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PHYSICS UNION MATHEMATICS
Physics II
Work & Energy
Student Edition
Supported by the National Science
Foundation (DRL-0733140).
PUM Physics II
Work & Energy
Most of the module activities were adapted from:
A. Van Heuvelen and E. Etkina, Active Learning Guide,
Addison Wesley, San Francisco, 2006.
Used with permission.
Contributions of: E. Etkina, T. Bartiromo, M. Blackman, A. Bourdeaux, S.
Brahmia, S. Kanim, H. Briggs, C. D’Amato, J. Flakker, J. Finley, J. Goett,
H. Lopez, R. Newman, J. Santonacita, E. Siebenmann, C. Soni, R. Therkorn,
K. Thomas, M. Trinh, A. Van Heuvelen, R. Zisk.!
This material is based upon work supported by the National Science Foundation under Grant
DRL-0733140. Any opinions, findings and conclusions or recommendations expressed in this
material are those of the authors and do not necessarily reflect the views of the National Science
Foundation (NSF).
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© Copyright 2014, Rutgers, The State University of New Jersey.
Table of Contents
LESSON 1: HOW DO WE EVER GET ANYTHING DONE AROUND HERE? ........ 4!
LESSON 2: HOW AM I SUPPOSED TO KEEP TRACK OF IT? ............................... 13!
LESSON 3: REASONING WITH ENERGY BAR CHARTS........................................ 19!
LESSON 4: SUCH GREAT HEIGHTS............................................................................ 31!
LESSON 5: GALILEO’S PENDULUM ........................................................................... 39!
LESSON 6: HOW TO CALCULATE KINETIC ENERGY .......................................... 41!
LESSON 7: THE ENERGY IN A SLINGSHOT AND OTHER PRACTICAL………45!
LESSON 8: SPRING INTO ACTION .............................................................................. 51!
LESSON 9: CALCULATING THE INTERNAL ENERGY CHANGE ....................... 54!
LESSON 10: POWER UP .................................................................................................. 58!
LESSON 11: PRACTICE & REVIEW ............................................................................ 61!
LESSON 12: WHEN WORK IS NOT EASY .................................................................. 66!
LESSON 13: OH BABY, DON’T LET ME GO .............................................................. 71!
SUMMARY: DEFINITIONS AND PRINCIPLES ......................................................... 73!
LESSON 14: SIMPLE MACHINES I .............................................................................. 75!
LESSON 15: SIMPLE MACHINES II ............................................................................. 80!
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Lesson 1: How Do We Ever Get Anything Done Around Here?
1.1 Observe and find a pattern
In these three experiments, we will study the ability for a group of objects to smash a piece of
chalk.
a) Consider a 1-kg block with a flat bottom with a string attached to the top and Earth as
your system. There is also a piece of chalk sitting below the block. You pull up on the
string so that the 1-kg block slowly rises 0.5 m above the piece of chalk. After this lifting
process, you release the block. It falls and breaks the chalk.
1"kg"
Lift"
1"kg"
chalk"
b) Consider a 1-kg dynamics cart that can roll on a low-friction horizontal dynamics track as
your system. There is also a piece of chalk that is taped to the fixed, vertical end of the
track. You push the cart so that it rolls faster and faster toward the chalk at the end of the
dynamics track and the cart breaks the chalk when it hits it.
wall"
Push"
wall"
c) Now consider a slingshot that holds a piece of chalk as your system. You slowly pull
back on the sling. When you release the sling, the chalk shoots out at a high speed and
hits the wall, causing the chalk to break.
Pull"
wall"
wall"
Complete this table:
Experiment
Draw an arrow indicating the direction of
the external force you exerted on each of the
system objects that you studied (
).
a)
b)
c)
Draw an arrow indicating the displacement
of the system object while you were
exerting the force ( ).
d) Look for a pattern of what was done to the systems that we studied to give them the
chalk-smashing potential. Then, devise a new physical quantity to describe this pattern.
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1.2 Observe and find a pattern
Now, suppose that a friend decides to save the chalk in the first two experiments by exerting,
with her hands, an opposing force on the block or on the cart after they are released. In each case,
she pushes on the moving object opposite to the direction of its velocity. Below, give the
direction of the external force your friend exerts on the moving object relative to its displacement
as she stops it, thus causing the system to lose its potential to break the chalk.
a) After lifting the block, you release the block and it starts falling. Your friend then starts
pushing upward on the falling block, slowing it down, and the block does not break the
chalk.
b) You push the cart so that it rolls faster and faster. You then stop pushing. Just before the
cart reaches the chalk, your friend pushes it in a direction opposite to its direction of
motion. This causes the cart to slow down and stop so that it does not break the chalk.
Complete this table:
Experiment
Draw an arrow indicating the
direction of the external force your
friend exerted on the system object
that you studied (
).
a)
b)
Draw an arrow indicating the
displacement of the system object
while your friend was exerting the
force ( ).
c) How could you modify the definition of the quantity you devised in the previous activity
to account for the system’s loss of the chalk-breaking potential due to your friend’s
intervention?
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1.3 Observe and find a pattern
Consider Earth and a 1-kg block as your system.
a) You hold a string tied to a block so that it stays about 1 cm
above a table. A piece of chalk is placed on the table under
the block. If you release the block and it falls on the chalk,
the chalk will not break (it’s too close to the chalk).
You are holding
the string
"Motion
Next you slowly walk about 2 m beside the table, continually
keeping the block 1 cm above the surface. After walking the
2 m, the block hangs over a second, identical piece of chalk.
Draw the force exerted by the string on the block and the
displacement of the block as you walked the 2 m.
b) Discuss whether the vertical force the string exerted on the
block while moving it horizontally above the tabletop caused Earth and block to have a better
chance of breaking the second piece of chalk than the first piece.
c) Revise the quantity you devised in the last two activities to account for this result. Your
revision will involve the angle between the external force exerted on the system and the
system object’s displacement. We call this quantity work.
1.4 Observe and find a pattern (if you know trigonometry)
a) Consider a 1-kg dynamics cart a system. You pull it at angle θ causing it to roll on a lowfriction horizontal dynamics track toward a piece of chalk that is taped to the fixed,
vertical end of the track. You pull the cart so that it rolls faster and faster toward the
chalk at the end of the dynamics track and breaks the chalk when it hits it. Draw the force
exerted by you on the cart and the displacement of the cart while you were pulling it.
"
θ"
"
θ"
b) Discuss whether the angled force exerted on the cart while moving it horizontally gave it
a better chance of breaking the piece of chalk than the force exerted in activity 1.1 part
(b).
c) What trigonometric function would help you determine the system’s increase in chalksmashing ability? Is this consistent with the increase, decrease, and no change in chalksmashing potential for activities 1.1-1.3?
d) Revise the quantity you devised in the last three activities to account for this result. Your
revision will involve the angle between the external force exerted on the system and the
system object’s displacement. We call this quantity work.
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Students!familiar!with!trig,!proceed!to!page!8;!those!who!are!not,!continue!here.!
Work done by a constant force
System
System
"
"
"
A woman pulls a box upwards.
Since the force exerted by the woman on
the box is in the same direction as the
displacement,
"
"
"
r
d"
W = FWoman On Box × d "
"
r
d"
A woman lowers a ball downwards.
Since the force exerted by the woman on
the ball is in the opposite direction as the
displacement,
"
W" = − FWoman On Ball × d "
"
"
"
System
r
d
"
A woman carries a box while walking at a
constant pace. Since the force exerted by
the woman on the box is perpendicular to
the displacement,
"
Did You Know?
Work done by a constant force on an object is equal to the product of the force magnitude and
displacement magnitude with a positive sign (W = FW on B *d) if the force and displacement are
in the same direction and the product of the force and displacement with the negative sign (W =
-FW on B * d) if they are in the opposite direction. If the force and displacement of the object are
perpendicular to each other, the force does NO work!
1.5 Regular Problem
Jeff did 573 J of work on a sled. He pulled the sled for a distance of 30 m. Represent the sled as a
box. Draw the direction of the force Jeff exerts on the sled and the direction of the sled’s
displacement. What is the average force that he exerted on the system?
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1.6 Regular Problem
Steve slowly lifts a 20 kg barbell 1 meter vertically. Represent the barbell as a box. Draw the
direction of the force Steve exerts on the barbell and the direction of the barbell’s displacement.
How much work does he do on the barbell? What assumptions did you make?
1.7 Regular Problem
If Natasha slows a moving grocery cart by pulling on it exerting a force of 23 N over 2.3 m, what
will be the work she does on it? Represent the cart as a box. Draw the direction of the force
Natasha exerts on the cart and the direction of the cart’s displacement. What assumptions did you
make?
Here’s an Idea!
In problem 1.7 Natasha did negative work on the box. Notice that when the force exerted on the
object and the object’s displacement are in the opposite directions, the force does NEGATIVE
work!
1.8 Regular Problem
Jessica, at a constant slow speed, moved a 1 kg book from a 2 m high shelf to the floor.
Represent the book as a box. Draw the direction of the force Jessica exerts on the book and the
direction of the book’s displacement. How much work did she do on the book?
1.9 Reason
Dylan is trying to lift his bike into the back of his pick-up truck of height h. He attaches a light
string to it and pulls it vertically upward. If the bike is the system, what kind of work do Dylan
and Earth do on the bike?
i. Dylan does positive work on the bike. Earth does positive work on the bike.
ii. Dylan does positive work on the bike. Earth does negative work on the bike.
iii. Dylan does negative work on the bike. Earth does positive work on the bike.
iv. Dylan does negative work on the bike. Earth does negative work on the bike.
1.10 Reason
Dylan can lift the bike either by pulling the string vertically upward or along a frictionless
inclined plane (see Figure). In both cases he exerts exactly the same magnitude force on the bike.
Which of the following statements is true?
(1)
h"
(2)
h"
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i. The work done by the string on the bike is the same in both cases.
ii. The work done by the string on the bike is larger in case (1) than case (2).
iii. The work done by the string on the bike is smaller in case (1) than case (2).
1.11 Reason
As Dylan is deciding how to lift his bike into the back of his pick-up truck, he also wants to
know how much work Earth does on his bike if Earth is not in the system.
i. The work done by Earth on the bike is the same in both cases.
ii. The work done by Earth on the bike is larger in case (1) than case (2).
iii. The work done by Earth on the bike is smaller in case (1) than case (2).
iv. Earth does not do any work on the bike.
"
1.12 Relate
Homework
Describe a situation when you have done:
a) +1 J of work on a system.
b) -1 J of work on a system.
c) 0 J of work on a system.
1.13 Regular Problem
While working out, a man lifts a 10-kg object a vertical distance of 0.80 m. He then carries it for
10 m where he sets it down a vertical distance of 0.80 m. How much work does he do on the
object when he picks the object up, when he carries it, and when he sets it back down? What is
the total work that he does on it?
1.14 Observe and Explain
In another situation, you stretch a block-spring system and then release the block. The block
slides toward the wall and smashes a piece of chalk. Label whether the ability of the blockspring-wall system to crush the chalk increases, decreases, or remains the same between each
step.
v"
"
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Is this process consistent with the pattern we observed today between the net force exerted on an
object and the displacement of the object?
1.15 Regular Problem
Sara drops a 1 kg ball from the top of a cliff. It falls vertically with a constant acceleration (9.8
m/s2). What object (if any) exerts a force that does work on the ball (system)? How much work
has been done on the ball if it falls 10 m? Sketch the situation. Draw the direction of the force
exerted on the ball and the direction of the displacement.
1.16 Regular Problem
Minh is riding the elevator up to the top floor of a shopping mall. He moves with a constant
velocity from the ground level to the floor above (5 m high). Estimate the work done on Minh by
the elevator. To calculate the force exerted on Minh, choose him as the system and draw a force
diagram.
1.17 Regular Problem
Shelly is now riding up the same elevator as Minh in the previous problem. She is going to the
fifth floor so as the elevator goes from the ground level to the first floor, it accelerates at a rate of
1.0 m/s2. Estimate the work done on Shelly by the elevator during this part of the trip if her mass
is 2/3 the mass of Minh. To calculate the force exerted by the elevator on Shelly, choose her as
the system and draw a force diagram.
1.18 Regular Problem
As Shelly goes from the fourth floor to the fifth floor the elevator slows to a stop from 2.0 m/s.
Estimate the work done on Shelly by the elevator as it goes from the fourth to fifth floor. To
calculate the force exerted by the elevator on Shelly, choose her as the system and draw a force
diagram.
1.19 Evaluate
Heather, Minh and Danny are arguing about the units of work. Minh says that the units of work
are poundsXinches, Heather thinks that the units of work are in Joules, and Danny thinks they
are NewtonsXmeters. Which of them is right? Explain your answer. What are your assumptions?
1.20 Regular Problem
Kate is waiting in the security line at the airport and has to keep picking up her suitcase, moving
forward, and setting it down again. Calculate the work done by Kate on her 15-kg suitcase when
(a) slowly lifting it 0.80 m upward, (b) when holding it and slowly moving forward 1.0 m, and
(c) when lowering it 0.80 m down again.
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Work-Trigonometry Section: In this section, students will use trigonometry to express
work done for a more general range of situations.
Work
"
System
"
"
"
"
θ
"
!
"
"
W"="Fp"on"s"cos(θ)"d"
"
Magnitude of displacement—always
positive"
Magnitude of force—always positive"
"
Angle between
" !
and"
Did You Know?
Work W: Work is a physical quantity that is equal to the product of the magnitude of the
average force FEx on O that an external environmental object exerts on a system object, the
magnitude of the system object’s displacement d, and the cosine of the angle between
FEx on O and d.
W = (FEx on O cos θ) d
1.21 Regular Problem
Suzanne is pulling a sled up a hill that makes a 24° angle with the horizontal. She keeps the rope
parallel to the hill and exerts a 150-N force on it. How much work will she do if she pulls the
sled 150 m?
1.22 Regular Problem
A 4 kg grocery cart rolls down a 3 m long incline with an angle of 10°. How much work does
Earth do on the cart?
1.23 Regular Problem
Juan pushes a box at an angle to the horizontal, doing 250 J of work over a distance of 10 m. If
the force exerted is 30 N, what is the angle between the force exerted by Juan on the box and the
horizontal?
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1.24 Regular Problem
To clean the floor, David exerts a 40 N force on a broom handle to push it 2 m. If the broom
handle makes a 40° angle with the floor, what is the work done by David on the broom? If the
broom handle were angled at 65° would David do more or less work? Explain your answer.
1.25 Regular Problem
Tara pushes a baby carriage at an angle 35° to the horizontal over a distance of 20 m on her way
to the playground. If she does 600 J of work on the carriage, what force does she have to exert
on the carriage?
