Unit 3: Uniformly Accelerated Particle Model · PDF fileUnit 3: Uniformly Accelerated Particle...

Unit 3: Uniformly Accelerated Particle Model

1. Lab Notes: Motion on an incline

Apparatus A wheel and axle made from a 4-inch hole saw cut-out, dowel, and golf tees to roll down a pair of

inclined rails made from two lengths of electrical conduit. With the narrow axle, depending on the

steepness, the disks take from 15 to 30 seconds to roll down the ramp.

Pre-lab discussion Let a ball roll down an inclined track and ask students for observations. Record all observations.

To proceed, they must mention something to the effect that the ball speeds up as it rolls down.

(If students say that the ball accelerates, tell them that we will wait to use this term until we have

formally defined it - ask for simpler way to describe the motion.)

To obtain a finer description, ask students which observations are measurable. These should

include the time, position and speed.

Ask them how they can measure speed directly. Lead them to the conclusion that they cannot,

but that they can measure position and time. Note: Since this lab is so similar to the BPV lab

and other motion labs, this should be a matter of review to most of them.

Lab performance notes We prefer to have the students control the time for this lab. They controlled the time in the

buggy lab and we would like the same for this lab. It is impossible to do this rolling a cart down

a ramp, the time is too short. The reduced speed of the wheel-and-axle apparatus allows for up

to 30 s of hand-collected data.

Time should be plotted as the independent variable. The graph should be scaled to completely fill

a sheet of paper to aid in later analysis.

One key difference between this lab and the past motion labs is that the wheel must start from

position 0 with a velocity of 0 at t = 0. (This is very important.)

Have different lab groups to use inclines of different steepnesses.

Students usually choose to mark the position of the axle or front of the disk on the conduit using

whiteboard marker. Stress that they need to be making their mark at the chosen time.

A good online metronome can be found at http://www.webmetronome.com/

Post-lab discussion

From the lab, the students have some variation of the following graph:

Focus the whiteboard discussion on their experimental procedure and the verbal interpretation of

the parabolic x-t graph. Students should be able to describe that the displacement during each

time interval increases over the previous time interval. Since the object travels greater distances

in each successive time interval, the velocity is increasing.

Post-lab extension 1 Discussion Contrast the x vs t graph for this lab with the one obtained in unit 2.

One can speak of the average velocity as the slope of the graph (above left) because the slope of

a straight line is constant. It doesn't matter which two points are used to determine the slope.

On the other hand, one could speak of the average velocity of the

object in the graph to the right, but since the object started very

slowly and steadily increased its speed, the term average velocity is

not particularly useful.

What would be more useful is to have a way of describing the object's speed at a given instant

(or as Arons terms it: clock reading). To develop this idea, you must show that, as you shrink

the time interval t over which you calculate the average velocity, the secant (line intersecting

the curve at two points) more closely resembles the curve during that interval.

x (cm)

t (s)

t

x

x

t

t

x

x

t

t

x

x

t

t

x

x

t

t

x

t

x

Unit 2 Unit 3

That is, the slope of the secant gives the average velocity for that

interval. As the interval gets shorter and shorter, the secant more

closely approximates the curve. Thus, the average velocity of this

interval becomes a more and more reasonable estimate of how fast

the object is moving at any instant during this interval.

As one shrinks the interval, t to zero, the secant becomes a tangent;

the slope of the tangent is the velocity at this instant, or simply the

instantaneous velocity at that clock reading.

Worksheet 1: Uniformly Accelerated Motion Worksheet Due to the fact that drawing tangents is a very imprecise activity, I choose to do worksheet 1

next. The purpose is to show that the instantaneous velocity can also be found mathematically

from a sequence of position-time data. The desired goal of the activity is to establish that for

uniformly accelerated motion, the average velocity for a time interval is equal to the

instantaneous velocity at the clock reading in the middle of the time interval. This idea is

important since it is the same algorithm used by computer motion analysis programs such as

Logger Pro and Science Workshop. Students can use the reasoning developed here to analyze

the motion of a picket fence falling through a photogate.

The students now have a mathematical way of producing a v-t graph from their measurements of

position and time. Analyzing the v-t graph allows determination of acceleration and displacement

with or without an initial velocity. With this background, students can begin the uniformly

accelerated motion worksheet.

After completing and whiteboarding the worksheet, students now have the tools to do a much

better graph of velocity vs. time than finding slopes of tangent lines. They should be assigned to

take their original position and time data and make a velocity vs. time graph and a position vs.

time2 graph.

