special Relativity

Following the logical consequences of universal relativity, we have so far discovered time dilation, length contraction and a universal speed limit. In this activity you will conduct several more thought experiments and use logic to discover something fascinating about energy. You will work in groups of three using chart paper or a large whiteboard. Each group will have a Reader (who keeps the group on task), a Recorder (who writes the answers on the chart paper), and a Reviewer (who asks questions to check for understanding).

Part A: A Head-on Collision

Imagine that two Super Balls collide head-on.

1. Both balls have speed v going in and speed v going out of the collision. How do their masses compare?

2. Both balls have speed v going in, but one gains speed as a result of the collision. How do their masses compare?

3. One ball has speed v going in and out of the collision, and the other has a greater speed, V, going in and out of the collision. How do their masses compare?

Part B: A Glancing Collision

Two identical Super Balls undergo a very fast glancing collision that is perfectly symmetrical. Alice is riding on the upper ball. She sees the dashed line whizzing by her with a very large horizontal speed as she drifts toward and then away from it with a very small vertical speed, V. Bob, riding on the lower ball, sees the same thing for himself.

1. Imagine that you are now moving to the left with enough speed that Bob has no horizontal speed relative to you (see lower Figure). Would your new perspective change the vertical speeds that Alice and Bob observe for themselves?

2. Alice is now moving even faster relative to you, and her time is dilated compared to you by a factor of γ. If it takes Alice one second to move the vertical distance to the dashed line according to her clock, will it take more or less time for her to cover the same vertical distance according to your clock? How does time dilation change Alice’s vertical speed, from your perspective?

3. Alice’s vertical speed does not change as a result of the collision, neither does Bob’s. If we ignore Alice’s horizontal motion, this collision is the same as #3 in Part A. How do their masses compare?

By time dilation, a moving object has greater “effective inertia” for sideways deflection: M = γm

Part C: All Forms of Energy have Inertia

Alice comes across a closed box “at rest” in deep space.

1. (a) Nothing enters or leaves the box. Can the mass of the box suddenly change from M to M’? Why or why not? (b) Bob is drifting by and sees the box moving at a constant speed v. How can Bob use conservation of energy and/or momentum to explain that such a change in mass is impossible?

36

2. Alice opens the box and finds two balls of mass m connected by a spring. The balls are oscillating back and forth very quickly. When they are moving fastest, their time dilation factor is γ.(a) Using the concept of “effective inertia” for sideways deflection, how does the motion of the balls affect the mass of the box? Does kinetic energy have inertia? (b) As the balls move outwards, their kinetic energy changes into elastic potential energy stored in the spring. The total mass of the box cannot change. How does this show that potential energy has inertia?

3. A brick is made of atoms connected by spring-like inter-atomic bonds. As the brick is heated, would you expect its mass to increase? Explain.

4. A brick can be heated with the energy in light. The “before” picture shows a box containing a brick of mass M and two pulses of light in midflight heading towards the brick. In the “after” picture, the block has absorbed the light energy and is warmer. Does the mass of the box change when the light is absorbed? What does this say about the inertia of light?

All forms of energy have one property in common: inertia. This is a powerful unifying principle in unravelling the mystery of what “energy” is.

Part D: E=mc2

We have just discovered that various forms of energy (kinetic, potential, thermal, and electromagnetic) inside a box contribute to the mass (or inertia) of the box. So any change in the energy, ΔE, inside a box must produce a corresponding change in its mass, Δm. The exact relationship is very simple: ΔE = Δmc2. This mass-energy equivalence applies to all physical processes, including chemical and nuclear. All forms of energy have inertia; but do all forms of inertia have energy? Does even mass at rest have energy?

1. The “before” picture shows a box containing two particles at rest, each of mass m; one is matter and the other is antimatter. In the “after” picture, the matter and antimatter have been transformed entirely into light. (a) The box is sitting on a weigh scale. Does its weight change? Explain. (b) We uncover a window on the box and let out all the light. What is the change in mass of the box? How is this related to the amount of energy that left the box?

