Download Physics II - PUM Energy Module

PHYSICS UNION MATHEMATICS
Physics II
Work & Energy
Student Edition
Supported by the National Science
Foundation (DRL-0733140) and Science
Demo, Ltd.
PUM Physics II
Work & Energy
Adapted from:
A. Van Heuvelen and E. Etkina, Active Learning Guide,
Addison Wesley, San Francisco, 2006.
Used with permission.
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).
2 PUM | Work & Energy |
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, 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? ............................... 11
LESSON 3: REASONING WITH ENERGY BAR CHARTS........................................ 16
LESSON 4: SUCH GREAT HEIGHTS............................................................................ 26
LESSON 5: GALILEO’S PENDULUM ........................................................................... 32
LESSON 6: HOW TO CALCULATE KINETIC ENERGY .......................................... 34
LESSON 7: THE ENERGY IN A SLINGSHOT AND OTHER PRACTICAL
THINGS ............................................................................................................................... 36
LESSON 8: SPRING INTO ACTION .............................................................................. 41
LESSON 9: CALCULATING THE INTERNAL ENERGY CHANGE ....................... 44
LESSON 10: POWER UP .................................................................................................. 48
LESSON 11: PRACTICE & REVIEW............................................................................. 51
LESSON 12: WHEN WORK IS NOT EASY .................................................................. 58
LESSON 13: OH BABY, DON’T LET ME GO............................................................... 62
SUMMARY: DEFINITIONS AND PRINCIPLES.......................................................... 64
LESSON 14: SIMPLE MACHINES I............................................................................... 66
LESSON 15: SIMPLE MACHINES: APPLICATIONS................................................. 70
LESSON 16: SIMPLE MACHINES II ............................................................................. 73
PUM | Work & Energy | 3
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
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 and a string attached to the top, the Earth,
and a piece of chalk. 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 and 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. 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 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 ( ).
4 PUM | Work & Energy | Lesson 1: How Do We Ever Get Anything Done Around Here?
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
d) Look for a pattern of what was done to the objects that we studied to give them the
chalk-smashing potential. Then, devise a new physical quantity to describe this
pattern.
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 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 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 ( ).
PUM | Work & Energy | Lesson 1: How Do We Ever Get Anything Done Around Here? 5
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
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?
1.3 Observe and find a pattern
Consider the Earth and a 1-kg block.
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 the 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 being pulled at angle θ that can roll on a low-friction
horizontal dynamics track and 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
chalk-smashing ability? Is this consistent with the increase, decrease, and no change
in chalk-smashing potential for activities 1.1-1.3?
6 PUM | Work & Energy | Lesson 1: How Do We Ever Get Anything Done Around Here?
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
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.
Students familiar with trig, proceed to page 8; those who are not, continue here.
Work
System
System
x
A woman catches a ball thrown at her.
Since the force exerted by the woman on
the ball is in the opposite direction as the
displacement,
A woman pulls a box upwards.
Since the force exerted by the woman on
the box is in the same direction as the
displacement,
System
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,
1.5 Practice
Jeff did 573 J of work on a sled. He pulled the sled for a distance of 30 m. What is the
average force that he exerted on the system?
1.6 Practice
Steve slowly lifts a 20 kg barbell 1 meter vertically. How much work does he do on the
barbell?
PUM | Work & Energy | Lesson 1: How Do We Ever Get Anything Done Around Here? 7
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
1.7 Practice
Jessica, at a constant slow speed, moved a 1 kg book from a 2 m high shelf to the floor.
How much work did she do on the book?
1.8 Practice
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?
Homework
1.9 Relate
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.
1.10 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?
1.11 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
Is this process consistent with the pattern we observed today between the net force exerted
on an object and the displacement of the object?
8 PUM | Work & Energy | Lesson 1: How Do We Ever Get Anything Done Around Here?
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
Work-Trigonometry Section: In this section, students will use trigonometry to express
work done for a more general range of situations.
Work
System

W = Fp on s cos() 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 ) d
1.12 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?
1.13 Regular Problem
A 4 kg grocery cart rolls down a 3 m long incline with an angle of 10°. How much work
does the Earth do on the cart?
PUM | Work & Energy | Lesson 1: How Do We Ever Get Anything Done Around Here? 9
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
1.14 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?
1.15 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?