Analyzing the v vs. t graph A plot of instantaneous velocity (v, instead of v ) vs. time should yield a straight line. The slope

of this line is

v

t a . That is, the change in velocity during a given time interval is defined to

be the average acceleration. Students need to understand the units for acceleration. In this lab,

the units for the slope of the graph will be cm/s/s. Students must be able to state that a slope of 5

cm/s/s, for example, means that the wheel’s velocity changed 5 cm/s in each second. Many

times, cm/s/s is written as cm/s2. It will reduce confusion if the acceleration is stated as

“centimeters per second each second” even if it is written as cm/s2.

Because the graph of v vs. t for the wheel lab (shown at right) is linear, one

can conclude that the acceleration is constant. If the acceleration is

constant, then the average acceleration for the entire time interval is equal

to the instantaneous acceleration at any given clock reading. [This is why

we use a instead of a-bar in the equations.] The equation for the line can be

written as v a t v

i, where v0 is the y-intercept.

Generalizing the linear equation from the velocity vs. time graph for any

time interval ti to tf yields v

f at v

i. The development of this expression is provided at the end of

this document to clarify the use of t (time interval) as opposed to t (clock reading).

t

x

x

t

t

v

Post-lab Extension 2: Linearizing the x vs. t graph Some students will recognize that the shape of the x-t graph is parabolic. If this is true, a test plot

of x vs. t2 will be linear.

Student Activity Have students add a third column to their inclined motion lab data

tables—a column for t2. Ask students in what units t

2 should be

measured. Fill in the column for t2 by squaring the clock readings in

the t column (this is easily done in Logger Pro by adding a new

calculated column). Students can now make a graph of position vs.

time2. The graph should look like the one to the right.

Discussion From the graph, it is clear that position is directly proportional to the square of the time. Upon

performing a mathematical analysis of the graph of x vs. t2 students will obtain the following

equation: x = kt2

where k is the slope of the graph. Students should be asked about the units of

the slope and what those units mean. A few students will notice that the units of the graph are

cm/s2. Ask what physical quantity has those units. Comparing the slope of this graph to the

slope of the

v-t graph from this unit should lead students to the understanding that the slope of the x-t2 graph

is equal to half of the acceleration of the object. Thus, the general form of the equation for the x-

t2 graph becomes x

12 at2 . Make it clear, however, that this relationship is only true if the

object starts from rest at position zero. For situations where the initial position is not zero, the

initial position can simply be added to the equation: x

f 1

2 at2 xi. For situations where the

initial velocity is not zero, the equation for the displacement can be developed by analyzing the

area under the v-t graph. (See Appendix 2 for all the gory details.)

Post-lab Extension 3: Graphing and linearizing a v vs. x graph Ask the students how the velocity is related to the position of the object. It is clear that as the

position increases, the velocity does too, but the details of the relationship elude casual

observation and require graphical analysis.

Student Activity Have the students prepare a graph of instantaneous velocity vs. the position of the object. Their

initial graph will be a side-opening parabola. A test plot of velocity squared vs. position yields a

linear result. Ask students to write the equation relating velocity and position and whiteboard

their results for discussion.

Discussion

Students will develop an equation of the form v2 kx where k is the slope of the graph. Ask

students what the slope of the graph means. The units will be (cm2/s

2)/cm, and students may need

to be encouraged to simplify the units, finding them to be cm/s2. Again, this slope is not equal to

the acceleration, even though the units are the same. However, when students compare this slope

to their accelerations, they will find the slope to be twice their acceleration. Thus, the general

form of the equation becomes v2 2ax . If the object had not started from zero velocity, there

would be a y-intercept, adding a vi2 term so that our final equation becomes:

v

f

2 2ax vi

2 .

t2

x

Post-lab Extension 4 (Development of Kinematic Expressions) After completing the lab and both lab extensions, students will

have developed the key kinematics equations. We do need to

complete the equation x

f 1

2 at2 xi for a situation where

the initial velocity is not zero. Remind students that the area

under a velocity-time graph represents the displacement. By

providing students with a quantitative v-t graph of accelerated

motion, students can determine the displacement of the object.

Using the graph below we see the rectangle has an area vit so

we complete the equation 212f i i

x v t at x or

212i

x v t at . Considering the shape of the graph as a

trapezoid we can come up with

x v

fv

i

2

t .

Deployment lab: (Increasing and Decreasing Speed) I then have the students perform a lab where a cart on a ramp is used to tie together the motion, the 3

graphs of the motion and motion maps are tied together. One section is shown on the next page.

Apparatus Motion detector-interface-computer

Ramp and books or bricks (to elevate ramp)

Cart and track

Pre-lab discussion Make sure students understand the format of the lab. They are to observe the motion and then

draw a motion map and predicted graphs. Then the students check their graph predictions using

the motion detector.