1.26 Regular Problem
Komila is driving her car alongside a river. Theo is water skiing in the river and holding onto a
rope attached to Komila’s car. The force exerted by the rope on Theo is 400 N, and the rope
makes an angle of 15° with the river. Calculate the work done by the rope in pulling Theo 300 m
up the river. Assume he does not fall into the water.
1.27 Regular Problem
Dustin exerts 200 N of force on a box sitting on a horizontal surface. He does not do any work
on the box. How is this possible? Explain your answer.
Reflect: How does work in physics compare to work in real life? What is the
same? What is different? Ask your relatives what they think the word “work”
means in physics and then try to teach them what you know. What difficulties
did they have? Share them tomorrow in class.
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Lesson 2: How am I Supposed to Keep Track of It?
2.1 Describe
You do work on a system to change its potential to do something (for example, to smash chalk or
to make the touching surfaces of two objects in a system warm). In lesson 1, the work done on
the system by the external force caused different types of changes in the system. Below, we
describe each type of change in the system as a result of the work done on it. Devise a name for
each type of change.
a) The external force caused the block to move higher above Earth’s surface.
b) The external force caused the cart to move faster and faster.
c) The external force caused the slingshot to stretch.
d) The external force caused the surfaces of the touching objects to warm
Did You Know?
These changes are called energy changes. Each type of energy has a formal name: kinetic
energy, gravitational potential energy, and elastic potential energy. All of these fall under a
larger category called mechanical energy. The change in the warmth of the surfaces of touching
objects or their shape is called internal energy of the system.
e) In parts (a) through (d), you came up with names for different types of energy. See if you
can match your answers to the traditional terms in the help box above.
f) Describe the change of energy of a system if someone does positive work on it? Negative
work?
2.2 Design an Experiment
"
Use materials on your desk to show an experiment consistent with each item below. Identify
your system for each experiment.
a) Positive work causes an increase in the gravitational potential energy of the system.
b) Positive work causes an increase in the kinetic energy of the system.
c) Positive work causes an increase in the elastic potential energy of the system.
d) Kinetic energy in the system is converted to gravitational potential energy.
e) Kinetic energy in the system is converted to elastic potential energy.
f) Gravitational potential energy in the system is converted to internal energy.
g) Gravitational potential energy in the system is converted to elastic potential energy.
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2.3 Relate
Describe one real-life situation that is consistent with the processes described below. Identify
your system for each situation.
a) Positive work causes an increase in the gravitational potential energy of the system.
b) Positive work causes an increase in the kinetic energy of the system.
c) Positive work causes an increase in the elastic potential energy of the system.
d) Kinetic energy in the system is converted to gravitational potential energy.
e) Kinetic energy in the system is converted to elastic potential energy.
f) Gravitational potential energy in the system is converted to internal energy.
g) Gravitational potential energy in the system is converted to elastic potential energy.
2.4 Reason
Examine the picture to the right. One of your classmates says, “When
the car gets to the edge it will have ‘the ability to fall’ or ‘falling
ability.’”
a) If Earth weren’t there, would the car still have “ability to fall?”
Explain.
b) Should we include or exclude Earth with the car in the system when we analyze this
problem? How would the analysis be different if we do? If we do not? (use the language
of work and energy for your analysis)
Did You Know?
We have been examining a series of systems and analyzing the changes that occur to them. A
system is an object or group of objects that we are interested in analyzing.
REMEMBER! When we determine the objects in our system, we might need to include objects
that aren’t in direct contact, like Earth.
c) Create a story for what happened to the cart using the language of energies.
d) Decide what to include in your system. How did you decide?
e) Consider the situation. Is there a way this could be the final state of a process? Could it
be the initial state of a process? Explain.
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2.5 Observe and Reason
Lift a box from the floor to a tabletop very, very slowly at a constant velocity. Assume that
during this process you do a total of 125 J of work. (There are no changes in kinetic energy or
internal energy of the system.)
a) Identify the objects included in your system. What is not in your system?
b) Draw a picture of the initial and final states
c) Complete the table below.
Portion of the Process
Work that has
been done so far
Gravitational Potential Energy
of the Box-Earth system
Before you start, the box is on
the floor.
0J
You have lifted the box ¼ of the
way.
You have lifted the box ½ of the
way.
You have lifted the box ¾ of the
way.
You have lifted the box all the
way to the table.
125 J
2.6 Reason
Go to the Physics Teaching Technology Resource website and investigate the experiment under
Newton’s Third Law (forces approach), where Eugenia and David push on each other: Push.
The above video depicts a situation where David and Eugenia are both on roller blades. First,
David pushes Eugenia. Then Eugenia pushes David.
a) Choose David as your system.
b) Describe how the energy of the system changes after each process.
c) Now choose Eugenia as your system. Describe how the energy of the system changes
after each process.
d) Compare your results from David as your system and Eugenia as your system. What
caused the change in energy of each system?
e) What if you chose both David and Eugenia as your system. Where did the kinetic energy
they both acquired come from?
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2.7 Observe and Reason
Pull back the spring on a Nerf gun very, very slowly at a constant velocity. Assume that during
this process there are no changes in kinetic energy or internal energy of the system. You then
point the Nerf gun vertically and release the trigger causing the dart to shoot into the air. The dart
then comes back down and lands on the ground.
a) Identify the objects included in your system. What is not in your system? Make sure to
include all relevant objects.
b) Assume you do 215 J of work in total to compress the spring. Could you determine the
work you do on the spring when the spring has been compressed half of the way? Why
or why not? Consider if the force you exert on the spring is constant.
c) Complete the table below. Draw a picture of each portion of the process. What are your
assumptions?
Portion of the Process
Before you start, the Nerf gun is
sitting in your hand 1m above
the ground.
Picture
Energy of the system
You have pulled the spring back
all the way.
You point the gun vertically and
release the trigger.
The dart flies up 3 m in the air.
The dart falls down and hits the
floor where it stops.
2.8 Observe and Describe
A system consists of a crate and a rough horizontal surface on which it sits (see the illustration
below). The rough surface is made of a special material that changes color when it changes
temperature.
You do positive work on the system by pulling the crate
for about 10 m at a constant velocity. You observe the
colors of the surface change indicating that the
temperature increased.
a) Draw a force diagram that explains why the crate is moving at constant velocity
b) Describe how the system (crate and surface) is different after you do the work than before
the crate started moving.
c) If the ground/surface were not there, would the crate have “warming potential?” Should
we include or exclude the ground as part of our system?
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d) Revisit the choice of a system. Do you want to make any changes? Write down your
system below.
e) You touch the bottom of the box and feel that it also increased in temperature. If the
ground/surface was not in the system, how would you account for this change in energy
of the box?
Homework
2.9 Reason
A hockey puck is sliding along the ice in an ice rink. Consider the puck and the ice as the system.
Jon changes the speed of the puck by exerting a force on the puck-ice system over a distance of
4.0 m. He does -54.0J of work. How has the energy of the system changed? Explain your answer.
i.
The kinetic energy of the system increased.
ii.
The kinetic energy of the system decreased.
iii.
The internal energy of the system increased.
iv.
The internal energy of the system decreased.
2.10 Represent and Reason
Eva is learning how to ski and starts at rest at the top of a hill. While she is skiing down the hill,
she gains speed and notices that the snow melts under her skis.
a) You need to analyze energy transformation during this process. Identify the objects
included in your system. What is not in your system?
b) Draw a picture of the initial and final states
c) Describe the types of energies and if there was work done on your system.
d) Describe any changes of energy of the system from the initial to final state.
2.11 Observe and Explain
a) Complete the table below.
Describe the system.!
Hector lifts a new television
off the ground and places it on
the TV stand.
Identify the
objects that are
part of the
system.!
Identify the initial and final state.!
!
Television
Earth
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!
Jeff starts at the top of a hill
and slides down on his snow
sled. At the bottom of the hill,
Jeff is moving really fast.
1 kg block
A spring
Earth
!
2.12 Reason
Describe a real-life situation in which an external force does the following and state explicitly
whether the system’s energy increases or decreases: Identify the system in each situation.
a) Positive work on a system;
b) Positive work on a system but with a value that is less than in part (a);
c) Negative work on a system;
d) Zero work on the system even though an object in the system moves.
Reflect: Throughout your day, listen to your family and friends and find three
examples of situations where they talk about energy.
a) Ask them what they think energy is.
b) Compare what they say with what you know about energy from your
physics class.
18" PUM"|"Work"&"Energy"|"Lesson"2:"How"am"I"Supposed"to"Keep"Track"of"It?""
© Copyright 2014, Rutgers, The State University of New Jersey.
Lesson 3: Reasoning with Energy Bar Charts
3.1 Represent and Reason
a) How can we use charts to represent data?
b) Create a chart that shows that you have $60 in your bank account, no money in your
pocket, and $20 on a gift card.
c) Imagine that you withdraw $20 from your bank account and put it in your pocket. Create
a new chart that represents your new situation.
d) If we place the two charts side-by-side, how does it express a process? Explain.
Need Some Help?
We can use a bar chart to represent transformations of a
quantity during some process. We do this by placing
the before bar chart next to the after bar chart.
We can also abbreviate the column headings in order
to make this easier to read. We just have to make sure to
include a key so that we know how to reading the
chart. See below.
•
•
•
•
P represents the amount of money in your pocket
CATM represents the amount of money in your ATM card
CGIFT represents the amount of money on a rechargeable Best Buy gift card.
Earn/Spend represents the amount of money that you gain or lose through transaction
with other people
For situations with no energy in a particular column, place a zero (0) to represent a zero bar.
e) What do you notice about the money before and after this process?
Did You Know?
The total amount of money you have remains the same before and after, unless you earn some or
spend some -- right? If the total amount of money you have does not change, we say it is
constant. If it changes in a predictable way due to the expenses, we say that it is conserved, as it
does not appear from nowhere and disappear to nowhere.
PUM"|"Work"&"Energy"|"Lesson"3:"Reasoning"with"Energy"Bar"Charts"" 19"
© Copyright 2014, Rutgers, The State University of New Jersey. "
3.2 Represent
Situation Before and After
Bar Chart Representation
a) You have $15 in your pocket, $60 in your ATM
account, and a gift card with $20 on it. You
withdraw $20 cash from the ATM.
Represent this transaction on the bar chart. Did you
earn or spend any money?
Represent this transaction with a mathematical
statement"
"
b) Next, you buy a snow shovel for $10 cash at
Jones Hardware.
Represent this transaction on the bar chart. Did you
earn or spend any money?
Represent this transaction with a mathematical
statement"
"
c) After returning from the hardware store, you
spend three hours shoveling snow for an old lady
who gives you $20 cash.
Represent this transaction on the bar chart. Did you
earn or spend any money?
Represent this transaction with a mathematical
statement"
"
d) When you are finished shoveling, you spend $20
cash to put gas in your car so you can drive to the
Best Buy.
Represent this transaction on the bar chart. Did you
earn or spend any money?
Represent this transaction with a mathematical
statement"
20" PUM"|"Work"&"Energy"|"Lesson"3:"Reasoning"with"Energy"Bar"Charts""
© Copyright 2014, Rutgers, The State University of New Jersey.
"
Situation Before and After
Bar Chart Representation
e) At Best Buy, you purchase a "Cher's Greatest
Hits" DVD Box Set for $40. You empty out your
gift card and use your ATM card to pay for the rest.
Represent this transaction on the bar chart. Did you
earn or spend any money?
Represent this transaction with a mathematical
statement"
"
f) What happens next? Continue the story and make
a graph to match."
"
g) Does this graph show something that could
happen? If not, explain why not. If so, describe a
situation it could match.
"
h) Draw the missing bar.
Write a mathematical statement to match the chart.
Describe a story that could match.
"
PUM"|"Work"&"Energy"|"Lesson"3:"Reasoning"with"Energy"Bar"Charts"" 21"
© Copyright 2014, Rutgers, The State University of New Jersey. "
Situation Before and After
Bar Chart Representation
i) Make a chart to match this mathematical
statement:
$20 + $0 + $0 + $40 = $20 + $40 + $0
Describe a story that could match."
"
j) For each problem, relate your money at the beginning to the money at the end with an
equation.
k) Which of the above follow the principle that money is constant? Conserved?
l) Compare money transfers and energy transfer. What similarities do you see?
m) What property of wealth is illustrated in this activity?
n) What other physical quantities (besides energy) exhibit this property? What quantities do
not?
3.3 Represent and Reason
While working on the following problem, Alan decided that he could represent work-energy
processes with a bar chart similar to the ones we used for money.
Problem: Jessica stands in an elevator on the first floor. The elevator doors close and the
elevator delivers Jessica to the fourth floor. Identify all the changes that occur.
a) Identify the system of interest.
b) Draw pictures of the initial and final states. (Make sure to include a description.)
c) Review the problem and create your own work-energy bar chart.
Need Some Help?
Work-energy bar charts provide a concrete way to represent work-energy processes. In a workenergy bar chart, a bar represents each type of energy initially in the system, as well as the final
energies of the system. If external objects do work on the system (positive or negative), then
there is a bar to represent work.
22" PUM"|"Work"&"Energy"|"Lesson"3:"Reasoning"with"Energy"Bar"Charts""
© Copyright 2014, Rutgers, The State University of New Jersey.
Across the top of the chart, you see several symbols for
different energies…
K – Kinetic energy
Ug – Gravitational potential energy
Us – Elastic or spring potential energy
W – Work
∆Uint – Change in internal energy
(Difference between final and initial)
The i and the f represent initial and final states
We don’t know the exact amount of energy or work usually but we can still make estimates
based on the situation. The column for the work bar is shaded to indicate that it is not a type of
energy but is instead a process involving an interaction between a system object and an object
outside the system.
3.4 Represent and Reason
You are investigating the energy of an oscillating pendulum. Use the materials in front of you to
take observations of the bob at different points along its path. Make sure you observe the bob
swinging for a while before you fill out the table.
a) Complete the tables below to describe all the energy transformations. Be sure to
identify the system in each step.
Initial State
The bob has no
velocity and is at
the highest point
of the swing.
System:
Final State
The bob is at the
bottom of the swing
and is going really
fast.
Construct the Work-Energy Bar Chart
Equation:
PUM"|"Work"&"Energy"|"Lesson"3:"Reasoning"with"Energy"Bar"Charts"" 23"
© Copyright 2014, Rutgers, The State University of New Jersey. "
Initial State
The bob is at the
bottom of the swing
moving really fast.
Final State
The bob gets to the
top of the swing
and slows down.