10 PUM | Work & Energy | Lesson 1: How Do We Ever Get Anything Done Around Here?
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
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 the 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 in ability are energies. Each type of energy has a formal name: internal
energy, kinetic energy, gravitational potential energy, and elastic potential energy. All
of these fall under a larger category called mechanical energy.
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 amount 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 in which for each item below, describe
one real-life situation that is consistent with the processes described below.
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.
PUM | Work & Energy | Lesson 2: How am I Supposed to Keep Track of It? 11
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
g) Gravitational potential energy in the system is converted to elastic potential energy.
2.3 Reason
Examine the picture to the right. One of your classmate says, “When the
car gets to the edge it will have ‘the ability to fall’ or ‘falling ability.’”
a) If the Earth weren’t there, would the car still have “ability to
fall?” Explain.
b) Should we include or exclude the Earth with the car when we analyze this problem?
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 the Earth.
c) Create a story for what happened to the cart.
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.
2.4 Explain
Why do we include the Earth in the system in some problems?
2.5 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.
a) On the picture to the right, identify objects in the
system. Explain why you made this decision
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.
b) Draw a force diagram that explain why the crate is moving at constant velocity
12 PUM | Work & Energy | Lesson 2: How am I Supposed to Keep Track of It?
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
c) Describe how the system (crate and surface) is different after you do the work than
before the crate started moving.
d) 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?
e) Revisit your choice of a system. Do you want to make any changes? Write down
your system below.
2.6 Observe and Explain
a) Complete the table below.
Describe the system.
Identify the
objects that are
part of the
system.
Hector lifts a new television
off the ground and places it on
the TV stand.
Television
Earth
Identify the initial and final state.
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
b) Eugenia slowly lifts a 5 kg box by exerting a constant force. She moves the box
from the ground up onto the table, which is 1 m high.
1. Draw a force diagram for the box while she is lifting it.
2. What is the force Eugenia exerts on the box?
3. Calculate the work Eugenia has to do in order to lift the box onto the table.
4. Suppose Eugenia exerted a larger force on the box; what would she have to do to
get the box to stop when it got to the table?
c) Mary rides in an elevator from the 1st floor to the 3rd floor. Answer the following
questions.
1. Sketch the initial and final states, and then identify the system.
2. Make a reasonable approximation for Mary’s mass and the distance between the
floors in the building.
PUM | Work & Energy | Lesson 2: How am I Supposed to Keep Track of It? 13
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
3. Using the approximations, write an integer statement for the work the elevator
does to lift Mary to the 3rd floor. Determine the work.
4. How much work would the elevator have to do if Mary’s twin sister joined her?
5. How much work would the elevator have to do if Mary decided to go to the 4th
floor instead? (Determine the work both with and without her sister.)
d) Mary needs to ride the elevator back down to the first floor.
1. Sketch the initial and final states, and then identify the system.
2. Write an integer statement for the work the elevator does to move Mary from the
3rd floor down to the first floor. Use your approximations from the previous
problem.
3. What if Mary were at the 4th floor instead? Write an integer statement for the
work the elevator does to move Mary from the 4td floor down to the first floor.
4. What if Mary’s twin sister joined her, how much work would the elevator have to
do to move them both from the 3rd floor to the first floor? Be sure to write an
integer statement.
Homework
2.7 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
Before you start, the box is on
the floor.
Work that has
been done so far
Gravitational Potential Energy
of the Box-Earth system
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.
14 PUM | Work & Energy | Lesson 2: How am I Supposed to Keep Track of It?
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
You have lifted the box all the
way to the table.
125 J
2.8 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:
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.
PUM | Work & Energy | Lesson 2: How am I Supposed to Keep Track of It? 15
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, 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 you
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
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.
16 PUM | Work & Energy | Lesson 3: Reasoning with Energy Bar Charts
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, 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
PUM | Work & Energy | Lesson 3: Reasoning with Energy Bar Charts 17
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
Situation Before and After
Bar Chart Representation
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
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.
18 PUM | Work & Energy | Lesson 3: Reasoning with Energy Bar Charts
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
Situation Before and After
Bar Chart Representation
h) Draw the missing bar.
Write a mathematical statement to match the chart.
Describe a story that could match.
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) Draw a comparison between money transfers and energy transfer. What similarities do
you see?
l) What property of wealth is illustrated in this activity?
m) 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.