It may be helpful to show students how to resize the graph axes so students can focus on the

relevant portions of the graph. Analysis should focus on the region of the graph in which the cart

is coasting and not on initial pushes or final stops.

Lab-performance notes Perform the lab ahead of time and create a template file the students can use. The template

should display x-t, v-t and a-t graphs each with appropriately scaled axes for your situation.

Make sure students complete the questions about the slope for each graph before moving on to

subsequent examples.

©Modeling Instruction - AMTA 2013 1 ws 1-Uniform Acceleration, v3.0

Name

Date Pd Uniformly Accelerated Motion Model Worksheet 1:

Development of Accelerated Motion Representations

1. The data to the left are for a wheel rolling from rest down an incline. Using the position/time data given

in the data table, plot the position vs. time graph.

t x

(s) (cm)

0.0 0.0

1.0 5.0

2.0 20.0

3.0 45.0

4.0 80.0

5.0 125.0

6.0 180.0


2. What is the significance of the slope of a position vs. time graph?

3. What is happening to the slope of your position vs. time graph as time goes on?

4. Explain what your answers to questions 2 and 3 tell you about the motion of the wheel.

5. On the position vs. time graph, draw a line which connects the point at t = 0 to the point at t = 6.0 s.

6. Calculate the slope of this line in the space below. Explain what the slope of this line tells you about the

motion of the wheel.

7. On the position vs. time graph, draw a line which connects the point at t = 2.0 s to the point at t = 4.0 s.



9. On the position vs. time graph, draw a line tangent to the graph at t = 3.0 s.



11. Compare the slopes you have calculated in questions 6, 8, and 10. Explain the results of your

comparison.

12. Consider an object accelerates uniformly. If you were to calculate the average speed of the object for a

given interval of time, would the object ever be traveling with an instantaneous speed equal to that

average speed? If so when? Explain!


13. Use your mathematical equation from the position vs. time squared graph to complete the data table

below. Using the completed table, plot a velocity vs. time graph using Logger Pro.

t x t x midt v

2. (s) 3. (cm) (s) (cm) (s) (cm/s)

4. 5. 6. 7.

8. 9. 10. 11.

12. 13. 14. 15.

16. 17. 18. 19.

20. 21. 22. 23.

24. 25. 26. 27.

28. 29. 30. 31.

32. 33. 34. 35.

36. 37. 38. 39.

40. 41. 42. 43.

44. 45. 46. 47.

48. 49. 50. 51.

52. 53. 54. 55.

14. Using your velocities at the mid-times above, calculate the disk’s position at that time. Use the equation

from the position vs. time squared graph for your calculation. Use the completed table, plot a velocity vs.

position graph using Logger Pro.

midt v x

(s) (cm/s) (cm)

56. 57. 58.

59. 60. 61.

62. 63. 64.

65. 66. 67.

68. 69. 70.

71. 72. 73.

74. 75. 76.

77. 78. 79.

80. 81. 82.

83. 84. 85.

86. 87. 88.

89. 90. 91.

92. 93. 94.

Mathematical Analysis:

Mathematical Analysis:

©Modeling Instruction 2013 1 U3 Uniform Acceleration – lab extension v3.1

Uniformly Accelerated Particle Model

Lab Extension: Increasing and Decreasing Speed

1. Increasing speed in the positive direction a. Without using the motion detector, observe the motion of the cart as it starts from rest and rolls

down the incline.

b. Draw a motion map of the cart’s motion along the ramp. Include velocity and acceleration

vectors.

c. Is the velocity positive or negative? d. Is the acceleration positive or negative?

e. Predict the graphs describing

the motion.

f. Record the graphs as displayed

by the motion detector.

g. The slope of the

position-time graph is (constant / increasing / decreasing)

and (positive / negative)

and represents

______________________.

h. The slope of the

velocity-time graph is (constant / increasing / decreasing)


and represents

______________________.

0

+


2. Decreasing speed in the positive direction a. Without using the motion detector, observe the motion of the cart slowing after an initial push.

Answer the following questions for the cart while coasting.

b. Draw a motion map of the cart’s motion along the ramp. Include both velocity and acceleration

vectors.



the motion.



g. The slope of the



and represents

______________________.

h. The slope of the



and represents

______________________.

0

+


3. Increasing speed in the negative direction a. Observe the motion of the cart starting from rest and rolling down the incline without using the

motion detector.


vectors.


e. Predict the graphs describing the

motion.



g. The slope of the



and represents

______________________.

h. The slope of the



and represents

______________________.

0

+


4. Decreasing speed in the negative direction a. Observe the motion of the cart slowing after an initial push without using the motion detector.