Experiments with elementary particles confirm Einstein’s intuition that even mass at rest has energy: E=mc2. The general form of Einstein’s mass-energy equivalence relation is: E=Mc2, where M=γm is the relativistic mass and m the rest mass. This mass-energy equivalence can be rewritten as a general relation between energy and momentum: E2 = m2c4 + p2c2, where p = γmv is the relativistic momentum of the system. When the system has no rest mass (e.g. a photon) the general relation reduces to E=pc, a result that agrees with both a wave model of light and a particle model of light. When the system is at rest (p=0), the general relation reduces to E=mc2.

It is a remarkable fact of nature that matter can transform into light, and vice versa, but notice that in such processes both the total mass and the total energy stay the same. Mass is not “converted” into energy, or vice versa. The energy in light has inertia, and the inertia in matter has energy—as described by E=mc2.

Putting It All Together:

1. Review the work that you have done as a group and discuss any points that need clarification. Summarize the concepts in your notebook. Be sure to address the following points: • How does time dilation result in moving objects having extra “effective inertia”? • How is an increase in the “effective inertia” different from an increase in the mass of the object itself? • How would you explain to your friend that kinetic and potential energy have inertia? • Does a cup of coffee weigh more when it is hot? Explain. • Why is it incorrect to say that mass is converted into energy, and vice-versa? • As the Sun emits light energy, what must happen to its mass? Where has the mass gone?

Kinetic Energy

37

SA: Scientific Revolution: Special Relativity

1. The scientists at CERN accelerate protons to 99.9999991% c as the protons travel around a 27 km long ring. How does this extreme speed create challenges for them?

2. One of the most famous equations in science is E=mc2. There are several ways to understand where this equation comes from. Here is one published by Albert Einstein:

(a) Start with a block of mass M at rest. Let it absorb two pulses of light, each with energy E/2, so the block heats up (see Figure).

(b) Consider this same experiment from the point of view of an observer drifting by with speed v. To this observer, the block has momentum Mv, and the pulses of light approach the block at an angle.

(c) Draw a diagram to show how this angle is related to the relative speed of the block and the speed of the light pulses. Remember that light moves at speed c.

(d) Express the sine of the angle as a ratio of these two speeds.

(e) The momentum of light is given by Maxwell’s equations as: p= . What is the momentum of each pulse of light approaching the block? What is the vertical component of this momentum?

(f) What is the change in momentum of the block when it absorbs the two pulses of light?

(g) The block does not speed up when it absorbs the light. Why not?

(h) If momentum changes but speed does not, then the mass of the block must increase from M to M’. Using conservation of momentum, find an expression for E in terms of the change in mass.

3. In the animation Where Does Energy Come From? Alice and Bob discover that we are literally eating the Sun!The Sun provides the energy that is the basis for virtually every food chain. We measure the energy output of the Sun by its luminosity and find that the Sun emits 3.8x1026 W.

(a) Use E=mc2 to determine how much mass the Sun is losing every second.

(b) The Sun has a total mass of 2x1030 kg. How long will it take to use up the mass of the Sun?

(c) The Sun is expected to survive for another 5 billion years. Will the mass loss be significant by then?

4. The general relation between energy and momentum in Special Relativity is E2 = m2c4 + p2c2, where m is the rest mass for the system and p is the relativistic momentum, p = γmv.

(a) Start with the general energy-momentum relation and use common factors to derive E = γmc2.

(b) Start with E = γmc2 and use the binomial expansion of to show that E ≈ mc2 + ½mv2.

(c) What does this last equation contribute to our understanding of energy?

38

SW

1: A

nsw

ers

Part

A:

1. (a

)

(b) A

lice

cann

ot te

ll th

at s

he is

mov

ing.

Both

pat

hs w

ill be

iden

tical

.

(c)

2. (a

)

(b)

(c) A

lice

can

tell

that

she

is m

ovin

g re

lativ

e to

the

wat

er b

ecau

se th

e tim

e ta

ken

for t

he w

ave

to re

turn

is lo

nger

whe

n sh

e is

mov

ing

than

whe

n sh

e is

at

rest

. The

tim

e is

long

er b

ecau

se th

e pa

th in

long

er, b

ut th

e sp

eed

of th

e w

ave

rela

tive

to th

e st

ill w

ater

is th

e sa

me

in b

oth

case

s.