System:
The bob has
intermediate
velocity moving to
the left.
Equation:
The bob is at the
bottom of the
swing and is going
really fast to the
left.
System:
The bob has come
to a stop at the top
of the swing’s
motion on the left
begins to swing
forward.
System:
Construct the Work-Energy Bar Chart
Equation:
After a few swings
the bob eventually
comes to a stop at
the center of the
swing.
Equation:
a) Describe how the system’s energy has changed in each step the bob swung back-andforth.
b) How has the total energy of the system changed?
24" PUM"|"Work"&"Energy"|"Lesson"3:"Reasoning"with"Energy"Bar"Charts""
© Copyright 2014, Rutgers, The State University of New Jersey.
c) If the rope from the pendulum is not in the system, how is accounted for in the bar
charts?
3.5 Reason
Adventurous Theo was testing out a stunt cannon. He compressed the spring inside the cannon
and placed himself in the barrel. He then released the spring and was pushed up. Which of the
following graphs accurately represents the situation where Theo and the cannon are the system?
(1)"
(2)"
(3)"
(4)"
a) Which graph accurately represents the situation where Theo and Earth are the system?
b) In which graph(s) is energy constant?
c) In which graph(s) is energy conserved?
Homework
3.6 Bar Chart Jeopardy
In the table that follows, invent a process using words and a sketch (the system, its initial and
final situations, and any work done on the system). Be sure both are consistent with the
qualitative work-energy bar chart shown below.
PUM"|"Work"&"Energy"|"Lesson"3:"Reasoning"with"Energy"Bar"Charts"" 25"
© Copyright 2014, Rutgers, The State University of New Jersey. "
Bar chart for a process.
before"
State what is in your system.
Describe in words one
possible consistent process.
Sketch the process just
described.
State what is in your system.
Describe in words one
possible consistent process.
Sketch the process just
described.
after"
Ki + Ug,i + Us,i + W = Kf + Ug,f + Us,f +∆Uint
+"
"
"""""0"
\"
Relate these quantities mathematically:
Bar chart for a process.
before"
after"
Ki + Ug,i + Us,i + W = Kf + Ug,f + Us,f +∆Uint
A car screeches to a halt, the
tires of the car start smoking.
+"
"""""0"
\"
Relate these quantities mathematically:
3.7 Explain
Read through the following problem and then read Alan’s solution below. Use it and your
response to activity 3.3 to answer the questions.
a) How does your answer compare with Alan’s?
b) Does Alan’s bar chart help him understand the problem? Explain your answer.
c) What do the lengths of the bars represent in Alan’s Work-Energy bar chart?
d) How do you think Alan decided to make the lengths of the bars in his bar chart? Explain.
26" PUM"|"Work"&"Energy"|"Lesson"3:"Reasoning"with"Energy"Bar"Charts""
© Copyright 2014, Rutgers, The State University of New Jersey.
Problem: Jessica stands in an elevator on the first floor. The elevator doors close and the
elevator delivers Jessica to the fourth floor. Identify all the changes that occur.
Alan’s solution: The system includes Jessica, who might be
moving so we may need to consider kinetic energy. The
system also includes Earth, so we will have to consider
gravitational potential energy, too. But because Jessica’s initial
and final velocity is zero, the kinetic energy does not change. I
put the initial energies of the system on the left side of the bar
chart and the final energies on the right side. However, I
System
Jessica
had no energy on the left but some on the right, which can’t
Earth
be possible. Then I remembered the elevator that pulled
Jessica up. The elevator is not in my system, so it must do
positive work on Jessica. I put the work done in the column
labeled W for work.
Initial State
Jessica is not
moving, standing
in the elevator on
the first floor.
Final State
Jessica is not
moving, standing
in the elevator on
the fourth floor.
e) Use the rubric below to assess you work-energy bar chart. How did you do? Describe
your difficulties.
Rubric!to!self;assess!your!work;energy!bar!charts!
Absent!
No work-energy
bar chart is
constructed. "
An attempt!
Work-energy bar chart is
constructed but is missing or
contains extra energy bars; the
initial and final states described do
not match the initial and final states
on the chart. The initial quantities
plus the work do not equal the final
quantities. "
Needs some
improvement!
Work-energy bar chart
lacks a key feature such
as labels, the zero energy
is not indicated, or
quantities are not drawn
to scale."
Acceptable!
The chart is labeled clearly
so that one can understand
the initial and final states
of the system. The relative
lengths of the bars are
correct. And the zero
energy is indicated."
3.8 Reason
a) Look back at the bar charts from the previous activity. If the elevator only went ½ as
high, which of the bars would change and by how much?
b) How would the work-energy bar chart look if we chose a final state when Jessica was still
moving?
c) How can we convert Alan’s bar charts into a mathematical statement?
d) Describe a situation with Jessica that could be represented by this energy bar chart.
PUM"|"Work"&"Energy"|"Lesson"3:"Reasoning"with"Energy"Bar"Charts"" 27"
© Copyright 2014, Rutgers, The State University of New Jersey. "
3.9 Represent and Reason
Use a bar chart to represent transformations of a quantity. This time, the exercise concerns the
food in your house, and we are going to make bar charts with estimated quantities instead of
exact numbers. Fill in the bar charts provided. Write a mathematical equation represented in each
chart that relates the quantities of food and shopping and discarding.
FP = food in your pantry
FR = food in the refrigerator
FS = food on the stove
ΔUate = food "U" ate
Situation Before and After
Bar Chart Representation
You have some food in your pantry and refrigerator
already.
You go shopping and come home with bags of
groceries.
You put some away in the pantry and some away in
the refrigerator.
before
FP,i +FR,i +
"
+"
after
buy or
FS,i+discard =FP,f +FR,f +FS,f +∆Uate
"
"
"
"
"""""0"
"
\"
"
Represent this on the bar chart.
The situation starts from where we left off in the
previous question.
You take some ingredients from the pantry and
some from the refrigerator and make a meal on the
stove.
Represent this on the bar chart.
before"
FP,i +FR,i +
"
+"
buy or
FS,i+discard
"
=FP,f +FR,f +FS,f +∆Uate
"
"
"
"""""0"
"
\"
"
28" PUM"|"Work"&"Energy"|"Lesson"3:"Reasoning"with"Energy"Bar"Charts""
© Copyright 2014, Rutgers, The State University of New Jersey.
after"
"
Situation Before and After
Bar Chart Representation
You eat half of the food you cooked and store the
rest in the refrigerator.
before"
FP,i +FR,i +
after"
buy or
FS,i+ discard =FP,f +FR,f +FS,f +∆Uate
+"
"""""0"
\"
"
Represent this on the bar chart.
The next day you come home with a takeout from
White Castle. You eat the stack of White Castles
and throw out your leftovers from the day before.
before"
FP,i +FR,i +
after"
buy or
FS,i+ discard =FP,f +FR,f +FS,f +∆Uate
+"
"""""0"
\"
"
Represent this on the bar chart.
a) Why do we show ΔUate on the "after" side but not on the "before" side?
3.10 Observe and Reason
Use the PhET simulation Energy Skate Park to answer the questions below. Go to the website:
http://phet.colorado.edu/ and search for “Energy Skate Park.” Click on the first Simulation and
then click Run Now!
a) Identify your system.
b) Draw an energy bar chart for the system as the skater goes from the top of the ramp down
to the bottom. Then draw a bar chart for the process as he continues from the bottom of
the track up the other side. Make sure your charts are consistent with your chosen
system.
c) Based on your observations, when does the system have the highest kinetic energy?
d) When does the system have the highest gravitational potential energy?
e) Is the skater your system? If so, is he able to have gravitational potential energy and
kinetic energy by himself?
f) Click on the Bar Graph tab on the right side of the simulation. Compare your bar charts
with the ones provided. How are they similar? How do they differ?
g) What do you notice about the “Total” column on their Bar Graph? What does this
represent?
PUM"|"Work"&"Energy"|"Lesson"3:"Reasoning"with"Energy"Bar"Charts"" 29"
© Copyright 2014, Rutgers, The State University of New Jersey. "
h) On the Bar Graph provided, what could “Thermal” stand for? What kind of energy would
that look like on our bar charts?
i) Click “Track Friction>>” and change the coefficient of friction between the skater and
the track. How does that change the bar chart? Use your knowledge of energy transfer to
explain what you observe.
3.11 Reason
In this lesson you have investigated the ideas of constant and conserved quantities.
a) Based on your observations, when is energy constant?
b) Based on your observations, when is energy conserved?
c) What role does work play in the bar charts? Does it affect whether the energy is constant
or conserved?
Reflect: What is the purpose of work-energy bar charts? How are they similar
to impulse-momentum bar charts? How are they different? Find a family
member or a friend and teach her/him about the bar charts. What difficulties
did they have? Prepare to share your experience in class.
30" PUM"|"Work"&"Energy"|"Lesson"3:"Reasoning"with"Energy"Bar"Charts""
© Copyright 2014, Rutgers, The State University of New Jersey.
Lesson 4: Such Great Heights
4.1 Represent and Reason
Complete the table that follows. Consider a 20 N brick sitting on a table which is 1 m high.
Word description
of a process.
a) Hector lifts the
brick 1 m up off of
the surface of the
table. He then
moves the brick
horizontally so it is
held over a piece
of chalk on the
floor.
Sketch the initial and Complete the work-energy bar chart for this
final state. Circle the process
system.
before
after"
Ki + Ug,i + Usi + W = Kf + Ug,f + Us,f +∆Uint
40"J"
20"J""
0"
\20"J"
\40"J""
Write an equation relating these quantities:
b) Eva lifts an
identical brick 2 m
from the floor to a
spot right next to
Hector’s brick. It
is also hanging
over a piece of
chalk on the floor.
before
after"
Ki + Ug,i + Usi + W = Kf + Ug,f + Us,f +∆Uint
40"J"
20"J""
0"
\20"J"
\40"J""
Write an equation relating these quantities:
c) When dropped, the bricks in parts (a) and (b) will both smash the chalk on the floor the same
amount. You found different initial potential energies for each system, though. How can this be?
Compare and contrast Hector’s situation with Eva’s situation to help you answer this.
The energy of each system measures its ability to smash a particular piece of chalk. For Hector’s
final state, the 20 J of energy actually measures the system’s ability to smash a piece of chalk on
the table. Since you have decided that when the block was on the table, the system begins with
no energy, you must have been describing the ability to smash the chalk if it were on the table.
PUM"|"Work"&"Energy"|"Lesson"4:"Such"Great"Heights"" 31"
© Copyright 2014, Rutgers, The State University of New Jersey. "
d) Redraw your work-energy
bar chart for Hector, but now
describe the ability to smash
chalk on the floor. Below,
relate the quantities
mathematically.
before
"
40"J"
"
after"
Ki + Ug,i + Usi + W = Kf + Ug,f + Us,f +∆Uint
20"J""
0"
\20"J"
\40"J""
e) Redraw your energy-bar
chart for Eva, but use it to
represent the system’s ability
to smash the chalk on the table.
Is the final state identical to
Hector’s final state in part (a)?
Below, relate the quantities
mathematically.
before
"
after"
Ki + Ug,i + Usi + W = Kf + Ug,f + Us,f +∆Uint
40"J"
20"J""
0"
\20"J"
\40"J""
Need Some Help?
When calculating the gravitational potential energy of a system, you must pick a reference level.
When an object is at this reference level, the gravitational potential energy of the system is zero.
4.2 Reason
You smash open walnuts on a picnic table by lifting a block from the table to a height of 1 m
above the walnuts and then dropping the block on the walnuts.
a) Draw a work-energy bar chart and a diagram for the initial and final states representing
the block-Earth system’s ability to smash these walnuts as you lift the block.
Your friend is in a tree house that is 10 m above the picnic table. Your friend has walnuts for a
snack in his tree house.
b) Draw a work-energy bar chart and a diagram for the initial and final states representing
the block-Earth system’s ability to smash these walnuts in the tree house as you lift the
block 1 m above the picnic table below.
c) In parts a) and b), you have a system with a negative gravitational potential energy. What
does a negative value represent about the ability of the system to accomplish a task?
32" PUM"|"Work"&"Energy"|"Lesson"4:"Such"Great"Heights""
© Copyright 2014, Rutgers, The State University of New Jersey.
4.3 Derive the Relationship
To develop a mathematical expression for gravitational potential energy, we analyze the
following situation. To build the foundation for a new skyscraper, a construction company needs
to drive metal poles into granite stone. To hammer the poles into the ground, a crane lifts a
massive block at a slow constant speed from a height yi above the pole to a height yf above the
pole. The crane then drops the massive block onto the top of the pole, which is at height y = 0.
The massive block has a mass m.
Below is a picture of the initial and final states of the process. The system for analysis is the
block and Earth.
\"yf"
\"yi"
\"0"
\"0"
a) Complete a work-energy bar chart for this process. Write a mathematical expression
representing this process.
b) Write an expression for the work the cable does on the block during its displacement yf –
yi. Substitute this into the expression in part a.
c) Draw a force diagram for the block during this process. Use it to find an expression for
the force that the cable exerts on the block in terms of its mass and the gravitational
constant g. Substitute this expression into the expression in part b.
d) Examine the expression that you derived in part c. Do you see that the work that the cable
did on the block equals the change in a quantity: mgyf – mgyi? Relate the work done on
the system to the change in gravitational potential energy of the system (∆Ug) in order to
find the expression for the gravitational potential energy of the block-Earth system at any
moment.
Did You Know?
The change in gravitational potential energy of the block-Earth system is caused by the work
done by the cable on the block. From that relationship, we find the accepted value for the
gravitational potential energy of a system at a given moment is equal to U g = mgy , where y
represents the height with respect to the reference level. The gravitational potential energy of the
system describes its ability to smash a piece of chalk at the reference level, where U g = 0 .
4.4 Test the Relation
You are the head engineer for the construction company discussed in the last problem. Before
you build the machine to drive the poles into the ground, you need to test whether the ability of
the block-Earth system to do something (to smash chalk or clay, to dent Styrofoam, to splash
water, etc.) depends on the mass of the block and the initial height of the block above the target.
PUM"|"Work"&"Energy"|"Lesson"4:"Such"Great"Heights"" 33"
© Copyright 2014, Rutgers, The State University of New Jersey. "
Describe two experiments that you can perform to test these relationships. How many variables
can you change in this experiment? What will be your dependent and independent variables?
Include a sketch. What does the relationship predict will happen? What are your assumptions?