Answer the following questions for the cart while coasting.


vectors.



the motion.



g. The slope of the



and represents

______________________.

h. The slope of the



and represents

______________________.

0

+


5. Up and down the ramp a. Observe the motion of the cart after an initial push without using the motion detector. Answer

the following questions for the cart while coasting.


vectors.


Does the direction of the velocity change? Does the direction of the acceleration change?


the motion.



0

+

g. Describe how the slope

of the position-time graph

changes:

h. Describe the slope of the

velocity-time graph:


6. Up and down the ramp a. Observe the motion of the cart after an initial push without using the motion detector. Answer

the following questions for the cart while coasting.


vectors.


Does the direction of the velocity change? Does the direction of the acceleration change?


the motion.



0

+

h. Describe the slope of the

velocity-time graph:

g. Describe how the slope

of the position-time graph

changes:

©Modeling Instruction - AMTA 2013 1 U5 Net Force - Teacher Notes v3.1

Unit 5: Unbalanced Force (Net Force) Particle Model

Instructional Goals

1. The amount by which the forces acting on an object are unbalanced is called the net force.

2. When the forces acting on an object are unbalanced, the object will accelerate. Because

acceleration is a change in velocity, and velocity includes both speed and direction, a net force

will change the speed and/or the direction of an object's motion.

3. Newton's 2nd Law:

The acceleration of an object is directly proportional to the net force acting on it,

a Fnet , and

inversely proportional to its mass,

a 1

m. While this relationship is usually expressed as

Fnet ma, we prefer to use

a Fnet

m because it more explicitly relates the dependent variable, a,

to the independent variables responsible for any change.

4. Force is measured in units of newtons. A one newton net force acting on a one-kilogram object

produces an acceleration of 1 m/s2. Therefore, a newton is the same as a kilogrammeter/second2.

(N = kgm/s2)

5. Use Newton's 2nd Law to qualitatively describe the relationship between m and a, F and a, m

and F. (For example, if you double the mass, the acceleration will be ½ as great.)

6. Solve quantitative problems involving forces, mass and acceleration using Newton's 2nd Law.

a. use force diagram analysis to find the net (unbalanced) amount of force.

b. list known and unknown force and motion variables:

force variables motion variables

acceleration mass

net force

acceleration initial velocity

final velocity

change in time

displacement

equation

a Fnet

m

equations

vf at v

i

x 12 at 2 v

it

vf

2 vi

2 2ax

c. The variable that ties both lists of variables together is acceleration. Depending on the

variables you know, use either the force or motion mathematical models to solve for

acceleration, then use the acceleration value to solve for the unknown quantity.


7. For a given pair of surfaces, the friction force is generally some fraction of the normal force,

where that fraction is called the coefficient of friction, (greek letter mu).

Therefore, Ffriction = FN. Every pair of surfaces has its own coefficient of friction.

Sequence

1. Pre-lab activity Human dynamics cart and acceleration

2. Modified Atwood's Machine lab

3. Elevator forces Lab

4. Friction Lab

5. Worksheet 1: Force Diagrams and Net Force

6. Worksheet 2: Newton's second law

7. Quiz 1: Modified Atwood's Machine Lab (Newton's Second Law relationships)

8. Worksheet 3: Kinematics & Newton's second law

9. Sliding on Slippery Roads Lab

10. Worksheet 4: Newton's 2nd Law and Component Forces

11. Quiz 2: Newton's second law and quantitative force diagrams

12. Worksheet 5: Newton's 2nd Law and Friction

13. Lab Practicum

14. Newton's Second Law Review Problems

15. Unit 5 test

Overview

In this unit, students learn the second half of Newton's Modeling Cycle:

a) from changes in velocity we infer unbalanced forces

b) from unbalanced forces we deduce changes in velocity

Changes in Velocity

Unbalanced Forces

infer

deduce


Students should be able to correctly describe the kinematic behavior of an object from the force

diagram. Typically, students are expected to be able to determine the net force, then the value of

the unknown applied force from a description of the object's kinematic behavior.

In the deployment worksheets reinforce the practice of drawing force diagrams as the first step in

preparing a solution to the problem. Make sure that these diagrams faithfully represent the forces

(long-range and contact) that act on the object. In whiteboard presentations try to induce students

to recognize multiple approaches to problems dealing with systems that consist of more than one

object.