3. (a

)

(b)

(c) T

his

prob

lem

is in

prin

cipl

e th

e sa

me

as #

2, a

nd s

tude

nts

mig

ht e

xpec

t th

at th

e re

turn

tim

e w

ill be

diff

eren

t—Al

ice

can

use

this

exp

erim

ent t

o te

ll th

at

she

is m

ovin

g. In

fact

she

can

not,

and

stud

ents

cha

lleng

e th

e re

ason

for t

heir

expe

ctat

ion

(abs

olut

e tim

e) in

Par

t B.

Part

B:

1. If

Alic

e m

easu

res

the

sam

e tim

e as

Bob

, whi

ch is

diff

eren

t fro

m w

hen

she

was

at r

est,

then

she

wou

ld k

now

that

she

is m

ovin

g. If

she

can

tell

that

sh

e is

mov

ing

then

rela

tivity

is n

ot u

nive

rsal

. Rel

ativ

ity w

ould

not

app

ly to

ex

perim

ents

don

e w

ith li

ght i

n a

clos

ed ro

om.

2. If

Alic

e m

easu

res

the

sam

e tim

e as

whe

n sh

e w

as a

t res

t, th

en s

he w

ould

no

t kno

w th

at s

he is

mov

ing.

But

sin

ce th

is ti

me

is d

iffer

ent f

rom

Bob

’s, t

ime

is n

ot a

bsol

ute.

Tw

o ob

serv

ers

mov

ing

rela

tive

to e

ach

othe

r will

disa

gree

on

how

muc

h tim

e ha

s el

apse

d.

3. (a

)

(b) T

he k

aons

are

trav

ellin

g 10

tim

es fu

rther

than

they

“sho

uld”

—27

cm

ve

rsus

2.7

cm

. Acc

ordi

ng to

New

ton,

whe

n 8.

9x10

-11 s

ela

pses

in th

e de

tect

or

fram

e, th

e sa

me

amou

nt o

f tim

e sh

ould

hav

e el

apse

d fo

r the

kao

ns. T

hey

“sho

uld”

dec

ay a

fter t

rave

lling

only

2.7

cm

. (c

) Ein

stei

n w

ould

inte

rpre

t the

se re

sults

by

sayi

ng th

at w

hen

8.9x

10-1

1 s

elap

ses

in th

e de

tect

or fr

ame,

less

tim

e el

apse

s fo

r the

kao

ns. W

hile

the

kaon

s do

n’t f

eel i

t, tim

e pa

sses

mor

e sl

owly

for t

hem

rela

tive

to th

e de

tect

or. T

he

kaon

s st

ill ha

ve ti

me

to tr

avel

furth

er (i

n fa

ct, 1

0 tim

es a

s fa

r) be

fore

they

dec

ay.

4. Th

e de

tect

or ru

shes

by

at 0

.995

c fo

r 8.9

x10-1

1 s. D

urin

g th

is ti

me

the

kaon

s w

ill “s

ee” 2

.7 c

m o

f the

det

ecto

r pas

s by

. But

act

ually

they

hav

e tra

velle

d th

roug

h 27

cm

of t

he d

etec

tor.

From

the

kaon

s’ p

ersp

ectiv

e, th

e de

tect

or

mus

t be

cont

ract

ed in

the

dire

ctio

n of

mot

ion

by a

fact

or o

f 10,

so

the

27 c

m

of d

etec

tor o

ccup

ies

a sp

ace

of o

nly

2.7

cm in

the

kaon

s’ fr

ame

of re

fere

nce.

Part

C:

1. If

d is

the

leng

th o

f the

pat

h (u

p an

d do

wn)

, the

n

2.

3.

4. B

ob s

ees

the

light

mov

ing

up a

nd d

own

mor

e slo

wly

than

spe

ed c

, and

so

it m

ight

see

m th

at A

lice

woul

d se

e a

verti

cal s

peed

of l

ess

than

c fo

r the

light

pu

lse in

her

fram

e. B

ut A

lice’

s tim

e is

also

pas

sing

mor

e slo

wly

rela

tive

to B

ob,

and

so s

he s

ees

the

norm

al s

peed

for l

ight

—c.

It is

the

natu

re o

f tim

e an

d sp

ace,

and

not

the

natu

re o

f lig

ht, t

hat m

akes

all o

bser

vers

mea

sure

the

sam

e sp

eed

for l

ight

(or a

nyth

ing

else

mov

ing

at th

e un

ivers

al s

peed

limit)

. Alic

e ca

nnot

use

the

retu

rn ti

me

or th

e sp

eed

of th

e lig

ht p

ulse

to d

etec

t her

mot

ion!