4.5 Evaluate the Relation
You are performing an experiment involving a track that you can tilt at different angles and a
marble. Tape a Styrofoam cup some distance from the base of your desk. Your goal is to let the
marble roll down the track and have it land in the cup. The only rule is that you cannot move the
cup.
a) Design and perform an experiment that will allow you to find a pattern in the location of
the marble from which you can release it and it lands in a cup (make sure that you try at
least 4 different locations that work).
b) Draw a picture of the experiment and describe what you did.
c) Record the data you collected and decide what the best way to represent it is.
d) Describe the pattern you found in words and mathematically.
e) Use your knowledge of energy to relate the location of the marble and its ability to land
in the cup. Compare your pattern to the definition of the gravitational potential energy of
the system. Use any necessary representations to help with your explanation. Are the two
relationships consistent? How do you know?
4.6 Reason
Imagine that you could use one of two ramps to slowly move a 7-kg cart to a position that is 2 m
above the ground.
3"m"
15"m"
2"m"
a) Using a work-energy bar chart; determine how much work you will have to do to move
the cart up each ramp slowly.
b) Determine the value of the force that you will need to exert on the cart to slowly push it
up each ramp.
4.7 Reason
You slowly lift a box from the floor to the top of a table and do 310 J of work on the box. Now
lift the same box quickly from the floor to the top of the table. It ends at rest on the table.
a) How does the work you did on the system now compare to the previous situation where
you lifted the box slowly?
34" PUM"|"Work"&"Energy"|"Lesson"4:"Such"Great"Heights""
© Copyright 2014, Rutgers, The State University of New Jersey.
i.
More work was done on the box when it was lifted quickly.
ii.
The same amount of work was done on the box when it was lifted quickly.
iii.
Less work was done on the box when it was lifted quickly.
iv.
No work was done on the box when it was lifted quickly.
a) In the second experiment identify the time intervals and displacements when you did
positive work on the box, negative work. What is the total work being done? How does it
compare to the work you do when lift very slowly? What if you lifter the box with even
higher acceleration at the beginning – will you more total work lifting the box to the same
height?
b) Think of how your answers are related to the change of gravitational potential energy of
the system box-Earth in all situations.
Here’s an Idea!
The work that you did not on the system box-Earth in the previous activity is equal to the change
in gravitational potential energy of that system. If the system’s energy change in two
experiments is the same, the external work done on it should the same too!
Homework
4.8 Represent and Reason
Imagine that a cart is rolling down an inclined plane. The initial state is when it is on top of the
incline, the final state is when it is moving fast at the bottom.
Case A: Our system is the cart only therefore, it does not have any potential energy. There is also
no initial kinetic energy. Earth pulls down on the cart; the surface exerts a force perpendicular to
the direction of motion. Because the cart rolls, its kinetic energy increases. Earth does positive
work and the surface of the plane does no work because the force is perpendicular to the
direction of motion.
Case B: The system is the cart and Earth together. It has initial gravitational potential energy. As
the cart rolls down, some of this energy is transformed into kinetic energy. Earth does not do any
work because it is internal to the system.
PUM"|"Work"&"Energy"|"Lesson"4:"Such"Great"Heights"" 35"
© Copyright 2014, Rutgers, The State University of New Jersey. "
Draw a diagram for the situation
and circle the system.
Complete the bar chart for this process and relate
the quantities mathematically.
Case A
before"
after"
Ki + Ug,i + Us,i + W = Kf + Ug,f + Us,f +∆Uint
+"
"""""0"
"
\"
Case B:
before"
after"
Ki + Ug,i + Us,i + W = Kf + Ug,f + Us,f +∆Uint
+"
"""""0"
"
\"
4.9 Equation jeopardy
Write a problem that would require the mathematical equation below to solve it.
10 J = m*9.8 (N/kg) (15 m – 3 m)
4.10 Reason
When you crushed the chalk with a block, you released the 5 kg block from a height of 1 m
above the chalk. What is the gravitational potential energy of the block-Earth system before you
released it? What did you set as your reference level?
4.11 Reason
A skier slides down an icy hill. He has a mass of 70 kg and begins 50 m above the bottom of the
hill. What is the skier’s kinetic energy at the bottom of the hill? What is his kinetic energy when
he is ¾ of the way down the hill? What system did you choose for analysis?
36" PUM"|"Work"&"Energy"|"Lesson"4:"Such"Great"Heights""
© Copyright 2014, Rutgers, The State University of New Jersey.
4.12 Reason
Jeff and Jim are both demolition experts skilled in using a wrecking ball to destroy old buildings.
The motion of the wrecking ball is shown below.
When asked to draw work-energy bar charts for the motion of the wrecking ball, Jeff and Jim
drew the bar charts below.
Jim’s"
Jeff’s
before
after"
before"
Ki#+##Ug,i#+##Us,i#+#####W###=##Kf#+##Ug,f#+#Us,f#+#∆Uint#
after"
Ki#+###Ug,i#+##Us,i#+###W###=##Kf#+##Ug,f#+#Us,f#+#∆Uint#
+"
+"
"""""0"
"""""0"
\"
\"
a) When would Jeff’s be correct and when would Jim’s be correct? Be sure to state the initial and
final states, what objects you are including in the system, and where you are defining the
reference level for zero gravitational potential energy.
b) How should Jim and Jeff label their work-energy bar charts to prevent any more confusion?
c) Shouldn’t they include the work due to the force that the rope exerts on the ball on both
charts? Explain your answer with a force diagram.
4.13 Represent and Reason
Imagine that you throw a baseball out of your dorm room window on the fifth floor to a friend
standing outside on the ground level. Determine which of the following work-energy bar charts
could represent this situation. Decide what is included in the system and state the reference level
for the gravitational potential energy for each correct bar chart.
PUM"|"Work"&"Energy"|"Lesson"4:"Such"Great"Heights"" 37"
© Copyright 2014, Rutgers, The State University of New Jersey. "
before"
"
after"
Ki#+####Ug,i#+#Usi#+####W###=##Kf#+###Ug,f#+#Us,f#+∆Uint#
before"
"
"
"
"
"
""
""
"
"
0"
"
"
before"
after"
Ki#+####Ug,i#+#Usi#+####W###=##Kf#+###Ug,f#+#Us,f#+∆Uint#
0"
"
"
""
"
""
"
"
"
""
""
"
"
0"
0"
after"
Ki#+####Ug,i#+#Usi#+####W###=##Kf#+###Ug,f#+#Us,f#+∆Uint#
before"
after"
Ki#+####Ug,i#+#Usi#+####W###=##Kf#+###Ug,f#+#Us,f#+∆Uint#
Reflect:
Look back at everything that you
learned about gravitational
"
"
potential
energy. Make a list of things that
are important to know about this
""
""
physical quantity. Compare your list to your classmate’s list.
38" PUM"|"Work"&"Energy"|"Lesson"4:"Such"Great"Heights""
© Copyright 2014, Rutgers, The State University of New Jersey.
Lesson 5: Galileo’s Pendulum
5.1 Test a Hypothesis
Suppose that you have a pendulum (a bob hung from a string) and a horizontal rod in its path
(the device is called Galileo’s pendulum). The pendulum is pulled to the side and released. When
the string swings into its vertical orientation, it hits a horizontal rod, causing the pendulum to
swing in an arc with a smaller radius (as pictured below).
Pivot
Horizontal bar"
Pendulum
Bob
The height of the horizontal rod can be adjusted.
Design and conduct an experiment that uses this pendulum to test whether energy is conserved in
any system or constant in an isolated system.
a) State clearly the hypothesis that you will test in the experiment.
b) Play with the pendulum and decide what features of its behavior can be explained using
the concept of energy.
c) Think of an experiment that you can perform whose outcome you can predict using the
ideas of energy conservation and energy constancy. Draw a picture. Decide what
quantities you will measure and what quantities you will calculate. Decide what objects
are in your system and whether any external objects do work on it.
d) Make a prediction of the outcome of the experiment based on the idea being tested (the
hypothesis). Make sure that you include the experimental uncertainties in your prediction.
e) What are the additional assumptions that you are making? Can you validate them? If
these assumptions are not valid, how will they affect your result?
f) Perform the experiment as many times as you think is necessary, collect the data, and
calculate the result. How close is it to your prediction?
g) What is your judgment about the hypothesis that you were testing?
PUM"|"Work"&"Energy"|"Lesson"5:"Galileo’s"Pendulum"" 39"
© Copyright 2014, Rutgers, The State University of New Jersey. "
h) Write a report describing your experiment, data, findings and judgment. Use the rubrics
below to improve your lab report.
An alternative version of the experiment can be found at the PAER website: Galileo's Pendulum.
Homework
"
Write a lab report for the first experiment you performed in class. Use the rubrics to guide your
writing.
Hypothesis-prediction-testing rubric (used for testing experiments)
Scientific
Ability
Is able to
distinguish
between a
hypothesis and
a prediction.
Missing
An attempt
No prediction is
made. The
experiment is not
treated as a testing
experiment.
A prediction is made,
but it is identical to the
hypothesis.
Is able to make
a reasonable
prediction
based on a
hypothesis.
Is able to make
a reasonable
judgment
about the
hypothesis.
No attempt is
made to make a
prediction.
A prediction is made
that is distinct from the
hypothesis but is not
based on it.
No judgment is
made about the
hypothesis.
A judgment is made but
is not consistent with
the outcome of the
experiment.
Is able to
record and
represent data
in a
meaningful
way
Is able to
identify
sources of
experimental
uncertainty
Is able to
evaluate
specifically
how identified
experimental
uncertainties
may affect the
data
Data are either
absent or
incomprehensible.
Some important data
are absent or
incomprehensible.
No attempt is
made to identify
experimental
uncertainties.
An attempt is made to
identify experimental
uncertainties, but most
are missing, described
vaguely, or incorrect.
An attempt is made to
evaluate experimental
uncertainties, but most
are missing, described
vaguely, or incorrect.
Or only absolute
uncertainties are
mentioned. Or the final
result does not take the
uncertainty into the
account.
No attempt is
made to evaluate
experimental
uncertainties.
40" PUM"|"Work"&"Energy"|"Lesson"5:"Galileo’s"Pendulum""
© Copyright 2014, Rutgers, The State University of New Jersey.
Needs some
improvement
A prediction is made
and is distinct from the
hypothesis but does not
describe the outcome
of the designed
experiment.
A prediction is made
that follows from the
hypothesis but does not
have an if-and-then
structure.
A judgment is made
and is consistent with
the outcome of the
experiment but
assumptions are not
taken into account.
All important data are
present, but recorded in
a way that requires
some effort to
comprehend.
Acceptable
A prediction is
made, is distinct
from the hypothesis,
and describes the
outcome of the
designed experiment.
A prediction is made
that is based on the
hypothesis and has
an if-and-then
structure.
A reasonable
judgment is made
and assumptions are
taken into account.
All important data
are present,
organized, and
recorded clearly.
Most experimental
uncertainties are
correctly identified.
All experimental
uncertainties are
correctly identified.
The final result does
take the identified
uncertainties into
account but is not
correctly evaluated.
The experimental
uncertainty of the
final result is
correctly evaluated.
Lesson 6: How to Calculate Kinetic Energy
6.1 Hypothesize (Derive a Mathematical Model)
"
In a car crash testing facility, engineers evaluate
the reaction of a car to an impact on its front. To
vi"
create such an impact, a rod pushes a block of
mass m on wheels over a distance d. This causes
vf"
the block to accelerate from an initial to a final
velocity. To measure the smashing potential of
this block, let’s determine the change in the
d#
block’s kinetic energy after the rod pushes it a
distance d. The initial and final states of the process are pictured to the right.
a) The block is your system; the rod is an external object. Draw a force diagram for the
block. Use it to find an expression for the force that the rod exerts on the block in terms
of its mass m and acceleration a.
b) Use a kinematics equation to convert the acceleration a in the equation from part (a) into
an expression involving the block’s initial and final speeds vi and vf. Substitute this into
the expression for force from part (a).
c) Substitute the expression for force from part (b) into the expression for work when the
force is parallel to the displacement, W = Fd, and then simplify.
d) Using the given work-energy bar chart,
develop a mathematical representation of this
process in terms of work, initial kinetic
energy, and final kinetic energy. Make sure
to check the consistency of the graph with
your system. Compare this expression to the
one from part (c).
before"the"block"
is"lifted"
after"the"block"is"
lifted"
Ki + Ug,i + Us,i + W = Kf + Ug,f + Us,f +∆Uint
+"
"""""0"
\"
e) What characteristics of an object do you
expect kinetic energy to depend on? Its mass? Velocity? Acceleration? Height?
f) By comparing your answers from parts (c) and (d), do you see a term that could represent
kinetic energy and that depends on the characteristics that you think kinetic energy
should depend on?
g) Show that the units of this quantity are equal to the units for energy, joules.
Did You Know?
This change in kinetic energy of the block system is caused by the work done by the piston on
the block. From that relationship, we find the accepted value for the kinetic energy of a system at
a given moment is equal to K =
1
2 mv
2
.
PUM"|"Work"&"Energy"|"Lesson"6:"How"to"Calculate"Kinetic"Energy"" 41"
© Copyright 2014, Rutgers, The State University of New Jersey. "
6.2 Test a Hypothesis
Use a Hot Wheels car and launcher. Design an experiment to test the mathematical model you
developed for kinetic energy (see the guidance below)
Then design a second experiment that uses kinematics instead of work-energy. Evaluate their
consistency.
a) State clearly the hypothesis that you will test in the experiment.
b) Play with launcher and the car and decide what features of their behavior you can explain
using the concepts of kinetic energy.
c) Think of experiments that you can perform whose outcomes you can predict using the
ideas of kinetic and energy conservation. What other ideas/concepts can you use? Draw a
picture. Decide what quantities you will measure and what quantities you will calculate.
Decide what objects are in your system and whether any external objects do work on it.
d) Make a prediction of the outcome of the experiment based on the idea being tested (the
hypothesis).
e) What are the additional assumptions that you are making? If these assumptions are not
valid, how will they affect your result?
f) Perform the experiment as many times as you think is necessary, collect the data, and
calculate the result. How close is it to your prediction?
g) What is another experiment that you can perform that does not use the ideas of energy to
determine the same quantity? Perform it and analyze the results. Are the results of two
experiments close within experimental uncertainty?
h) What is your judgment about the hypothesis that you were testing?
6.3 Regular Problem
Kelly drives a truck which has twice as much mass as Heather’s car. Kelly also drives twice as
fast as Heather. Which statement is true about Kelly and her truck’s kinetic energy (K) compared
to that of Heather and her car?
i.
The truck has 4 times the K of the car.
ii.
All that can be said is that the truck has more K.
iii.
The truck has twice the K of the car.
iv.
The truck has 8 times the K of the car.