Instructional Notes

2. Modified Atwood's Machine Lab Apparatus

dynamics carts and tracks

or glider

air tracks

or wheeled carts

wood ramps

pulleys with clamps

balance for mass measurement

10-50 g hooked masses or

slotted masses with hanger

0.2kg – 0.5 kg masses

spring scales

Motion detectors or photogates and flags/picket fence

Force sensor

Logger Pro Software (Vernier) or Data Studio Software

(PASCO)

Pre-lab discussion • Allow a suspended mass to tow a cart (glider) across the track; ask students to observe its motion.

We've already established that unbalanced forces produce accelerated motion – we just haven't

quantified the relationship. Rather than brainstorming general observations, ask students to identify

other factors that might affect the acceleration of the cart. To proceed, the list must include total

system mass, amount of friction, and amount of force used to tow cart.

Motion detector

cart

Hanging

weight

string

pulley

incline the track slightly to

balance friction

force sensor


• Ask the students for ideas to minimize the effect of friction. After some discussion, they will suggest

inclining the ramp slightly to compensate for friction. When properly inclined, a small push to the

cart will allow it to travel at constant speed toward the pulley.

• Ask the students how to measure the acceleration of the cart. Several techniques could be used

depending on the equipment you have available. Using either a motion detector or a photogate with a

cart picket fence, students can obtain a v-t graph, the slope of which is the acceleration.

• The dependent variable is the acceleration of the cart.

• There are two independent variables (the mass of the cart and the tension force acting on the cart) so

two experiments are needed.

Experiment 1: System acceleration vs. net force. The cart mass must be held constant and

recorded. When a force sensor is used, additional masses can be added to the hanger to increase

the towing force.1 Each group should use a different cart mass to make the analysis more robust.

Experiment 2: System acceleration vs. mass. The net force must be held constant and recorded,

while the mass of the cart is increased by adding additional lab masses. Again, if each group

uses a different hanging weight (which provides the towing force) the analysis becomes more

robust.

Lab performance notes • Use small mass hangers (e.g. 5 g) and change by 10 to 20 g increments.

• Increase cart mass by 0.2 - 0.5 kg increments.

• Adjust the angle of incline so that the cart can move at a constant speed with a very small initial

push.

• F-t and v-t graphs are shown below. Be sure to record the average force during the interval the cart is

moving. The slope of the v-t graph yields the acceleration

• Record at least five values of force and acceleration in experiment 1.

• In experiment 2, six or more values of mass should be used.

1 If a force sensor is not used, and the weight of the hanging mass is assumed to be the towing force, then the lab masses

used to vary the towing force must start on the cart and be transferred one at a time to the hanger in order to vary the force, while keeping the system mass constant. See diagram at end of the modified Atwood’s machine section.

accele

ration

force

accele

ration

1/mass

accele

ration

mass

Fig 1 Fig 2 Fig 3


Post-lab analysis and discussion • Students should obtain graphs for a vs. F like the one shown in Fig 1. For the second experiment,

they will need to linearize their graph of a vs. m (Figures 2 and 3).

• This lab is a good opportunity for “megaboarding” results: Divide the main board into a grid as

shown below. Each lab group fills out a row with units according to the column labels. Patterns in

the data are much easier to see in this format.

Exp. #1 acceleration vs. net force Exp. #2 acceleration vs. mass

Lab group Constant

total mass

Slope of

graph

Equation of

graph

Constant

Fnet

Slope of

linearized

graph

Equation for

graph

When all of the class data is posted, ask the students “what’s the pattern?”

1. The most obvious pattern is that in experiment #2, the net force and the slope of the graph have the

same numerical values. This is reasonable because the slope of a graph is usually some function of

the system variable held constant during the experiment. The units are different, however. Force is

measured in newtons, while the units of the slope are in kgm/s2. To reconcile the difference we

could tentatively propose that the unit of the newton is equivalent to units of kgm/s2.

If the slope is equal to the net force, the equation can be generalized:

acceleration = net force×1

system mass+ 0 or

a Fnet 1

m

2. With some prompting, students may recognize in experiment #1 that as the system mass increases,

the slope of the graph decreases. As before, remind students that the slope should be some function

of the variable held constant. Ask students to take the reciprocals of their cart masses and compare

the results to their slopes. Numerically, the values will be the same, but again, the units need to be

reconciled. An analysis of the units of the slope shows that they are equivalent to those of 1/mass.

m

s2

N

m

s2

kg m

s2

1

kg

Therefore, the equation can be generalized:

acceleration =1

system mass× net force + 0 or

a 1

m Fnet

We conclude that

a Fnet 1

m and define a newton to be equivalent to a

kg m

s2 .

Note: if a force sensor is not used and the weight of the hanging mass is presumed to be the net

force, then the system mass (cart and hanging masses) must be considered, as the towing force

accelerates both.