39

5. A

lice

and

Bob

will

disa

gree

on

how

far s

he tr

avel

s. D

urin

g th

e tim

e t B

ob, B

ob

will

see

Alic

e tra

vel a

dis

tanc

e L B

ob. S

ince

Alic

e’s

trave

l tim

e is

less

(tAl

ice <

t B

ob),

she

mus

t see

this

dis

tanc

e as

less

: LAl

ice <

LBo

b (in

ord

er th

at A

lice

see

Bob

mov

ing

at th

e sa

me

spee

d at

whi

ch th

at B

ob s

ees

Alic

e m

ovin

g). F

rom

Al

ice’

s pe

rspe

ctiv

e, B

ob’s

spa

ce—

and

ever

ythi

ng in

it is

con

tract

ed s

o sh

e w

ill no

t hav

e go

ne a

s fa

r as

Bob

thin

ks.

Part

D:

1. (a

)

b)

(c

)

2. B

ob w

ill m

easu

re A

lice’

s ch

ange

in s

peed

to b

e le

ss th

an 1

m/s

—he

r spe

ed

chan

ge w

ill be

redu

ced

by a

fact

or o

f γ2 . A

s th

e sp

eed

of A

lice’

s ro

cket

ap

proa

ches

c, γ

bec

omes

larg

er a

nd la

rger

so

her c

hang

es in

spe

ed w

ill ap

pear

to B

ob to

be

less

and

less

. Tha

t “no

thin

g go

es fa

ster

than

the

spee

d of

ligh

t” is

a c

onse

quen

ce o

f the

ver

y na

ture

of t

ime

and

spac

e.

3. O

ne s

econ

d fo

r Alic

e is

γ ti

mes

one

sec

ond

for B

ob. H

e w

ill ne

ed to

app

ly

the

forc

e fo

r a lo

nger

tim

e, a

ccor

ding

to h

is c

lock

, to

have

the

sam

e ef

fect

on

Alic

e. A

ltern

ativ

ely,

if he

app

lies

a fo

rce

F fo

r som

e tim

e ac

cord

ing

to h

is

cloc

k, A

lice

will

clai

m th

at h

e ap

plie

d th

at fo

rce

for l

ess

time.

Tim

e di

latio

n m

akes

his

effo

rt to

acc

eler

ate

Alic

e le

ss e

ffect

ive

than

he

thin

ks!

4. G

ivin

g m

ore

ener

gy to

a v

ery

fast

m

ovin

g pa

rticl

e ha

s lit

tle e

ffect

on

its

spee

d. If

we

catc

h up

with

th

e pa

rticl

e, g

ivin

g it

ener

gy w

ill dr

amat

ical

ly in

crea

se it

s sp

eed

in th

is

new

refe

renc

e fra

me,

but

this

spe

ed

incr

ease

is s

mal

l as

seen

in th

e fra

me

of th

e ac

cele

rato

r. (S

ee #

2.) A

s in

#3

, tim

e di

latio

n ca

uses

the

effe

ctiv

e in

ertia

of t

he p

artic

le to

incr

ease

as

its

spee

d ap

proa

ches

c.

SW

2:

Ans

wer

s

Part

A:

1. T

he s

peed

s go

ing

in a

nd o

ut a

re e

qual

so

the

mas

ses

mus

t als

o be

equ

al.

2. O

ne b

all g

ains

spe

ed a

s a

resu

lt of

the

collis

ion

so it

mus

t hav

e le

ss m

ass

than

the

othe

r.

3. If

two

balls

of d

iffer

ing

spee

ds c

ollid

e an

d m

aint

ain

thei

r orig

inal

spe

eds

then

th

eir m

asse

s m

ust b

e di

ffere

nt b

y th

e sa

me

ratio

. The

slo

wer

bal

l mus

t hav

e m

ore

mas

s th

an th

e fa

ster

mov

ing

ball.

Part

B:

1. A

cha

nge

in y

our p

ersp

ectiv

e w

ill no

t affe

ct th

e sp

eeds

obs

erve

d by

Alic

e

and

Bob.