PUM | Work & Energy | Lesson 3: Reasoning with Energy Bar Charts 19
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
Need Some Help?
Work-energy bar charts provide a concrete way to represent work-energy processes. In a
work-energy 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.
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
a) Kristen and her friend Peter go to the park to ride on the swings. Complete the tables
below to describe all the energy transformations. Be sure to identify the system in
each step.
Initial State
Peter has no
velocity and is at
the highest point
of his swing.
System:
Final State
Peter is at the
bottom of his
swing and is going
really fast.
Construct the Work-Energy Bar Chart
Equation:
20 PUM | Work & Energy | Lesson 3: Reasoning with Energy Bar Charts
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
Initial State
Peter is at the
bottom of the
swing moving
really fast.
Final State
Peter gets to the
top of his swing
and slows down.
System:
Equation:
Peter has no
velocity and is at
the highest point
of his swing.
Peter is at the
bottom of his swing
and is going really
fast backwards.
System:
Equation:
Peter has come to
a stop at the top
of the swing’s
motion and
begins to swing
forward.
System:
Construct the Work-Energy Bar Chart
After a few swings
Peter eventually
comes to a stop at
the center of the
swing.
Equation:
b) Describe how the system’s energy has changed in each step Peter swung back-andforth.
c) How has the total energy of the system changed?
PUM | Work & Energy | Lesson 3: Reasoning with Energy Bar Charts 21
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
d) If the rope from the swing is not in the system, how is accounted for in the bar
charts?
Homework
3.5 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.
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:
3.6 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.
22 PUM | Work & Energy | Lesson 3: Reasoning with Energy Bar Charts
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
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.
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 the 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 had no
System
energy on the left but some on the right, which can’t be Jessica
Earth
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.
3.7 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?
PUM | Work & Energy | Lesson 3: Reasoning with Energy Bar Charts 23
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
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 an integer statement?
d) Describe a situation with Jessica that could be represented by this energy bar chart.
3.8 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.
before
FP,i +FR,i +
after
buy or
FS,i+discard =FP,f +FR,f +FS,f +∆Uate
+
You go shopping and come home with bags of
groceries.
You put some away in the pantry and some away in
the refrigerator.
0
-
Represent this on the bar chart.
24 PUM | Work & Energy | Lesson 3: Reasoning with Energy Bar Charts
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
Situation Before and After
Bar Chart Representation
The situation starts from where we left off in the
previous question.
before
FP,i +FR,i +
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.
-
after
buy or
FS,i+discard
=FP,f +FR,f +FS,f +∆Uate
0
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
buy or
after
FP,i +FR,i + FS,i+ discard =FP,f +FR,f +FS,f +∆Uate
+
0
-
Represent this on the bar chart.
e) Why do we show ΔUate on the "after" side but not on the "before" side?
PUM | Work & Energy | Lesson 3: Reasoning with Energy Bar Charts 25
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
Lesson 4: Such Great Heights
4.1 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.
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
-
26 PUM | Work & Energy | Lesson 4: Such Great Heights
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
4.2 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
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.
-20 J
before
after
Ki + Ug,i + Usi + W = Kf + Ug,f + Us,f +∆Uint
-40 J
40 J
20 J
0
Write
an equation relating these quantities:
-20 J
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
-40 J
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 27
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, 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.
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
before
after
Ki + Ug,i + Usi + W = Kf + Ug,f + Us,f +∆Uint
4.3 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?
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.
28 PUM | Work & Energy | Lesson 4: Such Great Heights
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
4.4 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 yf = 0.
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? Discuss how this
expression can be used to write an expression for the gravitational potential energy of the
block-Earth system.
4.5 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.
Describe two experiments that you can perform to test these relationships. Include a sketch.
What does the relationship predict will happen? What are your assumptions?
PUM | Work & Energy | Lesson 4: Such Great Heights 29
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
4.6 Reason
Imagine that you could use one of two ramps to slowly move a cart to a position that is 2 m
above the ground.
3m
15 m
2m
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.
Homework
4.7 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)
4.8 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?
4.9 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?
4.10 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.
30 PUM | Work & Energy | Lesson 4: Such Great Heights
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
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.