Note: In the resources folder is a follow-up lab in which students are asked to test whether the

equation they have developed in this lab still applies when the cart is placed on an inclined ramp.

3. Elevator Forces Lab

Three versions of this activity are provided in the unit folder.

02a U5 ElevatorLab-actual can be done in schools with easy access to an elevator. It provides

students kinesthetic reinforcement of the difference between weight and normal force.

In 02b U5 ElevatorLab-movies, students can do a similar analysis of a QuickTime movie (Elevator-

cues) which can be downloaded from the link provided on the Physics:Mechanics page at the AMTA

website. Links to two additional movies which can serve as extensions are also provided.

The third option is 02c Feelings in the Elevator, which draws on student recollections and allows

students to test them by raising and lowering a weight attached to a scale or sensor. Teachers who

use 02b can have their students do the 2nd

part of 02c so that their students can sense changes in the

force on the "elevator" for themselves. Below is a Logger Pro graph of force vs. time for the

hanging weight as it starts motionless (2s), then is accelerated briefly, moves at constant velocity

upwards (3.5s), then slows to a stop.

Motion detector

cart

Hanging

weight

string

pulley

incline the track slightly to

balance friction

Lab masses

©Modeling Instruction 2013 1 U5 Net Force – elevator lab v3.1

Name

Date Pd

Net Force Particle Model:

Elevator Lab

In this activity you will analyze the forces acting on a person riding in an elevator.

Before you watch the video clip answer the following questions:

1. Describe the times in the elevator when you feel your “normal” weight.

2. Describe the times in the elevator when you feel heavier than your “normal” weight.

3. Describe the times in the elevator when you feel lighter than your “normal” weight.

Activity: Watch the video clip: Elevator-cues. Record the scale readings you see.

Force (pounds) Force (newtons) (1 pound = 4.5 Newtons)

Scale reading at rest: _______________ _______________

Maximum scale reading: _______________ _______________

Minimum scale reading: _______________ _______________

Label the following as equal to, greater than, or less than the scale reading at rest.

_____________ At rest at the bottom

_____________ Starting to go up

_____________ Going up at constant speed

_____________ Slowing to stop at the top

_____________ Stopped at the top

_____________ Starting to go down

_____________ Going down at constant speed.

_____________ Slowing to stop at the bottom.

Calculate the mass of the person on the scale in kilograms: _____________________


Force Analysis: Draw a quantitative force diagram for the passenger in each of the

following situations during the elevator ride. Label the forces in newtons. To the right of

each diagram draw a velocity and acceleration vector that describes the motion of person

in the elevator. Calculate the net force and the acceleration of the person.

1. At rest at the bottom

Quantitative force diagram

velocity vector:

acceleration vector:

net force =

acceleration =

2. Starting to go up


velocity vector:


net force =

acceleration =


3. Going up at constant speed


velocity vector:


net force =

acceleration =

4. Slowing to stop at the top


velocity vector:


net force =

acceleration =

5. Stopped at the top


velocity vector:


net force =

acceleration =

6. Starting to go down


velocity vector:


net force =

acceleration =


7. Going down at constant speed.


velocity vector:


net force =

acceleration =

8. Slowing to stop at the bottom.


velocity vector:


net force =

acceleration =

9. How do the upward accelerations compare to the downward accelerations? Explain why.

Extension:

Watch the video clips Elevator-1 and Elevator-2. From changes in the scale readings during the

rides, determine whether the elevator was ascending or descending in each clip. Justify your

conclusions.

Preliminary Investigation: Determine the relationship of the force applied to a spring and the distance the spring stretches

We begin by doing a lab where students determine the relationship of the force applied to a spring and

the distance the spring stretches. Students gather their data and graph it on the computer. After that

lab the students share their results on whiteboards. They have to figure out why the graphs have

different slopes. After examining different springs used in the lab they notice that the springs with

larger slopes are more difficult to stretch. They come to the conclusion that the slope of the graph has

something to do with how strong the spring is. We then name the slope the spring constant and give it

the variable “k”. In further discussion they are asked what the spring might contain after it was

stretched that was not there before it was stretched. Usually someone says energy. I tell them the unit

of energy is the newton meter and ask them how they could use their graph to get a quantity with this

unit. From their study of motion students are familiar with obtaining values using the area of a graph.

Therefore it is not surprising for a student to propose the area of the spring graph would give us newton

meters. I relay to the students that yes, the area of a force-position graph is the called the energy

transferred by an external force moving something through a distance. (In this case an external force is

moving the end of the spring the distance stretched.) We go about analyzing the area of the spring

graph and notice that Fx/2 can also be calculated as ½ kx2. If the spring has a preload we add the area of

the rectangle under the triangle. We name this energy account elastic potential energy or simply elastic

energy. We are now ready to do an investigation of energy.