2. It

will

take

Alic

e lo

nger

from

you

r per

spec

tive

to c

over

the

sam

e ve

rtica

l di

stan

ce, s

o yo

u se

e he

r mov

ing

mor

e sl

owly

than

she

doe

s.

3. If

Alic

e’s

verti

cal s

peed

in a

nd o

ut is

lowe

r tha

n Bo

b’s,

then

she

mus

t hav

e m

ore

mas

s (o

r ine

rtia)

. Not

e to

teac

hers

: Ign

orin

g th

e ho

rizon

tal m

otio

n of

the

ball m

ay b

e ch

alle

ngin

g fo

r stu

dent

s. T

his

diffi

culty

is a

llevia

ted

in P

art C

by

effe

ctive

ly pu

tting

the

ball i

n a

box

and

letti

ng it

mov

e sid

e-to

-sid

e ve

ry ra

pidl

y.

Part

C:

1. (a

) The

mas

s of

the

box

cann

ot c

hang

e w

ithou

t gai

ning

or l

osin

g so

met

hing

to

the

surro

undi

ngs.

(b

) The

mas

s of

the

box

cann

ot c

hang

e be

caus

e th

at w

ould

vio

late

bot

h en

ergy

and

mom

entu

m c

onse

rvat

ion

law

s. F

or e

xam

ple,

the

kine

tic e

nerg

y of

th

e m

ovin

g bo

x is

½ M

v2 . Th

e ki

netic

ene

rgy

can’

t cha

nge

beca

use

ther

e is

no

out

side

forc

e ac

ting

on th

e bo

x. S

o if

v do

esn’

t cha

nge,

M c

an’t

chan

ge.

2. (a

) The

mov

ing

balls

hav

e m

ore

effe

ctiv

e in

ertia

for s

idew

ays

defle

ctio

n,

whi

ch m

eans

the

box

will

pres

ent a

gre

ater

resi

stan

ce to

upw

ard

acce

lera

tion.

(It c

an b

e sh

own

that

this

effe

ct is

the

sam

e fo

r all

dire

ctio

ns o

f ac

cele

ratio

n.) T

he k

inet

ic e

nerg

y of

the

balls

incr

ease

s th

e in

ertia

(or m

ass)

of

the

box.

(b

) The

mas

s (o

r ine

rtia)

of t

he b

ox c

anno

t cha

nge

so th

e po

tent

ial e

nerg

y st

ored

in th

e sp

ring

mus

t hav

e an

equ

al a

mou

nt o

f ine

rtia

as th

e ki

netic

en

ergy

that

was

pre

viou

sly

stor

ed in

the

mov

ing

balls

.

3. A

s th

e br

ick

is h

eate

d th

e at

oms

will

vibr

ate,

muc

h lik

e th

e ba

lls in

#2,

so

the

iner

tia o

f the

sys

tem

will

incr

ease

and

the

hot b

rick

will

have

a g

reat

er m

ass.

0255075100

08.3

16.7

25.0

Ener

gy (G

eV)

Speed (%c)

40

4. T

he b

ox c

onta

ins

the

bric

k an

d th

e lig

ht (t

he li

ght i

s no

t bei

ng a

dded

from

ou

tsid

e th

e bo

x) s

o th

e to

tal m

ass

of th

e bo

x ca

nnot

cha

nge

whe

n th

e br

ick

abso

rbs

the

light

. Acc

ordi

ng to

#3

the

war

mer

bric

k ha

s m

ore

mas

s, w

hich

m

eans

that

ligh

t mus

t hav

e in

ertia

! A b

ox w

ith li

ght b

ounc

ing

arou

nd in

side

w

ill re

sist

acc

eler

atio

n m

ore

than

the

sam

e bo

x w

hen

it’s

empt

y.

Part

D:

1. (a

) The

mas

s of

the

box

will

not c

hang

e. T

he in

ertia

of t

he li

ght m

ust b

e th

e sa

me

as th

e m

ass

of th

e pa

rticl

e/an

ti-pa

rticl

e pa

ir. M

ass

is c

onse

rved

. N

ote

to te

ache

rs: W

hile

stu

dent

s m

ight

not

mak

e th

e co

nnec

tion,

the

fact

th

at th

e w

eigh

t of t

he b

ox a

lso

does

not

cha

nge

follo

ws

from

the

natu

re

of g

ravi

ty: t

he w

eigh

sca

le is

acc

eler

atin

g up

in c

urve

d sp

acet

ime,

and

is

mea

surin

g th

e re

sist

ance

of t

he b

ox to

acc

eler

atio

n!