42" PUM"|"Work"&"Energy"|"Lesson"6:"How"to"Calculate"Kinetic"Energy""
© Copyright 2014, Rutgers, The State University of New Jersey.
Here’s an Idea!
Now that you have derived and tested mathematical expressions for both gravitational potential
energy and kinetic energy, you can start solving problems. Below is the problems solving
strategy that you might want to follow to learn how solve those problems as a physicist.
Problem-Solving Strategy: Work-Energy Problems
Sketch and Translate:
• Read the problem 3 times and visualize the situation/process.
• Sketch the physical process described in the problem. Include an initial state and a final
state and a reference frame. Put all givens on the sketch. Make sure they are in consistent
units.
• Decide on your system. Objects such as Earth, springs, and surfaces of interacting objects
are usually included in the system. Objects that belong to the system do no work on each
other but do possess different types of energy. External objects can do work on the
system objects, thus causing the system’s energy to change.
Simplify and represent using the work-energy bar chart:
• Decide what internal or external interactions you can ignore.
• Construct a work-energy bar chart. Use the bars to represent the initial energies in the
system, the work done on the system by any external objects, and the final energies in the
system. Consider whether the following change:
• A system object’s elevation above Earth (gravitational potential energy);
• A system object’s speed (kinetic energy);
• An elastic system object (like a spring) stretches or compresses (elastic potential energy);
• The surface temperature of system objects increase as they rub against each other while
one moves relative to the other (internal thermal energy change);
• A system object(s)’s shape during a collision changes (internal potential energy).
Represent Mathematically:
• Apply the generalized work-energy principle;
• Convert the bars in the bar chart into a mathematical description of the process (one term
for each bar in the bar chart).
Solve and Evaluate:
• Use the mathematical description of the process to determine the unknown. Evaluate the
results (units, magnitude, and limiting cases) to make sure they make intuitive sense.
6.4 Regular Problem
If you drop a 0.3 kg baseball from a window 20 m above the ground, how fast will the ball be
moving the instant before it hits the ground? Use the problem solving strategy.
6.5 Practice
If a stretched slingshot has 100 J of elastic potential energy, how fast will a 0.5 kg softball be
moving right after the launcher fires it? Using energy representations, how high will the softball
go? Use the problem solving strategy.
PUM"|"Work"&"Energy"|"Lesson"6:"How"to"Calculate"Kinetic"Energy"" 43"
© Copyright 2014, Rutgers, The State University of New Jersey. "
6.6 Reason
You may have seen one of these in your classroom or at home, it is called a Newton’s Cradle.
Use your knowledge of energy to explain the simulation of this device: Newton's Cradle. Specify
the system, its initial and final states, and any assumptions you made.
Homework
6.7 Regular Problem
A crane lifts a 50-kg crate so that the crate’s speed increases from 0 m/s to 5.0 m/s over a vertical
distance of 10.0 m. Draw a bar chart representing this process. What is the force that the crane
exerts on the crate? Use the problem solving strategy. Specify the system, its initial and final
states, and any assumptions you made. Explain how these assumptions affect your answer.
6.8 Regular Problem
A man throws a 0.4-kg softball uward at an initial speed of 10 m/s. How fast will it be traveling
when it passes 1/3 of its maximum elevation? Use the problem solving strategy.
6.9 Reason
Two identical water balloon slingshots are stretched the same distance so that they both have the
same elastic potential energy. The mass of one water balloon is 2/3 of the mass of the other water
balloon.
a) Which water balloon leaves the slingshot traveling at a faster speed?
b) How much faster is this water balloon traveling?
6.10 Equation Jeopardy
Write a problem and draw an energy bar chart that would require the mathematical equation
below to solve it.
Reflect: What did you learn about kinetic energy? How did you learn it? If
you were to ask two questions about kinetic energy to find out if a person
understands this concept, what would those questions be?
44" PUM"|"Work"&"Energy"|"Lesson"6:"How"to"Calculate"Kinetic"Energy""
© Copyright 2014, Rutgers, The State University of New Jersey.
Lesson 7: The Energy in a Slingshot and Other Practical Things
7.1 Represent and Reason
Complete the table that follows for three different processes. The goal is to devise a graphical
method to determine the work done by an external force on a system object. Note: P = person
and O = object.
Word description of a
process.
a) Rona lifts a backpack from
the floor to the desk, exerting
a constant upward force. The
backpack and Earth (not
Rona) are the system.
Draw FP on O –versus-Δy
(for vertical motion) or
Δx (for horizontal
motion) graphs.
Describe how to use
the graph to find the
work done by the
force.
If an object moves a
distance Δy or Δx,
what is the expression
for the work done on
the object by the force
on the graph?
FP on O
y
b) Kruti catches a medicine
ball in the gym. The ball and
Earth are the system (but not
Kruti). Her hands move back
toward her body while
stopping the ball.
FP on O
c) Carlos stretches a
horizontal rubber cord (it
behaves like a spring) with a
spring constant k. The spring
and Earth are in the system
but not Carlos.
FP on O
d) Two men push a stalled
car. For the first 50 m, they
exert a force of 1000 N on the
car. For the second 50 m, they
exert a force of 500 N on the
car.
FP on O
x
Δx
Δx
x
x
Summarize: Summarize how you can determine the work done by an external force using the
force versus displacement graph.
PUM"|"Work"&"Energy"|"Lesson"7:"The"Energy"in"a"Slingshot"and"Other"Practical"Things"" 45"
© Copyright 2014, Rutgers, The State University of New Jersey. "
7.2 Hypothesize (Derive a Mathematical Model)
Recall that the magnitude of the force exerted by an elastic spring on an object is F = kx where x
is the distance that the spring has been stretched (or compressed) from its relaxed position.
a) Graph the force that must be exerted on an elastic spring by the object stretching the
spring from its relaxed state to an extension xf. How does this force relate to the force that
the spring exerts on the object?
b) Determine the work done by external forces exerted on the elastic spring to stretch it this
distance. Use the graphical method.
c) Show that the units of this quantity are equal to the units of energy, joules.
d) Draw a work-energy bar chart for this process.
e) Write a mathematical expression for this process represented by the bar chart.
f) Use this mathematical expression in variable form (using k and x) to find an expression
for elastic potential energy.
Did You Know?
This change in elastic potential energy of the object-spring system is caused by the work done by
you on the system. From that relationship, we find the accepted value for the elastic potential
energy of a system at a given moment is equal to U s =
1
2 kx
2
and is the energy stored in a
stretched or compressed object. It depends on the force constant k (stiffness) of the elastic object
and the distance x that the elastic object is displaced from its equilibrium position.
7.3 Represent and Reason
A spring with a spring constant k = 80 N/m is compressed 0.3 m. A 0.2 kg book is placed on top
of the compressed spring.
a) Draw a picture of the process. What are the initial and final states of the process? What is
the reference level you are using?
b) What will happen when the spring is released?
c) Draw a picture of the process. What are the initial and final states of the process? What is
the reference level you are using?
d) Represent the process with an energy bar chart.
e) After the spring is released, how high will the book fly?
Need Some Help?
When calculating the elastic potential energy of a system, you must pick a reference level. When
an object is at this reference level, the elastic potential energy of the system is zero. Typically
this reference level is also referred to as the equilibrium position of the spring.
46" PUM"|"Work"&"Energy"|"Lesson"7:"The"Energy"in"a"Slingshot"and"Other"Practical"Things""
© Copyright 2014, Rutgers, The State University of New Jersey.
7.4 Reason
Go to the following ActivPhysics website: Activ Physics. Click on number 5.5: SpringLaunched Bowler. Use your knowledge of elastic potential energy to solve the problem.
Make sure to predict what will happen before you run the simulations. What are your
assumptions?
Watch the estimation video at the end of the simulation. Based on the given data and your
observations, estimate the force constant of the spring. What did you assume?
7.5 Equation Jeopardy
Create a problem where the following is the solution:
½ (50 kg) x (10 m/s)2 + m x (9.8N/m) x 20 m = ½ k * (1m)2
Draw a picture of the process. What are the initial and final states of the process? What is the
reference level you are using?
7.6 Regular Problem
How much work must be done on a spring with spring constant 100 N/m to stretch it from
equilibrium at 0.15 m to 0.25 m? Draw a picture of the process. What are the initial and final
states of the process? What is the reference level you are using? Include a work-energy bar chart.
7.7 Reason
Which quantity is greater?
i.
The amount of elastic potential energy of a system in which a spring is compressed
by 0.10 m.
ii.
The amount of elastic potential energy of a system in which a rubber band is stretched
by 0.20 m.
iii.
The two quantities are about equal.
iv.
There is not enough information to say which quantity is greater.
7.8 Reason
Which quantity is greater?
i.
The amount by which the rubber band in a slingshot is stretched in order to launch a
0.05 kg marble vertically to a maximum height of 10 m.
ii.
The amount by which the same rubber band in the same slingshot is stretched to
launch a 0.1 kg steel ball vertically to a maximum height of 8 m.
iii.
The two quantities are about equal.
iv.
There is not enough information to say which quantity is greater.
PUM"|"Work"&"Energy"|"Lesson"7:"The"Energy"in"a"Slingshot"and"Other"Practical"Things"" 47"
© Copyright 2014, Rutgers, The State University of New Jersey. "
Homework
7.9 Jeopardy
Complete the table that follows and formulate a problem.
Problem:
Sketch with the reference level:
Force-displacement Graph:
F
Work-energy bar chart:
before"
50 N –
after"
Ki + Ug,i + Us,i + W = Kf + Ug,f + Us,f +∆Uint
+"
"""""0"
25 N -
\"
""""""""I" "
0m
10 m
"I"
20 m
x
Mathematical Representation and Solution:
Us,i + W = Us,f
W= Us,f – Us,i
7.10 Regular Problem
You are the coach of the two-man U.S. Olympic Bobsled Team. At the beginning of a race, one
of the team members pushes the bobsled and its driver for 50 meters along the level track. For
the first 20 meters, the athlete exerts a 400 N force in the horizontal direction on the sled and
driver. For the next 20 meters, the member exerts a force of 350 N on the two. For the final 10
meters, he exerts a force of 300 N on the two. The total mass of the bobsled and driver is 330 kg.
Let’s calculate the total work that the team member does on the bobsled. The bobsled and driver
are the system.
48" PUM"|"Work"&"Energy"|"Lesson"7:"The"Energy"in"a"Slingshot"and"Other"Practical"Things""
© Copyright 2014, Rutgers, The State University of New Jersey.
a) Describe the motion of the bobsled, use the quantities initial speed, acceleration, final
speed in your description.
b) Over the first 20 meters, how much work does the teammate do on the system?
c) Over the next 20 meters, how much work does the teammate do on the system?
d) Over the last 10 meters, how much work does the teammate do on the system?
e) What is the total amount of work that the teammate does on the system?
f) On the axes below, graph the force exerted on the system by the teammate versus the
position of the system.
F (N)
x#(m)#
g) What property on the graph is equal to the work done by the teammate on the system?
7.11 Regular Problem
Kristen pushes her little sister on a sled on a packed icy surface. Her little sister and the sled are
the system and have a combined mass of 20
kg. Ignore the frictional forces exerted on
the sled.
a) Sketch a diagram of the situation
and identify an initial and final state.
b) Represent the process with a workenergy bar chart
c) Determine the final velocity of
Kristen’s sister.
7.12 Regular Problem
Danny does 3 J of work to compress the spring in his Nerf gun so it is ready to shoot. The barrel
of the Nerf gun is 16 inches long. Sketch the initial and final states.
a) What is the average force Danny has to exert on the spring?
b) What is the elastic potential energy of the spring-dart system once Danny has compressed
the spring 10 inches?
PUM"|"Work"&"Energy"|"Lesson"7:"The"Energy"in"a"Slingshot"and"Other"Practical"Things"" 49"
© Copyright 2014, Rutgers, The State University of New Jersey. "
Reflection: Why did we use the graphical method to derive the expression for
elastic potential energy? Why couldn’t we just multiple the magnitude of the
force we exerted to stretch the spring by the change in its length to find the
work and then set up work equal to the change in elastic potential energy?
50" PUM"|"Work"&"Energy"|"Lesson"7:"The"Energy"in"a"Slingshot"and"Other"Practical"Things""
© Copyright 2014, Rutgers, The State University of New Jersey.
Lesson 8: Spring into Action
8.1 Observe
You have a spring and a vertically mounted metal pole and other regular lab equipment. You can
stretch the spring along the metal pole and release it to fire it upwards. Play with the spring and
decide what features of its behavior can be explained using the concept of energy. What do you
notice when you stretch the spring the same length each time?
Caution: Protect your eyes with goggles! Only stretch the spring when nobody is close to its
pathway.
"
8.2 Hypothesize a Relation
Consider all of the activities that you have done so far. What is the relationship between the
initial energy of the system, the external work done on the system, and the final energy of the
system? Clearly state the hypothesis/relation that you will test in the experiment.
8.3 Test the Relation
a) Think of an experiment that you can perform whose outcome you can predict using the
relation being tested. Draw a picture. Decide what quantities you will measure and what
quantities you will calculate. Decide if you need to perform some additional experiments
to determine the unknown quantities.
b) Use your relation to predict the outcome of the experiment. Make sure that you include
the experimental uncertainties in your prediction.
c) What are the additional assumptions that you are making? Can you validate them? If
these assumptions are not valid, how will they affect your results?
d) Perform the experiment as many times as necessary, collect the data, and calculate the
results. How close is it to your prediction?
e) What is your judgment about the relation that you are testing?
f) Write a report about your experiment so that a person who did not see you perform the
experiment can repeat it and can also understand your results and conclusions.
g) Use the rubrics below to improve your lab report.
PUM"|"Work"&"Energy"|"Lesson"8:"Spring"into"Action"" 51"
© Copyright 2014, Rutgers, The State University of New Jersey. "
Hypothesis-prediction-testing rubric (used for testing experiments)
Scientific
Ability
Is able to
distinguish
between a
hypothesis/relat
ion and a
prediction.
Is able to make
a reasonable
prediction
based on a
hypothesis/
relation.
Is able to
identify the
assumptions
made in making
the prediction.
Needs some
improvement
A prediction is made and
is distinct from the
hypothesis/relation but
does not describe the
outcome of the designed
experiment.
A prediction is made
that follows from the
hypothesis/relation but
does not have an if-andthen structure.
Missing
An attempt
No prediction is
made. The
experiment is not
treated as a testing
experiment.
A prediction is made but
it is identical to the
hypothesis/relation.
No attempt is made
to make a
prediction.
A prediction is made that
is distinct from the
hypothesis/relation but is
not based on it.