Students gathering data to determine relationship of force vs. stretch. The data taking can also be done

with the spring horizontal and stretching it by pulling the spring with a dual force sensor.

Spring Energy

The area of the blue triangle is the change in the force times the stretch distance

divided by two: , however therefore: .

The area of the brown rectangle is the intercept times the stretch distance:

Graph of force vs stretch

The total energy is ½kx2 + bx

If your spring has no preload the energy is ½kx2

If was doing this lab with general physics kids I would make sure the springs I was using begin stretching with a very small force. To do this if your spring links press against each other you gently stretch the spring until they do not. Then the intercept on the graph will be zero. With my honors kids I don’t mind having a preload for the spring.

FxE =

2e

F=kx

212

E = kxe

E = bxe

Investigation : Determine the relationship of the energy stored in a spring and the velocity of a cart.

Stephanie is demonstrating how the equipment is set up for the energy in spring vs. velocity of cart lab.

In this investigation the students hook a spring to a cart, pull the cart back to stretch the spring and let

go. After the spring has pulled the cart and has no tension in it, the cart goes through a photogate

where the time for a flag to pass through the photogate beam is measured by the computer. Students

then calculate the energy stored in the spring before release and the velocity of the cart at the end.

Next they graph energy vs. velocity and get a top opening parabola. Students have learned that to

linearize a top opening parabola they need to square the x-variable, in this case the velocity. This

produces a linear graph so the students are now able to write an equation for the lab and state the

relationship. Their statement is the energy is proportional to the square of the velocity. After they write

the equation they then try to determine what the slope represents. By this lab students have become

adept at doing this. When they simplify the units of the slope they find the units simplify to kg. They

look at their lab and realize that they kept the mass of the cart constant and it was measured in kg. It’s a

little tricky but with a bit of thinking they usually come up with the slope being one half the mass of the

cart. They now have the general equation E = ½ mv2.

photogate

spring string Cart with blocker

The next step in the process is for the students to place what they discovered in the lab on a

whiteboard. (Above is an example of a completed whiteboard for this lab.) They get their thoughts

together as they make the whiteboard and then present the board to their classmates. The teacher’s

job in this presentation is to assist them by asking questions if they have not figured out everything, and

to put names to what they have discovered. In this lab I ask the students, “Where is the energy that

started out in the spring?” When they reply in the moving car, I tell them that we call the energy

account for moving objects is called kinetic energy. To check for their understanding I ask them how

could they do the lab again and get a different slope. Would this change make the slope larger or

smaller? In the lab setup each group was given a cart with a different mass and students had springs of

two different strengths. The springs made no difference in the slope and the mass of the cart was twice

the slope no matter what the mass was.

Another picture of setup

Actual whiteboard of this lab from an honors class.

Investigation 2: Determine the relationship of the energy stored in a spring and the change in height.

In this investigation the students make a ramp out of a track, and hook a spring to a cart and the top of

the track. (See picture above) They place the cart at a position where the spring is not stretched, but is

about to and release the cart. They measure the distance the spring stretched when it stopped the cart

and the height of the cart before and after release. They do seven more trials, each time with the track

at a different angle. Students then calculate the energy stored in the spring at maximum stretch. They

then graph energy vs. change in height and obtain a linear graph.

The students then prepare a whiteboard to share their results (Picture of an actual whiteboard is on the

next page.). They are able to make a statement that the energy is directly proportional to the change in

height. They write the equation of their lab. They then analyze the slope and see that it has the unit

newtons so it represents a constant force. The only constant force in their lab close to their slope is the

weight of the cart. They now have a general equation of E = Wh. In the whiteboard session we name

the energy account gravitational potential energy or more simply, gravitational energy.

These two investigations also allow the teacher to introduce conservation of energy. In the first

investigation the energy moved from the spring to the cart and the two are equal. In the second

investigation the energy moved from the gravitational field to the spring and the amount was equal.

©Modeling Instruction - AMTA 2013 1 U8 Energy - ws 3 v3.1

Name

Date Pd

Energy Model Worksheet 3:

Qualitative Energy Storage & Conservation with Bar Graphs

For each situation shown below:

1. List objects in the system within the circle. **Always include the earth’s gravitational field in your system.

2. On the physical diagram, indicate your choice of zero height for measuring gravitational energy.

3. Sketch the energy bar graph for position A, indicate any energy flow into or out of the system from position A to

position B on the System/Flow diagram, and sketch the energy bar graph for position B.