4.11 Represent and Reason
Imagine that you throw a baseball out of your dorm room window 5 stories up 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.
before
after
before
Ki + Ug,i + Usi + W = Kf + Ug,f + Us,f +∆Uint
0
before
0
after
Ki + Ug,i + Usi + W = Kf + Ug,f + Us,f +∆Uint
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
0
PUM | Work & Energy | Lesson 4: Such Great Heights 31
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, 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?
h) Use the rubrics below to improve your lab report.
32 PUM | Work & Energy | Lesson 5: Galileo’s Pendulum
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
5.2 Test a Relation
Using the available equipment, you will conduct an experiment to test the relation you
developed for the gravitational potential energy of a system. To do this, you will use this
relation to predict the positions that two plumb bobs must be moved to in the drop chute to
give each Earth-bob system the same initial gravitational potential energy. Then, you will
use the apparatus to measure whether or not the system has the same initial potential energy.
a) State clearly the relation that you will test in the experiment.
b) For the two plumb bobs, determine the height of each in the Earth-bob system so
that both systems have the same potential to smash the clay. Be certain to state the
reference level that you are using.
c) Explain how you can use the apparatus to measure whether each system has the
same initial gravitational potential energy. State your assumptions.
d) 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?
e) Repeat the experiment one more time but use a different reference level.
f) What is your judgment about the relation that you were testing?
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.
Is able to
make a
reasonable
prediction
based on a
hypothesis.
Is able to
make a
reasonable
judgment
about the
hypothesis.
Needs some
improvement
A prediction is made
and is distinct from
the hypothesis but
does not describe the
outcome of the
designed experiment.
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.
No attempt is
made to make a
prediction.
A prediction is made
that is distinct from
the hypothesis but is
not based on it.
A prediction is made
that follows from the
hypothesis but does
not have an if-andthen structure.
No judgment is
made about the
hypothesis.
A judgment is made
but is not consistent
with the outcome of
the experiment.
A judgment is made
and is consistent with
the outcome of the
experiment but
assumptions are not
taken into account.
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-andthen structure.
A reasonable
judgment is made
and assumptions
are taken into
account.
PUM | Work & Energy | Lesson 5: Galileo’s Pendulum 33
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
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 1000 kg
block 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 piston pushes it a
distance d. The initial and final states of the process are pictured to the right.
a) Draw a force diagram for the block. Use it to find an expression for the force that the
piston 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 a work-energy bar chart, develop a
mathematical representation of this process
in terms of work, initial kinetic energy, and
final kinetic energy. 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.
6.2 Practice
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 mathematical and visual
representations for energy in solving the problem. Disregard the force exerted by the air on
the ball.
34 PUM | Work & Energy | Lesson 6: How to Calculate Kinetic Energy
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
6.3 Practice
If a stretched slingshot has 100 J of 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?
Homework
6.4 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? Specify the system, its initial and final states, and any
assumptions you made. Explain how these assumptions affect your answer.
6.5 Regular Problem
A man throws a 0.4-kg softball vertically into the air with an initial speed of 10 m/s. How
fast will it be traveling when it passes 1/3 of its maximum elevation?
6.6 Reason
Two identical water balloon slingshots are stretched the same distance so that they both
having the same 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?
6.7 Equation Jeopardy
Write a problem and draw an energy bar chart that would require the mathematical equation
below to solve it.
PUM | Work & Energy | Lesson 6: How to Calculate Kinetic Energy 35
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, 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 the Earth (not
Rona) are the system.
Draw FP on O –versus-y
(for vertical motion) or
x (for horizontal
motion) graphs.
Describe how to use
the graph to find the
work done by the
force.
If an object moves a
distance y or x,
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
the 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 the 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
y
Δy
Δy
y
y
36 PUM | Work & Energy | Lesson 7: The Energy in a Slingshot and Other Practical Things
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, 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.
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.
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 frame 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 frame you are using?
d) Represent the process with an energy bar chart.
e) After the spring is released, how high will the book fly?
7.4 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 frame you are using?
PUM | Work & Energy | Lesson 7: The Energy in a Slingshot and Other Practical Things 37
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
7.5 Regular Problem
How much work must be done on a spring with spring constant 100 N/m to change its
stretch from 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 frame you are using? Include a workenergy bar chart.
Homework
7.6 Jeopardy
Complete the table that follows and formulate a problem.