(b) W

hen

all t

he li

ght h

as e

scap

ed, n

othi

ng re

mai

ns, a

nd th

e ch

ange

in th

e m

ass

of th

e bo

x is

Δm

= 2

m, w

here

2m

is th

e m

ass

of th

e or

igin

al p

artic

le/

anti-

parti

cle

pair.

Usi

ng Δ

E =

Δm

c2 , the

cha

nge

in th

e en

ergy

of t

he b

ox is

Δ

E =

(2m

)c2 . B

ut s

ince

ene

rgy

is c

onse

rved

, we

mus

t hav

e Δ

E =

E,

whe

re E

is th

e en

ergy

in th

e es

cape

d lig

ht. S

o E

= (2

m)c

2 in th

is c

ase.

In

gene

ral,

an o

bjec

t with

mas

s m

at r

est h

as e

nerg

y E=

mc2 . E

ven

mas

s at

re

st h

as e

nerg

y!

SA:

Ans

wer

s1.

The

pro

tons

are

trav

ellin

g so

fast

that

ther

e is

a h

uge

incr

ease

in th

eir e

ffect

ive

iner

tia fo

r sid

eway

s de

flect

ion

( γ=7

500)

. The

y ne

ed a

mag

netic

fiel

d of

ove

r 8

Tesl

a to

ben

d th

eir t

raje

ctor

ies

into

a c

ircle

—th

is re

quire

s su

perc

ondu

ctin

g m

agne

ts.

2. Th

e sp

eed

of th

e bl

ock

does

not

cha

nge

in it

s re

st fr

ame.

The

spe

ed o

f the

bl

ock

in th

e m

ovin

g fra

me

is d

ue to

the

mot

ion

of th

e ob

serv

er, a

nd s

o ob

viou

sly

it w

ill no

t cha

nge.

This

thou

ght e

xper

imen

t sho

ws

that

add

ing

an a

mou

nt o

f ene

rgy

E to

a s

yste

m

will

incr

ease

the

mas

s of

the

syst

em b

y E/

c2. A

s a

gene

ral f

orm

ula

we

writ

e Δ

E =

Δm

c2. A

cha

nge

in th

e en

ergy

of a

sys

tem

is a

lway

s ac

com

pani

ed b

y a

corre

spon

ding

cha

nge

in it

s m

ass.

Ene

rgy

has

iner

tia.

3. (a

)

(

b)

(

c)

Even

afte

r 5 b

illion

yea

rs th

e am

ount

of m

ass

lost

by

the

Sun

is n

eglig

ible

.

4. (a

) (b) B

inom

ial E

xpre

ssio

n:

(c) E

inst

ein

show

ed th

at e

ven

whe

n an

obj

ect i

s at

rest

and

has

no

ki

netic

ene

rgy

(½m

v2 =0),

it st

ill ha

s en

ergy

—a

new

kin

d of

ene

rgy

calle

d “re

st e

nerg

y”.

CREDITS

Authors Executive Producer Scientific Advisor

Dr. Richard Epp Greg Dick Dr. Niayesh Afshordi Manager of Educational Outreach Director of Educational Outreach Associate Faculty Perimeter Institute for Theoretical Physics Perimeter Institute for Theoretical Physics Perimeter Institute for Theoretical Physics Dave Fish Senior Physics Teacher Sir John A. Macdonald Secondary School

Educational Advisors

Joan Crawford Rob Crawford Kevin Donkers Elisa Gatz Glenview Park Secondary School Turner Fenton Secondary School Preston High School Sterling High SchoolCambridge, Ontario Brampton, Ontario Cambridge, Ontario Sterling, Illinois Philip Freeman Olga Michalopoulos Dennis Mercier Richmond Secondary School Georgetown District High School Turner Fenton Secondary School Richmond, British Columbia Georgetown, Ontario Brampton, Ontario Dr. Damian Pope Melissa Reist David Vrolyk Manager of Educational Outreach Elizabeth Ziegler Public School Sir John A. Macdonald Secondary School Perimeter Institute for Waterloo, Ontario Waterloo, Ontario Theoretical Physics

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