No attempt is made
to identify any
assumptions.
An attempt is made to
identify assumptions but
the assumptions are
irrelevant or are confused
with the hypothesis.
Relevant assumptions
are identified but are not
significant for making
the prediction.
Is able to
determine
specifically the
way in which
assumptions
might affect the
prediction.
Is able to make
a reasonable
judgment about
the hypothesis/
relation.
No attempt is made
to determine the
effects of the
assumptions.
The effects of the
assumptions are
mentioned but are
describe vaguely.
The effects of the
assumptions are
determined but no
attempt is made to
validate them.
No judgment is
made about the
hypothesis/relation.
A judgment is made but
is not consistent with the
outcome of the
experiment.
A reasonable
judgment is made and
assumptions are taken
into account.
Is able to
identify
experimental
uncertainty.
No attempt is made
to identify
experimental
uncertainty.
Is able to
evaluate
specifically how
experimental
uncertainties
will affect the
data and
calculations.
No attempt is made
to evaluate
experimental
uncertainties.
An attempt is made to
identify experimental
uncertainty but most are
missing, described
vaguely, or incorrect.
An attempt is made to
evaluate uncertainties,
but most are missing,
described vaguely, or
incorrect.
A judgment is made and
is consistent with the
outcome of the
experiment but
assumptions are not
taken into account.
Most uncertainties are
correctly identified.
The final result does
take uncertainties into
account but they are not
correctly evaluated.
The experimental
uncertainty of the
final result is correctly
evaluated.
52" PUM"|"Work"&"Energy"|"Lesson"8:"Spring"into"Action""
© Copyright 2014, Rutgers, The State University of New Jersey.
Acceptable
A prediction is made,
is distinct from the
hypothesis/relation,
and describes the
outcome of the
designed experiment.
A prediction is made
that is based on the
hypothesis/relation
and has an if-and-then
structure.
All relevant
assumptions are
identified and their
effects on the
accuracy of the
prediction are
correctly determined.
The effects of the
assumptions are
determined and the
assumptions are
validated.
All experimental
uncertainties are
correctly identified.
Homework
Write up your lab using the rubric as a guide.
8.4 Regular Problem
Instead of traditional brakes, a spring is used to slow down a new type of roller coaster. The
coaster has 5.0 x 105 J of kinetic energy before it compresses the spring and comes to a stop. The
spring constant for the spring is 2.0 x 104 N/m. How far is the spring compressed? What are the
system and the initial and final states that you chose for the situation? (When you are solving the
problem, make sure that draw a picture of the process, identify the initial and final states of the
process and specify the reference level you are using.)
8.5 Regular Problem
A model airplane launcher uses an elastic cord to accelerate a small wood and paper airplane to
flight speed. The 0.005-kg plane must be moving at 1 m/s to fly. If the elastic band has a spring
constant of 120 N/m, how far should you stretch the elastic band so that the plane will accelerate
to its flight speed? (When you are solving the problem, make sure that draw a picture of the
process, identify the initial and final states of the process and specify the reference level you are
using.)
Reflection: How is a testing experiment different from an observational
experiment? Can you explain it to someone who has never taken physics?
Give two examples of each from this module.
PUM"|"Work"&"Energy"|"Lesson"8:"Spring"into"Action"" 53"
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Lesson 9: Calculating the Internal Energy Change
9.1 Evaluate
Review the experiment report of your classmate using the rubric. Write a report describing your
review without giving any scores. Think of the strong aspects of the report. Think of what could
be improved. After you receive the review of your report written by your classmate, revise it
based on the review and hand it in to your teacher.
Did You Know?
In the last four lessons, you have derived mathematical representations for gravitational potential
energy, kinetic energy, and elastic potential energy. These mechanical forms of energy and the
change in internal energy can be summarized mathematically in the generalized Work-Energy
Principle.
Generalized Work-Energy Principle:
The initial energy of the system Ui plus any work W done on the objects in the system by objects
outside the system equals the final energy Uf of the system:
or
The energy can take many different forms: kinetic K, gravitational potential Ug, elastic
potential Us, internal energy change ∆Uint, and others introduced in later chapters. The unit
of energy is the joule (J), where 1 J = 1 N•m.
9.2 Hypothesize (Derive a Mathematical Model)
Determine an expression for the change in internal energy due to friction in a system that
consists of a crate and a rough horizontal surface on which it slides. You, outside of the system,
pull on a rope attached to the crate
so that it moves slowly at constant
System"
velocity. At the end of the process,
the bottom of the block and the
surface on which it was moving
have became warmer.
a) Write an expression for the work done on the system by the external force of the rope on
the crate as the rope pulls the block a distance s across the surface.
b) Choose the crate alone as the system (a different system than in the sketch above). Draw
a force diagram for the crate. Apply Newton’s Second Law for the horizontal x-axis. How
are FR on C (rope on crate) and Fs on C (surface friction on crate) related?
c) Now, combine (a) and (b) to write an expression for the work done by the force exerted
by the surface through friction on the crate. Is it positive or negative?
54" PUM"|"Work"&"Energy"|"Lesson"9:"Calculating"the"Internal"Energy"Change""
© Copyright 2014, Rutgers, The State University of New Jersey.
before"
after"
Ki + Ug,i + Us,i + W = Kf + Ug,f + Us,f +∆Uint
+"
"""""0"
\"
d) Represent the process with a bar chart. The
system is the crate, the surface, and Earth. The
rope is outside. In the initial state, the crate is
moving but the surfaces are cool; in the final
state, the crate is still moving with the same
velocity but the surfaces are warmer. Think of
what force does work on the system and what
happens to the internal energy of the system as
a result of this process.
e) Examine the bar chart. Write an expression for the change in internal energy and decide
whether it increases or decreases.
f) Show that the units of this quantity are equal to the units of energy, joules.
Did You Know?
It is important to understand that the bottom surface of the crate is hotter, as is the rough surface
on which it moves. Also, there may be parts of the surface that are rubbed off - a form of
chemical internal energy change (such as the skid marks caused by a car coming to an abrupt
stop). The system’s internal energy increase due to friction between two surfaces is:
.
Where fS on O is the magnitude of the friction force and s is the magnitude of th displacement –
those quantities are always positive, thus the change in internal energy due to friction between
the surfaces is always positive
9.3 Regular Problem
After sledding down a hill Tanya is moving at 7.0 m/s. Tanya and her sled have a mass of 58kg.
On a horizontal surface Tanya slides to rest after 5m.
a) Sketch the situation; identify the initial and final states of the process and specify the
reference frame you are using.
b) Use a bar chart to represent the process.
c) Where does all the kinetic energy go? Explain.
d) Determine the coefficient of friction between the snow and the sled.
9.4 Regular Problem
When a regular car slows down, all of its kinetic energy is converted into internal energy through
work done on the car by frictional forces. In a hybrid car, an electrical generator exerts a force on
the spinning wheels to slow them down. If the wheels don’t slip on the road, the generator can
transform 20% of the car’s initial kinetic energy into reusable electrical energy. Later on, the
car’s electrical motor can use this electrical energy to spin the wheels of the car.
PUM"|"Work"&"Energy"|"Lesson"9:"Calculating"the"Internal"Energy"Change"" 55"
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a) If a hybrid car slows down from 18 m/s to 1/3 of its speed, what is the amount of kinetic
energy that the electrical generator converts into electrical energy? The car has a mass of
1200 kg. Remember that in order to express the energy in J, all quantities should be in SI
units.
b) Soon afterwards, the hybrid car is traveling at 15 m/s. Over what distance can the car
maintain the speed? Notice that the force exerted on the car is not the force of the motor;
it is the force exerted by Earth’s surface. When the motor rotates the wheels, they push
against the ground and the ground in turn pushes back on them, making the car go
forward.
c) The total force exerted on the car at this speed is 350 N. What is the work done by the
ground during a 1-hour trip? Why doesn’t the hybrid accelerate if the surface exerts a
constant forward force on it all the time?
Homework
9.5 Observe and Reason
Open the following PhET simulation: Friction.
a) Rub the Chemistry book back and forth across the top of the Physics book. What do you
observe?
b) How does this relate to the change in internal energy of the two books? Is the change
positive or negative? Explain your reasoning.
9.6 Reason
A hockey puck slides across an ice rink at a speed of 2 m/s. The frictional force exerted by the
ice on the hockey puck is 0.6 N. The puck has a mass of 0.4 kg.
Describe the motion of the object in words and
sketch the situation. Specify the reference
frame.
Find the distance the puck will travel using
Newton’s laws and kinematics.
56" PUM"|"Work"&"Energy"|"Lesson"9:"Calculating"the"Internal"Energy"Change""
© Copyright 2014, Rutgers, The State University of New Jersey.
Find the distance that the puck will travel using
your knowledge of energy. Include the surface
of the ice in your system.
Use the energy approach again, but this time,
do not include the surface of the ice in your
system.
Discuss whether the distance traveled by the puck is the same for all three methods. Explain.
9.7 Observe and Reason
Go to the Physics Teaching Technology Resource website and investigate the following
experiment: Happy and sad balls.
The above video depicts a situation where two balls of equal volume exhibit different properties.
Observe the differences between the two balls and come up with an explanation for why the
result of the interaction of the “happy” ball with the plank is different from the interaction of the
“sad” ball.
a) What is the difference between the happy and sad balls when they collide with the table?
b) Which ball is heavier?
c) If you choose the table and the ball as a system for the moment of collision only, what
happens to the energy of the system?
d) What is the difference between the happy and sad balls when they collide with the plank?
Shouldn't the sad ball have a bigger impact on the plank because of its mass?
e) Suggest possible explanations for this difference. Use your knowledge of energy to
answer the questions.
Reflection: What did you learn about internal energy today? Why do you
think on a bar chart there are no spaces for bars for initial and final internal
energy but is a space for a bar for internal energy change?
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Lesson 10: Power Up
10.1 Reason
Hans, a weightlifter, can bench press 100 kg (220 lbs). Hans can lift the 100 kg, from a height of
0.8 m above the ground to a height of 1.3 m in 0.2 seconds. Hans wants to determine the rate at
which work is done on the barbell and weights. What would you tell Hans to do, to determine the
rate at which he does work on the barbell and weights?
10.2 Develop a relation
Your friend Heather brings in two motors to class. She wants to install one of them in her
lawnmower but wants to make sure that she picks the best motor. Your job is to determine a
rating system for the motors to determine which one will cut her grass most efficiently.
You set the motors on top of a table and attach a string and 10.0 kg bob to each of them. You see
that Motor 1 pulls the bob from the floor to the top of the table in 0.032 seconds. You see that
Motor 2 pulls the bob from 5 cm above the floor to the top of the table in 0.020 seconds.
a) Estimate the height of the table. Devise a physical quantity that can describe the motor’s
ability to pull the bob up. What is it equal for the first motor? Second motor? Are the
numbers reasonable? How d you know?
b) Read the definition for power below. How did your definition compare?
Did You Know?
The physical quantity of Power (P) describes how much work is done on a system per time
interval or the energy that is transferred into or out of a system each time interval.
The units of power are joules per second (J/s). Joules per second are often called watts (W),
named after the 18th century Scottish engineer James Watt.
10.3 Regular Problem
Corrine is training for the annual javelin throwing contest. Determine how much power she
exerts while lifting the following objects. Draw a picture of the initial and final states.
a) a 1.5-kg javelin 1.0 m in 1.0 s
b) a 1.5-kg javelin 1.0 m in 0.5 s
c) a 1.0-kg javelin 2.0 m in 1.0 s
d) a 2.0-kg javelin 1.0 m in 1.0 s
58" PUM"|"Work"&"Energy"|"Lesson"10:"Power"Up""
© Copyright 2014, Rutgers, The State University of New Jersey.
10.4 Regular Problem
A crane lifts an I-beam up the side of a building. The crane’s power output is 1750W for 20
seconds. After 20 seconds the I-beam was moving at 2 m/s and the mass has 200 kg. Use the
work-energy process to determine the change in height of the I-beam.
10.5 Explain
The luminosity of the Sun is the amount of power the Sun emits in the form of electromagnetic
radiation. The Sun’s luminosity is 3.8 x 1026 (W). If you were able to collect all of the Sun’s
energy, estimate how long you have to collect it in order to light all of Earth’s light bulbs.
Explain how you came to this conclusion.
Homework
10.6 Reason
There are two types of light bulbs, incandescent and CFLs. Incandescent light bulbs are
commonly rated at 60 Watts. CFL bulbs are rated at 14 Watts.
a) In one minute how many Joules of energy would be converted for each bulb? In one
hour?
b) If you are charged $0.11 per Kilowatt●Hr estimate how much money you could save per
year using a CFL compared to an incandescent light bulb. What assumptions did you
make in your calculations?
Did You Know?
The physical quantity of Kilowatt●Hour (kWh ) describes the Power multiplied by time. If you
examine the units, we can find out that how much energy is transferred.
10.7 Real World Applications
Power plants supply electrical potential energy to be used in our households. Think of electrical
potential energy as the capacity for a power plant to do work to light a light bulb. Consider the
different types of power plants, describe various the energies a power plant uses that are
converted into electrical potential energies. Do some research and find out how these energies
are converted to electrical potential energy.
10.8 Regular Problem
Tahreen, a 52 kg hiker, climbs to the summit of Mount Mitchell in western North Carolina.
During one 2.0 hr period, her vertical elevation increases 540 m.
a) Draw a picture of the process, identify the initial and final states of the process and
specify the reference level you are using.
b) Determine the change in gravitational potential energy of the system Tahreen-Earth.
PUM"|"Work"&"Energy"|"Lesson"10:"Power"Up"" 59"
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c) Determine the power generated to increase the gravitational potential energy of the
system.
10.9 Regular Problem
You have developed superpowers overnight. You can now stretch your arms to reach ridiculous
heights. You decide to become a hero and on your way home from school, you do the following
heroic acts.
Determine how much power you exert while lifting the following objects. Draw a picture of the
initial and final states.
a) You rescue a 0.05-kg bird from the sidewalk and place it back in its nest, 5.2 m up in a
tree, in 1.0 s.
b) You lift a 6-kg bag of Oreos 3.0 m up to your tree house in 6.0 s for your slumber party
with your friends (you have been saving up for all the Oreos).
c) You lift a 10.4-kg bag of rice 2.6 m to the top of the pantry for your mom in 2.3 s.
d) Your 70-kg sister twisted her ankle so you lift her from the foyer to the second floor 4.0
m straight up in 10.0 s.
Reflection: How did you learn about power? Give three examples from your
day today where the concept of power helps explain what happened.