Problem:
Sketch with the reference frame:
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
y
Mathematical Representation and Solution:
Ig,i + W = Ug,f
W= Ug,f - Ug,i
38 PUM | Work & Energy | Lesson 7: The Energy in a Slingshot and Other Practical Things
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
7.7 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.
a) Over the first 20 meters, how much work does the teammate do on the system?
b) Over the next 20 meters, how much work does the teammate do on the system?
c) Over the last 10 meters, how much work does the teammate do on the system?
d) What is the total amount of work that the teammate does on the system?
e) On the axes below, graph the force exerted on the system by the teammate versus
the position of the system.
F (N)
x (m)
f) What property on the graph is equal to the work done by the teammate on the
system?
7.8 Regular Problem
Kristen pushes her little sister on a sled on a packed icy surface. Her little sister and the
sled 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.
PUM | Work & Energy | Lesson 7: The Energy in a Slingshot and Other Practical Things 39
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
40 PUM | Work & Energy | Lesson 7: The Energy in a Slingshot and Other Practical Things
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, 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 41
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
Hypothesis-prediction-testing rubric (used for testing experiments)
Scientific
Ability
Is able to
distinguish
between a
hypothesis/rela
tion and a
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.
Missing
An attempt
Acceptable
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.
Is able to make
a reasonable
prediction
based on a
hypothesis/
relation.
Is able to
identify the
assumptions
made in
making the
prediction.
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.
A prediction is made
that follows from the
hypothesis/relation but
does not have an ifand-then structure.
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.
Is able to
identify
experimental
uncertainty.
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/relatio
n.
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.
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.
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-andthen 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.
42 PUM | Work & Energy | Lesson 8: Spring into Action
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
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 frame 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 frame you are using.)
PUM | Work & Energy | Lesson 8: Spring into Action 43
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
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, elastic potential energy and a change in internal energy.
These mechanical forms of 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
System
constant 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?
44 PUM | Work & Energy | Lesson 9: Calculating the Internal Energy Change
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
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?
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
the 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 is:
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 45
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
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 the 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 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.
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.
46 PUM | Work & Energy | Lesson 9: Calculating the Internal Energy Change
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
9.6 Regular Problem
A sled slides down a frictionless hill that is 17.6 m high. At the bottom of the hill, the sled
hits a long patch of rough snow that slows down the sled by exerting an average force of
190 N on the sled. The mass of the sled and riders is 86 kg.
a) Draw a picture of the process, identify the initial and final states of the process and
specify the reference frame you are using.
b) How fast is the sled traveling after sliding for 10 m on the rough snow?
c) How far must the sled travel on the rough snow before it is traveling at ½ of its
maximum speed?
d) How far will the sled slide on the rough snow until it comes to rest?
PUM | Work & Energy | Lesson 9: Calculating the Internal Energy Change 47
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
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 Represent and reason
a) A skier with a mass of 70 kg is pulled up a slope by a motor driven cable. Assume
the ski slope is frictionless. What is the smallest work required to pull him to the top
of the hill, 60 m high?
b) How much power does the motor need to pull the skier to the top of the hill in 30
seconds?
c) How fast does the skier move up the hill? What assumption did you make about the
skier’s motion?
d) Read the definition for power. How did your definition compare? Correct your
answer and decide if you need to revisit the questions above now that you have this
information.
Did You Know?
The physical quantity of Power (P) describes how much work is done on a system per time
interval or the energy is transferred into or out of a system each time interval.
Power has units of 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
Determine how much power you exert while lifting:
a) a 10-kg object 1.0 m in 1.0 s
b) a 10-kg object 1.0 m in 0.5 s
c) a 10-kg object 2.0 m in 1.0 s
d) a 20-kg object 1.0 m in 1.0 s
48 PUM | Work & Energy | Lesson 10: Power Up
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, 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 the 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. 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.
PUM | Work & Energy | Lesson 10: Power Up 49
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
10.8 Regular Problem
An 82 kg hiker climbs to the summit of Mount Mitchell in western North Carolina. During
one 2.0 hr period, the climber's 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 frame you are using.
b) Determine the change in gravitational potential energy of the system climber-Earth
c) Determine the power generated to increase the gravitational potential energy of the
system.