4. Write a qualitative energy equation that indicates the initial, transferred, and final energy of your system.

1a. In the situation shown below, a spring launches a roller coaster cart from rest on a

frictionless track into a vertical loop. Assume the system consists of the cart, the earth, the

track, and the spring,

1b. Repeat problem 1a for a frictionless system that includes the cart, the earth, and the track,

but not the spring.

1c. Use the same system as problem 1a, but assume that there is friction between the cart and

the track.

Position A

En

erg

y (

J)

0

Ek Eg Eel

Position B

En

erg

y (

J)

0

Ek Eg Eel Eth System/Flow

Qualitative Energy Conservation Equation:

A

B

Position A

En

erg

y (

J)

0

Ek Eg Eel

Position B

En

erg

y (

J)

0



A

B

Position A

En

erg

y (

J)

0

Ek Eg Eel

Position B

En

erg

y (

J)

0



A

B


1d. This situation is the same as problem 1a except that the final position of the cart is lower on

the track. Make sure your bars are scaled consistently between problem 1a and 1d. Assume

the system consists of the cart, the earth, the track, and the spring.

2a. A moving car rolls up a hill until it stops. Do this problem for a system that consists of the

car, the road, and the earth. Assume that the engine is turned off, the car is in neutral, and

there is no friction.

2b. Repeat problem 2a for the same system with friction.

3a. A person pushes a car, with the parking brake on, up a hill. Assume a system that includes

the car, the road, and the earth, but does not include the person.

Position A

En

erg

y (

J)

0

Ek Eg Eel

Position B

En

erg

y (

J)

0



A

B

y

yA = 0 vA > 0

yB > 0 vB = 0 A

B

Position A

En

erg

y (

J)

0

Ek Eg Eel

Position B

En

erg

y (

J)

0



y

yA = 0 vA > 0

yB > 0 vB = 0 A

B


Position A

En

erg

y (

J)

0

Ek Eg Eel

Position B

En

erg

y (

J)

0


y

hA = 0 vA = 0

hB > 0 vB = 0 A

B

Position A

En

erg

y (

J)

0

Ek Eg Eel

Position B

En

erg

y (

J)

0




3b. Repeat problem 3a for a system that includes the person.

4a. A load of bricks rests on a tightly coiled spring and is then launched into the air. Assume a

system that includes the spring, the bricks and the earth. Do this problem without friction.

4b. Repeat problem 4a with friction.

4c. Repeat problem 4a for a system that does not include the spring.

y

hA = 0 vA = 0

hB > 0 vB = 0 A

B

Position A

En

erg

y (

J)

0

Ek Eg Eel

Position B

En

erg

y (

J)

0



y

A

B

hA = 0 vA = 0

hB > 0 vB > 0

Position A

En

erg

y (

J)

0

Ek Eg Eel

Position B

En

erg

y (

J)

0


Energy Equation:

y

A

B

hA = 0 vA = 0

hB > 0 vB > 0

Position A

En

erg

y (

J)

0

Ek Eg Eel

Position B

En

erg

y (

J)

0


Energy Equation:

y

A

B

hA = 0 vA = 0

hB > 0 vB > 0

Position A

En

erg

y (

J)

0

Ek Eg Eel

Position B

En

erg

y (

J)

0


Energy Equation:


5a. A crate is propelled up a hill by a tightly coiled spring. Analyze this situation for a

frictionless system that includes the spring, the hill, the crate, and the earth.

5b. Repeat problem 5a for a system that does not include the spring and does have friction.

6a. A bungee jumper falls off the platform and reaches the limit of stretch of the cord. Analyze

this situation for a frictionless system that consists of the jumper, the earth, and the cord.

6b. Repeat problem 6a if the cord is not part of the system.

y

A

B

hA = 0 vA = 0

hB > 0 vB > 0

0 Energy Equation:

Position A

En

erg

y (

J)

0

Ek Eg Eel

Position B

En

erg

y (

J)

0


y

A

B

hA = 0 vA = 0

hB > 0 vB > 0

0 Energy Equation:

Position A

En

erg

y (

J)

0

Ek Eg Eel

Position B

En

erg

y (

J)

0


y A

B

hA > 0 vA = 0

hB > 0 vB = 0

0

y

0

B Position A

En

erg

y (

J)

0

Ek Eg Eel

Position B

En

erg

y (

J)

0


Energy Equation:

Energy Equation:

y A

B

hA > 0 vA = 0

hB > 0 vB = 0

0

y

0

B

Position A

En

erg

y (

J)

0

Ek Eg Eel

Position B

En

erg

y (

J)

0


Energy Equation:

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