Ask your relatives or friends what they think the word power means in
physics. Then teach them what you know. What difficulties did they have?
Prepare to share with your classmates in class.
60" PUM"|"Work"&"Energy"|"Lesson"10:"Power"Up""
© Copyright 2014, Rutgers, The State University of New Jersey.
Lesson 11: Practice & Review
Set 1: Internal Energy
11.1 Regular problem
As a traffic cop investigating a car accident, you want to determine how fast a car was moving
before its driver began to brake. While braking, the car left skid marks that are 70 m long.
According to your reference book, the mass of the car is 1600 kg and the coefficient of kinetic
friction between the tires and road is 0.2. Was the car traveling faster than the 40 mph speed
limit? Determine the car’s initial speed. What are your assumptions? Use the problem solving
strategy.
Homework
11.2 Regular problem
As part of your new job as a car safety engineer, you have been asked to predict the average
force exerted on a crash test dummy during a simulated car crash. The car accelerates to a speed
of 20 m/s and then collides with a piston that stops the car. The crash test dummy moves a total
of 0.7 m as the car comes to a stop. The dummy has a mass of 70 kg. Determine the average
force that the seat belt exerts on the dummy. Use the problem solving strategy. What
assumptions did you make? Is the force exerted by the seat belt on the dummy a safe amount?
Set 2: Elastic Potential Energy
11.3 Regular Problem
A popular new spring hockey game that Jay got for his birthday uses springs to move a 0.0030kg puck. It works like a regular table hockey game but instead of hitting the puck, the players use
small springs. Each spring has a 120-N/m spring constant and can be compressed up to 0.020 m.
How fast does the puck move when it departs a spring that was initially compressed this
distance? Use the problem solving strategy.
Homework
11.4 Regular Problem
You are designing a new Bungee-jumping system for beginners. An 80-kg cart (including its
passenger) is to start at rest near the top of a 30° incline. The uphill side of the cart is attached to
a spring. The other end of the spring is attached securely to a post farther up the hill. The spring
is initially relaxed at equilibrium. After you are secure in the cart, it is released and you coast 40
m down the hill before coming to a stop. For every 1 m that you coast down the hill, the height
of the cart above the ground decreases by 0.5 m. What is the spring constant of the spring that
you should buy for this invention? Follow the problem-solving strategy.
PUM"|"Work"&"Energy"|"Lesson"11:"Practice"&"Review"" 61"
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Set 3: A Trigonometric Expression of Work
11.5 Regular Problem
Alan exerts a downward force of 50 N on his wagon of vegetables at an angle 35o above the
horizontal. He exerts this force over a distance of 15 m to move his wagon closer to the center of
the market. Use the problem-solving strategy to determine the change in energy of the wagonEarth system.
11.6 Regular Problem
A 10 kg cart is traveling at 4 m/s. Aneta exerts a force of 10 N on the cart at an angle of 145°
below its direction of motion. Over what distance must Aneta exert this force before the cart
comes to rest? Sketch the picture and identify the initial and final states.
11.7 Regular Problem
A skier starting from rest slides down a slope that is 55 m long. The slope makes an angle of 32o
with the horizontal.
a) Consider only the skier to be in the system. What is the total energy of the system when
the skier reaches the bottom of the hill?
b) Now, consider the skier and Earth to be in the system. What is the total energy of the
system when the skier reaches the bottom of the hill?
11.8 Evaluate the Solution
Problem: You are traveling in your 2000-kg Chevy at 20 m/s up a hill with a 6.0o incline when
you see a goose crossing the road 24 m in front of you. You know from previous experience that
when you hit the brakes, a 16,000-N friction force opposes your motion. Will you hit the goose?
Solution: (1/2)(2000 kg)(20 m/s)2 = (16,000 N)x or x = 25 m. Oops!
a) Identify any errors in the solution.
b) Provide a corrected solution if you find any errors.
11.9 Evaluate the Solution
Problem: A 40.000-N/m spring initially compressed 0.50 m is released and launches you and
your cart (100 kg total) up a 30o incline. What distance along the incline do you travel before
coming to a stop?
Solution: (1/2)(40,000 N/m)(0.50 m) = (100 kg)(9.8 m/s2)y or y = 10.2 m.
a) Identify any errors in the solution.
b) Provide a corrected solution if you find any errors.
62" PUM"|"Work"&"Energy"|"Lesson"11:"Practice"&"Review""
© Copyright 2014, Rutgers, The State University of New Jersey.
Set 4: Bar Charts
11.10 Represent and Reason
Fill in the table that follows.
Experiment: Describe the
system and process.
Draw a sketch showing
the initial and final states.
Circle the object(s) in the
system.
Construct a quantitative work-energy bar
chart and mathematically relate the quantities
to each other.
A motor pulls a roller
coaster up the first hill of
the track via a chain.
Initial state: The roller
coaster is at rest at the
bottom of the hill.
Final state: The roller
coaster is moving at a
moderate speed at the top
of the hill.
System: Includes the
roller coaster, chain, and
Earth but excludes the
motor that pulls the chain
up the hill.
11.11 Represent and Reason
Repeat the previous activity with a different system.
Experiment: Describe the
system and process.
Draw a sketch showing
the initial and final states.
Circle the object(s) in the
system.
Construct a quantitative work-energy bar
chart and mathematically relate the quantities
to each other.
System: Includes the
roller coaster and the
chain but excludes Earth
and the motor that pulls
the chain up the hill.
11.13 Equation Jeopardy
The first column in the table that follows applies the generalized work-energy equation to two
different processes (in fact, there are many possible processes described by each equation). For
each mathematical description, construct a sketch, a word description, and a bar chart that is
consistent with the equation.
PUM"|"Work"&"Energy"|"Lesson"11:"Practice"&"Review"" 63"
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Generalized work-energy
equation applied to a process.
Sketch a process
that might be
described by the
equation.
Construct a bar chart.
Describe the process
in words.
a)
Ki Ugi Usi
W
Kf Ugf Usf∆Uint
0"
b)
Ki Ugi Usi
W
Kf Ugf Usf∆Uint
0"
Set 5: Experimental Design
11.14 Design an Experiment
You have a Hot Wheels® car, a Hot Wheels® track, a loop-the-loop piece for the track, a Hot
Wheels® car launcher, a surface that can be inclined at different angles, masking tape, and a
meter stick. Use any or all of this equipment to design an experiment to test whether the total
energy of a Hot Wheels®-Earth system is constant if there are no external forces exerted on it by
other objects.
64" PUM"|"Work"&"Energy"|"Lesson"11:"Practice"&"Review""
© Copyright 2014, Rutgers, The State University of New Jersey.
Describe the
experiment in words.
Sketch the apparatus.
List quantities
that you will
measure.
Use the generalized
work-energy principle
and other principles (if
needed) to make a
prediction.
List
assumptions
that you made.
11.15 Design an experiment
You have a flexible track that can be tilted at different angles with the horizontal and a small
metal ball. Use them to test the following idea: “The kinetic energy of the ball is directly
proportional to the distance it travels along a tilted track.”
11.16 Design an Experiment
Go outside and find skid marks on the pavement. Using the skid marks estimate the initial speed
of the car and the amount of its kinetic energy that went into the internal energy of the carpavement system. Clearly state all assumptions you made in your estimation.
Reflect: Write a note to your past self about productive things to think about
when solving work-energy problems. What are the things that would have
helped you?
PUM"|"Work"&"Energy"|"Lesson"11:"Practice"&"Review"" 65"
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Lesson 12: When Work is not Easy
12.1 Derive a Relation
To launch satellites into space quickly and inexpensively, NASA wants to design an “elevator”
spacecraft to slowly pull new satellites from the ground into space along a special tether. The
new satellite will move at a constant velocity upwards along the tether. As a scientist working for
NASA, you need to determine the amount of work the elevator spacecraft must exert on the new
satellite to move it into space.
Below are sketches of the initial and final states of this process.
Elevator spacecraft"
Elevator spacecraft
v#
v
New satellite
New satellite
"
"
"
Ri
"
Rf
"
"
"
"
Earth"
Earth"
a) As the new satellite moves higher, the spacecraft will be able to exert a smaller force on it
to keep it moving at a slow, constant speed. This graph shows the force exerted by the
spacecraft on the satellite-Earth system as the satellite is pulled into space.
Fcraft on sat
"
|"
"
Ri
"
|"
Rf
Separation between Earth and satellite
b) Which of the following graphs could represent the gravitational potential energy of Earthsatellite system as a function of the separation between the two?
66" PUM"|"Work"&"Energy"|"Lesson"12:"When"Work"is"not"Easy""
© Copyright 2014, Rutgers, The State University of New Jersey.
If a graph cannot represent Ug,
explain why not.
Ug
"
|"
"
"
|"
Ri
Rf
Separation between Earth and satellite
Ug
"
|"
"
"
|"
"
Ri"
"
"
Rf"
Separation between Earth and"satellite""
Ug
Separation between Earth and satellite
Ri
"
|"
Rf
"
"
|"
"
c) Consider the interaction between the satellite and Earth. At what separation should the
gravitational potential energy of the Earth-satellite system be equal to zero?
d) Which of these mathematical expressions could represent universal gravitational potential
energy? If one cannot, explain why not.
Ug = −
GM m
e s
r
Ug = −
Gms
r
Ug = GM E ms r
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12.2 Reason
The two expressions for gravitational potential energy look very different. The first one (Ug =
mgy) was developed for processes with elevation changes on or near Earth’s surface. Does the
new expression produce a similar result for such a change? Suppose you lift a pile driver of,
mass m from, position y, to a higher position y + ∆y, where ∆y is a relatively small elevation
change:
a) Use the first expression for gravitational potential energy to write an expression for the
gravitational potential energy change.
b) Now, does the new expression for gravitational potential energy produce the same result?
The pile driver starts at distance r from Earth’s center and ends at distance r + ∆y from
the center. You will have expressions for the initial and the final energies. Find a
common denominator and combine the two expressions. Note that g = GM/r2. Can you
get this expression to be the same as the expression in part (a)?
Did You Know?
The change in the physical quantity of Gravitational Potential Energy (Ug) describes the work
an external object has to do on the object of interest to move this object from the surface of Earth
to infinity. This work is equal to the change of the gravitational potential energy of the system
object-Earth. The final energy of this system when the object is infinitely far away form Earth is
GM m
e s . Thus Ugi + W = 0 or Ugi = −W
zero. The work done by the external force is W =
r
.
GM m
e s .
Therefore we can write an expression for gravitational potential energy as U g = −
r
Think of how this expression is different from the expression Ug = mgh that you learned before.
How are they the same? Hint: think of the zero level assumed in both expressions. "
12.3 Evaluate a Relation
You have developed a relationship describing the gravitational interactions of two objects. Open
the PhET simulation “Gravity Force Lab” at the following website: Gravity Force Lab.
a) Investigate whether your relationship makes sense. Before changing anything in the
simulation, what do you observe? How does the force exerted by m1 on m2 compare to
the gravitational potential energy of the system?
b) Identify the system necessary in order to have gravitational potential energy.
c) Neither m1 nor m2 are Earth. How can they have gravitational interactions?
d) How many variables can you change in this experiment? What will be your dependent
and independent variables? Include a sketch. What does the relationship predict will
happen?
e) How do the results of the simulation compare to your relationship?
68" PUM"|"Work"&"Energy"|"Lesson"12:"When"Work"is"not"Easy""
© Copyright 2014, Rutgers, The State University of New Jersey.
Reflect: You learned a new expression for gravitational potential energy. Why
wasn’t the old one sufficient? Explain how these two expressions give you the
same result for the gravitational potential energy of the system Earth-apple on
an apple tree.
Homework
12.4 Reason
Instead of moving the satellite into space using the space elevator, NASA could also fire it from
a cannon on the ground. To move the satellite from the surface of Earth (6.3*106 m from the
center of Earth) to an altitude of 3.58*107 m from the center of Earth, how fast would the
satellite have to be shot out of the cannon? At its final height, the satellite should not be moving
relative to Earth.
a) Below is a sketch of the initial and final states of the system.
v=0
v
New satellite
New satellite
"
"
" Rf
Ri
"
"
"
"
"
Earth"
Earth"
b) Represent this process using the work-energy bar chart below.
before
after
Ki + Ug,i + Us,i + W = Kf + Ug,f + Us,f +∆Uint
+"
"""""0"
\"
c) Represent this process mathematically.
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d) Determine the initial velocity of the satellite. (G = 6.67*10-11 Nm2/kg2, ME = 5.97*1024
kg)
e) Should NASA move the new satellite into space using the elevator method from activity
11.1 or the cannon method from this problem? Why?
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Lesson 13: Oh Baby, Don’t Let Me Go
13.1 Reason
a) Imagine a brick in a very deep well. At the top of the well, the gravitational potential energy
of the system brick-Earth is equal to zero. This is where we set our reference level. Above
the well, you have positive energy; below the well, there is negative energy. We are going
to draw an analogy to a person with credit card debt. A person in debt must earn money to
pay back the credit collectors, similar to the brick-earth system, which needs a certain
amount of energy in order to escape the well.
1. List a number of ways, in terms of work and energy, that the brick can obtain the amount
of energy needed to escape the well.
2. Using one of the examples in part (a), draw an energy bar chart such that in the final state
the object has enough energy to have escaped the energy debt of the well.
3. As the brick goes up the well, which energies increase and which decrease?
b) Now imagine that a satellite sits on Earth waiting to get into space. Think about the
interaction between Earth and the satellite. Where is the force exerted by Earth on the
satellite the greatest? Where is it non-existent? Now imagine that the bottom of the well in
part (a) is Earth and we know the satellite is waiting on Earth to get into space. Similar to
the person in debt, or the brick in the well, the satellite is in “energy debt” to Earth.
1. Consider the place where there is no interaction between Earth and the satellite. Where is
the value for gravitational potential energy zero?
2. List a number of ways, in terms of work and energy, that the satellite can obtain the
amount of energy needed to escape the clutches of Earth.
3. Using one of the examples in part (b), draw an energy bar chart such that in the final state
the satellite has enough energy to have escaped the energy debt it has on Earth’s surface.
4. As the satellite goes up the well, which energies increase and which decrease?
13.2 Reason
(Neglect the interaction with the Moon.)
In the movie Armageddon, a motley crew of
hard-nosed oil drillers rendezvous with a
Mass of the Sun = 2.0 x 1030 kg
menacing meteoroid just as it passes the orbit
Mass of Earth = 5.96 x 1024 kg
of the moon. Imagine that an asteroid fell in
Radius of Earth = 6.37 x 106 m
from the Oort Cloud, a region of space in the
"
depth of our Solar System which is very far away. What
is the speed of the meteoroid as it passes
the orbit of the Moon? The Moon orbits at a distance of approximately 60 Earth radii from the
center of Earth.
a) Use multiple representations to show the meteoroid’s journey from the Oort Cloud to
Earth.