10.9 Regular Problem
Determine how much power you exert while lifting:
a) a 3-kg object 1.2 m in 1.0 s
b) a 6-kg object 10.0 m in 6.0 s
c) a 10.4-kg object 2.6 m in 2.3 s
d) a 200-kg object 0.2 m in 10.0 s
50 PUM | Work & Energy | Lesson 10: Power Up
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
Lesson 11: Practice & Review
Problem-Solving Strategy: Work-Energy Problems
Sketch and Translate:
 Sketch the physical process described in the problem. For work-energy processes,
the sketch should include an initial state and a final state and a reference frame.
 Decide on your system. Objects such as the 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 the 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.
PUM | Work & Energy | Lesson 11: Practice & Review 51
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
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? Fill in the table below to determine the car’s initial speed.
Sketch and
translate
Represent using a workenergy bar chart
Represent
mathematically
Solve and evaluate
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 1.7 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. What assumptions did
you make? Is the force exerted by the seat belt on the dummy a safe amount?
52 PUM | Work & Energy | Lesson 11: Practice & Review
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
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
0.0030-kg 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? Fill in the table below.
Sketch and
translate
Represent using a workenergy bar chart
Represent
mathematically
Solve and evaluate
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. 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 53
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
Set 3: A Trigonometric Expression of Work
11.5 Regular Problem
A man exerts a force of 50 N on a wagon at an angle 35o above the direction of motion. He
exerts this force over a distance of 15 m. What is the change in energy of the wagon-Earth
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?
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 the 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.
54 PUM | Work & Energy | Lesson 11: Practice & Review
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, 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 the
Earth and the motor that
pulls the chain up the hill.
PUM | Work & Energy | Lesson 11: Practice & Review 55
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
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.
Generalized work-energy
equation applied to a process.
Sketch a process
that might be
described by the
equation.
Describe the process
in words.
Construct a bar chart.
a)
Ki Ugi Usi
W
Kf Ugf Usf∆Uint
0
b)
Ki Ugi Usi
0
56 PUM | Work & Energy | Lesson 11: Practice & Review
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
W
Kf Ugf Usf∆Uint
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.
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
car-pavement system. Clearly state all assumptions you made in your estimation.
PUM | Work & Energy | Lesson 11: Practice & Review 57
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
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
Rs
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
58 PUM | Work & Energy | Lesson 12: When Work is not Easy
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
b) Which of the following graphs could represent the gravitational potential energy of
the Earth-satellite system as a function of the separation between the two?
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
Separation between Earth and satellite
Ug
Ri
Rf
|
|
PUM | Work & Energy | Lesson 12: When Work is not Easy 59
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
c) Which of these mathematical expressions could represent universal gravitational
potential energy? If one cannot, explain why not.
Ug = 
GM m
e s +C
r
Ug = 
Gms
r
Ug = GME msr
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 the 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 the 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)?
60 PUM | Work & Energy | Lesson 12: When Work is not Easy
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
Homework
12.3 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 the Earth (6.3*106 m
from the center of the earth) to an altitude of 3.58*107 m from the center of the Earth, how
fast would the cannon have to fire the new satellite? At its final height, the satellite should
not be moving relative to the earth.
a) Sketch the initial and final states of the system.
v=0
v
New satellite
New satellite
Rf
Ri
Earth
Earth
b) Represent this process using a work-energy bar chart.
before
after
Ki + Ug,i + Us,i + W = Kf + Ug,f + Us,f +∆Uint
+
0
-
c) Represent this process mathematically.
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?
PUM | Work & Energy | Lesson 12: When Work is not Easy 61
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
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. 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 money in
order to escape debt.
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 the final
position the object has enough energy to escape 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 the Earth waiting to get into space. Think about the
interaction between the Earth and the satellite. Where is it the greatest? Where is it
non-existent? Now imagine at the bottom of well is the Earth (and the well has
disappeared) and the satellite is waiting on the Earth to get into space. Similar to the
person in debt, the satellite is in “energy debt” to the Earth.
1. Consider the place where there is no interaction between the Earth and the satellite.
Where is the value for zero gravitational potential energy?
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 the Earth.
3. Using one of the examples in part (b), draw an energy bar chart such that the final
position the satellite has enough energy to escape 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
Mass of the Sun = 2.0 x 1030 kg
drillers rendezvous with a menacing meteoroid just as it
Mass of the Earth = 5.96 x 1024 kg
passes the orbit of the moon. Imagine that an asteroid fell in
Radius of the 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 the Earth.
62 PUM | Work & Energy | Lesson 13: Oh Baby, Don’t Let Me Go
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
a) Use multiple representations to show the meteoroid’s journey from the Oort Cloud
to the Earth.
b) Determine the speed at which it passes the Moon’s orbit.
c) Let us say that Armageddon did arrive in that movie. With what speed would the
meteoroid have collided into the 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.
PUM | Work & Energy | Lesson 13: Oh Baby, Don’t Let Me Go 63
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
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.
Ki Ugi Usi
W
Kf Ugf Usf ∆Uint
0
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 the Earth. When the object is near the Earth’s surface,
we calculate the Earth-object’s gravitational potential energy using
where y is the object’s elevation relative to a chosen zero reference level. When far from the
Earth, we use the expression
where r is the object’s distance from the center of the Earth.
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
64 PUM | Work & Energy | Summary: Definitions and Principles
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
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:
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.
PUM | Work & Energy | Summary: Definitions and Principles 65
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
Lesson 14: Simple Machines I
14.1 Observe and Represent
Design an experiment to determine the amount of work it takes to move the cart from the
bottom to the top of the 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 carts 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?
66 PUM | Work & Energy | Lesson 14: Simple Machines I
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
Observational Experiment Rubric
Ability
Absent
An attempt
Is able to decide
what is to be
measured and
identify
independent and
dependent
variables.
It is not clear
what will be
measured.
It is clear what will
be measured but
independent and
dependent variables
are not identified.
Is able to use
available
equipment to
make the
measurements.
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 . 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?
PUM | Work & Energy | Lesson 14: Simple Machines I 67
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
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 the Earth?
14.4 Test Your Idea
a) Given the dimensions of another groups 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 you 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.
Homework
68 PUM | Work & Energy | Lesson 14: Simple Machines I
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
14.6 Represent and Reason
Integer statement
Bar chart for a process
Describe a
Process
3+2=5
4−2=1+1
200+800=500+500
8.2+4.1=12.4
PUM | Work & Energy | Lesson 14: Simple Machines I 69
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
Lesson 15: Simple Machines: Applications
15.1 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.)
(i)
2m
16m
(v) 1:10
(ii)
4m
46m
(vi) 3:42
(iii)
3m
36m
(iv)
0.5m
4m
(vii) 2:28
d) Calculate the smallest amount of work that is required to push a 20 kg object up
each of the ramps (i–iv). (lift the box)
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
lifting? What is the magnitude of this force? How can we determine this?”
70 PUM | Work & Energy | Lesson 15: Simple Machines: Applications
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
15.2 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 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. You need to make a ramp with a height of
1.5 m. What should the length of the ramp be to meet regulations?
c) The controller says that is too long. Give two reasons why it is important not to
make the ramp any steeper.
15.3 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.
Homework
15.4 Regular Problems
a) 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?
b) 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?
c) A meteoroid, moving with high speed, enters the Earth’s atmosphere and falls
toward the 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.
PUM | Work & Energy | Lesson 15: Simple Machines: Applications 71
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
d) 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 the 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, the 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.
72 PUM | Work & Energy | Lesson 15: Simple Machines: Applications
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
Lesson 16: Simple Machines II
16.1 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!.
c) Is it possible to reduce the amount of work needed to lift an object to a certain
height? Explain.
a) Write an H-D Statement using your experiment and Cadences hypothesis Use the
rubrics below to help you answer the questions in an informal lab report.
b) 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
Needs some
improvement
A prediction is made
and is distinct from
the hypothesis but
does not describe the
outcome of the
designed experiment.
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.
A prediction is made
that follows from the
hypothesis but does
not incorporate
assumptions.
No judgment is
made about the
hypothesis.
A judgment is made
but is not consistent
with the outcome of
the experiment.
A judgment is made
and is consistent with
the outcome of the
experiment but
assumptions are not
taken into account.
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.
PUM | Work & Energy | Lesson 16: Simple Machines II 73
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.
16.2 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.
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
16.3 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.
74 PUM | Work & Energy | Lesson 16: Simple Machines II
Adapted from A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, 2006
© Copyright 2009, Rutgers, The State University of New Jersey.