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b) Determine the speed at which it passes the Moon’s orbit if Earth exerts a gravitational
force on the asteroid.
c) Let us say that Armageddon did arrive in that movie. With what speed would the
meteoroid have collided into Earth?
d) Are there any other real-life scenarios where people may consider the idea of a negative
potential energy?
13.3 Reason
In 1865, Jules Verne wrote a novel where he imagined a space rocket fired to the Moon from
Earth by using a cannon. If Verne wanted to fire the rocket into the depths of space, with what
speed must the rocket have in order to escape Earth’s gravitational pull? Why is using a cannon
impractical for space exploration? How do space explorations get around this problem today?
13.4 Reason
If we assume that no object can move faster than the speed of light in a vacuum, then what is the
radius of an object of mass m, so that even light cannot escape? How big should the sun be to
become a black hole? The speed of light is 3 x 108 m/s.
13.5 Reason
A satellite is moving around Earth in a circular orbit at
constant speed (see Figure). Earth is the only external
object exerting a force on the satellite, pulling the satellite
directly towards Earth’s center. Which of the following
statements is true as the satellite moves from Point A to
Point B in the orbit?
A"
B"
Earth"
a) The gravitational potential energy of the satellite
changes as it moves from A to B.
b) The work done on the satellite by Earth is negative for the motion from A to B.
c) The work done on the satellite by Earth is zero for the motion from A to B.
d) The velocity of the satellite remains unchanged as it moves from A to B.
Reflect: Summarize what you learned about work, energy and power in this
module. Write a two-page summary. The best summary will be included in
the next year’s module.
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Summary: Definitions and Principles
Work W: Work is the product of the magnitude of the average force FEx on O that an external
environmental object exerts on a system object, the magnitude of the system object’s
displacement d, and the cosine of the angle between FEx on O and d.
The system gains energy if the work done on it is positive and loses energy if the work is
negative.
Kinetic energy K of a system object is one-half times the product of its mass m and the square of
its speed v:
Gravitational potential energy Ug of the system object-Earth depends on the relative
separation of an object of mass m and Earth. When the object is near Earth’s surface, we
calculate Earth-object’s gravitational potential energy using
where y is the object’s elevation relative to a chosen zero reference level. When far from Earth,
we use the expression
where r is the object’s distance from the center of Earth. The zero level of gravitational potential
energy is at infinity.
Elastic potential energy Us is the energy stored in a stretched or compressed object and depends
on the force constant k (stiffness) of the elastic object and the distance x that the elastic object is
displaced from its equilibrium position:
Increase in internal energy due to friction ∆Uinternal: When an object moves across a surface
with friction, the contacting surfaces warm slightly. If the surfaces are included in the system, the
increase in the system energy due to this friction is the product of the magnitude of the average
kinetic friction force Fk between the object and the surface and the distance d that the object
moves relative to the surface:
If the surface is not in the system, then the work done by the external friction force is:
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Generalized work-energy equation (energy conservation): The total energy of a system is the
sum of all the energies. In the initial state Ui = Ki + Ugi + Usi and in the final state Uf = Kf + Ugf
+ Usf + ∆Uint. If work is done on the system object(s), then the energy can change. Expressed
quantitatively as the generalized work-energy equation:
Constancy of energy principle: If, during a process, the net sum of the work done on the
objects in a system is zero, then the total energy of the system is constant (the same at the
beginning as at the end). However, the types of energy in the system can change.
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Lesson 14: Simple Machines I
14.1 Observe and Represent
Design an experiment to determine the amount of work it takes to move a cart from the bottom to
the top of an incline. You have the following materials: two inclined planes that go to the same
height but have different slopes, force probe or spring scale, a cart, meter sticks, and a scale. The
goal of this experiment is to find patterns relating work and energy.
a) Determine what the system of interest is and the initial and final states.
b) Decide what quantities you will have to measure to find the work it takes to move the
cart up the incline.
c) Perform the experiment and record your data.
d) Decide if you can make any assumptions that will help to simplify the problem.
Here’s an Idea!
You may notice that when the cart is pulled really fast up the incline, the force measured by the
spring scale was hard to read. Try pulling the cart fast and then slow and steady. Which way
makes it easier to take the force measurement? Why are the measurements different (Hint: Think
of whether the cart is accelerating)?
e) Draw a work-energy bar chart that represents the process. Think of what is included in
your system.
14.2 Observe and Represent
a) Repeat the same steps for a second set of trials; this time pull the cart straight up the side
of the incline, starting from the ground and finishing at the top.
b) Look for patterns in the data you collected.
c) How does the work-energy bar chart for each set of trials differ? How do the heights of
the bars compare between the diagrams?
d) How does the amount of work it takes to move the cart to the top differ in each set of
trials (assume that the cart moves very slowly)?
e) Was one method of getting the cart to the top “easier” than the other? Describe how.
f) What was the gravitational potential energy of the system when the cart was at the top of
each of the inclines?
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Observational!Experiment!Rubric!
Ability
Absent
An attempt
Is able to decide
what is to be
measured and
identify
independent and
dependent
variables.
Is able to use
available
equipment to make
the measurements.
It is not clear what
will be measured.
It is clear what will be
measured but
independent and
dependent variables
are not identified.
At least one of the
chosen
measurements
cannot be made
with the available
equipment.
No description is
mentioned.
All chosen
measurements can be
made, but no details
are given about how it
is done.
No attempt is made
to construct a
relationship that
represents a trend
in the data.
An attempt is made,
but the relationship
does not represent the
trend.
Is able to describe
what is observed in
words, pictures,
and diagrams.
Is able to construct
a mathematical (if
applicable)
relationship that
represents a trend
in data.
A description is
mentioned but it is
incomplete. No
picture is present.
Needs some
improvement
It is clear what will
be measured and
independent and
dependent variables
are identified but the
choice is not
explained.
All chosen
measurements can be
made, but the details
of how it is done are
vague or incomplete.
A description exists,
but it is mixed up
with explanations or
other elements of the
experiment. A
labeled picture is
present.
The relationship
represents the trend
but no analysis of
how well it agrees
with the data is
included (if
applicable), or some
features of the
relationship are
missing.
Acceptable
It is clear what will
be measured and
independent and
dependent variables
are identified and
the choice is
explained.
All chosen
measurements can
be made and all
details of how it is
done are clearly
provided.
It clearly describes
what happens in the
experiments both
verbally and by
means of a labeled
picture.
The relationship
represents the trend
accurately and
completely and an
analysis of how well
it agrees with the
data is included (if
applicable).
14.3 Hypothesize
a) In this experiment you determined the work needed to lift the cart straight up so it covers
the distance Δx. Look back to your bar charts from the experiment. What type of energy
increases as a result of doing work on the system?
b) Write an equation for the work-energy relationship in this problem.
c) How did we mathematically define work in the beginning of the unit? What is the force
that is doing the work? What is the magnitude of the force that you need to exert on the
cart to lift it up very slowly? To answer this question, draw a force diagram for the cart
and decide which force on the diagram is doing the work.
d) How can we combine the two expressions above in terms of gravitational potential
energy change?
e) How can we write the expression for the gravitational potential energy of the system that
involves an object of mass m at the height x (displacement!) above the surface of Earth?
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14.4 Test Your Idea
a) Given the dimensions of another group’s ramp, predict the amount of energy a cart would
have when at the top of the ramp. Perform the experiment and record your results. How
did the outcome and your prediction compare?
b) Revise your hypothesis if necessary.
14.5 Reason
You, Brianna, and Doug just finished bowling and are ready to put the bowling balls away. Doug
uses a ramp to roll the ball to the top shelf while Brianna says it is easier to just lift the heavy ball
straight up.
a) What are the pros and cons of both methods?
b) If both bowling balls have the same mass, does Doug do less work than Brianna to get the
ball to the top shelf assuming that they both lift the balls very slowly? Why is this
assumption important?
Did You Know?
Simple Machine: A simple machine is a device that only requires one force to do work on a
system.
Mechanical Advantage: A simple machine or a compound machine (two or more simple
machine combined to make one device) work in such a way to require a smaller force (usually
over a greater distance) to perform the same amount of work.
c) How does Doug’s ramp work like a simple machine? How could you improve Doug’s
ramp to make it a better simple machine?
d) Doug says his method is better for moving the bowling balls to the top shelf. Brianna
disagrees. Which method do you think is better? Explain.
Homework
14.6 Represent and Explain
You are a member of a team hired by an architectural firm to design a wheelchair accessible
entrance to a building. The Americans with Disabilities Act requires the steepness of wheelchair
ramps to be less than a 1:12 ratio of vertical change to horizontal change.
a) Explain why the Americans with Disabilities Act is concerned with the steepness of
accessibility ramps.
b) Draw three ramps that meet their requirement.
c) Decide which of the following ramps meet the recommendation. (Some are represented
by a picture and some are represented by a ratio.)
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2m
(i)
16m
(v) 1:10
(ii)
4m
46m
(vi) 3:42
(iii)
3m
36m
(vii) 2:28
(iv) 0.5m
4m
d) Calculate the smallest amount of work that is required to push a 20 kg object up each of
the ramps (i–vii).
Need some help?
Remember from our dynamics unit that we can draw force diagrams to solve this problem. Then
we can begin to ask ourselves, “What objects are exerting a force on a box that you’re pushing
up a ramp? What is the magnitude of this force? How can we determine this?”
14.7 Reason
A notice came in the mail that revealed the staircase height, 2.5 meters. Now that we have this
piece of information we can begin working on building the ramp. If the staircase is 2.5 meters
high,
a) How long does a ramp need to be to fit the regulations?
b) The company controller says that is too long. The controller argues that regardless of the
length of the ramp, a person pushing the chair must do the same amount of work to get a
wheelchair to the top. He suggests that we can save money by using less construction
material by making the ramp as short and steep as possible. He wants to make a ramp
with a length of 4.0 m.
c) Give two reasons why it is important not to make the ramp any steeper.
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14.8 Explain
One of your classmates is having trouble understanding the difference between the amount of
effort it takes to lift or move something (the amount of force you exert on an object or system)
and the amount of work required to lift or move something. He says, “If it is easier to do, it must
be less work.”
a) Describe how you would help your classmate to understand this idea better.
b) Use examples like the ramps from the previous problem.
14.9 Regular Problem
A rollercoaster pulls the cart to the top of a hill by doing 150,000 J of work on the system. The
internal energy of the system changes by 50,000 J. How much more kinetic energy does the
system have when the cart makes it back to the bottom of the hill?
14.10 Regular Problem
You do 50 joules of work to compress a system that includes an object on a spring. The spring
launches a 30-g object straight up into the air. How high does the object go? What if the object
were 30 kg; how high would it go?
14.11 Regular Problem
A meteoroid, moving with high speed, enters Earth’s atmosphere and falls toward Earth. As it
passes through the atmosphere, it warms due to friction with the air. When it hits the ground, it
creates a giant crater. Explain the process using any or all of the ideas of work and energy
changes (consider all possible types of energy change). Include a work-energy bar chart.
14.12 Regular Problem
A skydiver who jumps out of an airplane at 3500 m above sea level accelerates towards the
ground for about 15 seconds before the upward force exerted by the air on the skydiver is about
equal to the downward force exerted by Earth on the skydiver.
a) Draw a force diagram for the skydiver after she/he has been falling for about 15 seconds.
b) The skydiver’s speed remains constant at 55 m/s until he or she opens the parachute.
Choose the skydiver, Earth, and the air as the system. Draw a work-energy bar chart that
describes the skydiver’s fall from when he or she leaves the plane until just before the
parachute opens.
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Lesson 15: Simple Machines II
15.1 Test an Idea
In the previous lab, we found that using an inclined plane does not reduce
the amount of work you have to do to lift an object. Your friend Cadence
says, “I have a simple machine that allows me to lift an object by doing less
work than if I lifted the same object without the machine.” She claims her
machine reduces the amount of work that is needed to lift an object. Her
simple machine is called a movable pulley. She draws a schematic of this
machine for you.
Lift
Pulley
Hanging
object
a) Design an experiment to test her idea. You have the following
equipment: Single moveable pulley, block, and spring scale
b) Describe your experiment; include all the details about what you will measure and how
you will measure it: variables! What are your independent and dependent variables?
c) Is it possible to reduce the amount of work needed to lift an object to a certain height?
Explain.
d) Make a prediction for the outcome based on Cadence’s hypothesis. Use the rubrics below
to help you answer the questions in an informal lab report.
e) Perform the experiment and describe the outcome of the experiment
Scientific
Ability
Is able to
distinguish
between a
hypothesis and
a prediction.
Is able to make
a reasonable
prediction
based on a
hypothesis.
Is able to make
a reasonable
judgment
about the
hypothesis.
Absent
An attempt
No prediction is
made. The
experiment is not
treated as a testing
experiment.
A prediction is made
but it is identical to the
hypothesis.
No attempt is
made to make a
prediction.
A prediction is made
that is distinct from the
hypothesis but is not
based on it.
No judgment is
made about the
hypothesis.
A judgment is made but
is not consistent with
the outcome of the
experiment.
Needs some
improvement
A prediction is made
and is distinct from the
hypothesis but does not
describe the outcome
of the designed
experiment.
A prediction is made
that follows from the
hypothesis but does not
incorporate
assumptions.
A judgment is made
and is consistent with
the outcome of the
experiment but
assumptions are not
taken into account.
15.2 Evaluate
a) Describe your data and highlight any important patterns you noticed.
b) What judgment can you make about Cadence’s hypothesis? Explain.
c) Write a revised hypothesis.
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Acceptable
A prediction is
made, is distinct
from the hypothesis,
and describes the
outcome of the
designed
experiment.
A prediction is made
that follows from the
hypothesis and
incorporates
assumptions.
A reasonable
judgment is made
and assumptions are
taken into account.
d) Does Cadence’s idea represent a mechanical advantage? Explain why it is easier to pull
the block up using Cadence’s pulley.
e) Compare this device to the inclined plane. What is different and what is the same? Think
about the purpose of both devices.
Homework
15.3 Evaluate
Using your textbook, the Internet, or encyclopedias, find more examples of simple machines.
Compare them to the machines we worked with in class (movable pulley and the inclined plane).
Create one problem dealing with a simple machine and calculating work-energy.
Answer the problem with as many representations as you can.
Tomorrow in class you will exchange this with a classmate. You will answer your partner’s
question and then discuss the answers.
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