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PUM Physics II
Dynamics
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).
Lesson 1: Force as an Interaction
1.1 Observe and Represent
a) Pick up a tennis ball and hold it in your hand. Now pick up a medicine ball and hold it. Do you feel the
difference? How can you describe what you feel in simple words?
b) Think of how we represented the motion of objects in the last module. What are some possible ways of
representing the interaction between your hand and the tennis ball?
c) Let’s choose the ball as our object of interest. Represent the medicine ball with a dot and label the dot
with “Ball” Draw an arrow to show how your hand pushes the ball. Connect the tail of the arrow to the
dot. This arrow represents the force that your hand exerts on the ball.
Did You Know?
The word “force” is used in physics for a physical quantity that characterizes the interaction of two objects. A
single object does not have a force by default, as the force is defined through the interaction of two objects.
Remember that all physical quantities are measured in units. The unit of force is called the newton (N), where
1 N = (1 kg)(1 m/s2).
d) How could you label this force arrow to show that it is the force your hand exerts on the ball? Add this
label to your representation.
Here’s An Idea!
To show that the force arrow represents the push that the hand exerts on the ball, we can use a symbol F with
two little words at the bottom on the right. These are called subscripts.
For example: If we look at the interaction of a golf ball and a golf club while the club is hitting the ball. Then if
we choose the golf ball as the object of interest, the golf club exerts a force on the golf ball. As a label for an
arrow on a force diagram, this would be written as Fclub on ball.
e) What do you think would happen to the ball if your hand were the only object interacting with it? What
does this tell you about other objects interacting with the ball?
f) What other objects are interacting with the ball? List each object and the direction of the push or pull.
2 PUM | Dynamics | Lesson 1: Force as an Interaction
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.
1.3 Represent and Reason
a) In activity 1.1, did you say that gravity interacts with the ball? Gravity is not an object; you cannot hold
or touch it. So when we use the word gravity to note the pull down on all objects on Earth, what is the
object that exerts this downward pull?
b) Add another arrow on your diagram in 1.1 (c). Label the arrow with the appropriate subscripts.
c) What do you notice about the length of the arrows in your diagram? What do you think would happen if
the arrow representing the interaction with your hand were longer than the arrow due to the interaction
with the Earth? If it were the other way around?
d) Now draw a diagram for the heavy ball. How are the force arrows different from the arrows on the
diagram for the tennis ball?
1.3 Part B: How is the pull of the Earth on an object calculated?
The Earth pulls downward on objects so that they all have an equal acceleration (9.8 m/s/s). Two objects, one
with twice the mass of the other, will both accelerate at this rate under free-fall, even though common sense
tells us its harder to speed up a more massive object.
This constant acceleration occurs because the Earth pulls twice as hard on the twice as massive object. In other
words the force of the Earth on an object is proportional to its mass. As a matter of fact, we can calculate the
force needed to get a 1 kg object to accelerate at 9.8 m/s/s
Force of the Earth on an object = mass of object * 9.8
a) How hard does the Earth pull on a 20 kg object?
b) How hard does the Earth pull on a 35 kg object?
c) If a person weighs about 150 pounds, what is their approximate mass in kg?
d) How hard does the Earth pull on that person?
PUM | Dynamics | Lesson 1: Force as an Interaction 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.
Did You Know?
The diagrams you created in activity 1.1 through 1.3 are called force diagrams. Force diagrams are used to
represent the forces exerted on an object of interest (system) by other objects.
A system is an object or group of objects that we are interested in analyzing. Everything outside the
system is called the environment and consists of objects that might interact with and affect the system object’s
motion. These are external interactions. When we draw force diagrams, we only consider the forces exerted on
the system object(s).
1.4 Represent and Reason
a) Think of a word to describe the force arrows in each force diagram.
Did You Know?
When the forces exerted on an object of interest are balanced, we say that the object is in EQUILIBRIUM
(equilibrium does not necessarily mean rest).
b) How might we represent our force diagrams with a mathematical representation or math statement?
Write a math statement for the medicine ball.
Need Some Help?
Imagine putting an axis next to the force diagram with the origin at the dot. You can use + for the up direction
and – for the downward direction.
For example: Let’s take the situation of a puppy curled up in your lap. Then we can write the total force exerted
on the puppy by your legs and the Earth as: Flegs on dog + FEarth on dog = 0.
c) For your math statement, does it matter whether you chose up as positive or down as positive? How
would this affect the math statement you wrote? What happens to the total force exerted on the ball if we
switched the axis?
Did You Know?
Notice that depending on the orientation of the axis, either FHand on Ball or FEarth on Ball has a negative value, thus the
sum of a positive and a negative number can be zero. How do we know which force is positive and which one is
negative? If the force arrow points in the positive direction of the chosen axis, we consider the force positive. If
the y axis points down, for example, then FEarth on Ball >0 and FHand on Ball <0.
d) Look at your force diagrams for the tennis ball and medicine ball? What is the same about the diagrams?
What is different?
4 PUM | Dynamics | Lesson 1: Force as an Interaction
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.
1.5 Observe and Explain
a) Perform the experiments described in the first column. Then record your data and fill in the empty cells.
Remember that the scale, as a measuring instrument, has an uncertainty of measurement associated with
it.
Experiment
Draw a
picture of the
apparatus.
List objects
interacting
with the
object of
interest.
Draw a
force
diagram
for the
object.
Discuss what objects
exert forces balancing
the force that the
Earth exerts on the
object.
Write a
mathematical
expression for
the forces
exerted on the
object.
(a) Hang an object from
a spring scale. Record
reading of the scale here
______________
(b) Lower the object onto
a platform scale so it
touches the scale. Record
new reading of the spring
scale _______
Record the reading
platform scale
_______________
(c) Remove the spring
scale and leave the object
on the platform scale.
Record new reading
platform scale
_______________
(g) You place the block
on the platform scale and
then tilt the scale at a
small angle.
Record what happens
____________________
a) Some people think that only alive (animate) objects can exert forces. The table is not alive. How can a
table push on an object?
b) A book rests on top of a table. Jim says that the force exerted by the table on the book is always the
same in magnitude as the force exerted by the Earth on the book. Why would Jim say this? Do you
agree or disagree with Jim? If you disagree, how can you argue your case?
1.6 Reason
a) Summarize in what direction the force is exerted on an object of interest by the supporting object.
PUM | Dynamics | Lesson 1: Force as an Interaction 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.
b) Is this force always equal in magnitude and direction to the force that the Earth exerts on the object?
Provide experimental evidence and reasoning to support your opinion.
c) Look at the force diagram shown in the “Did You Know?” below. How would the force diagram change
if instead of dragging the box on a smooth floor, you dragged it on the carpet?
Did You Know?
The diagrams we constructed above are force diagrams. A force diagram is a physical representation used to
analyze and evaluate processes involving forces.
In order to create a force diagram, follow the 6 steps below.
SKETCH
FORCE DIAGRAM
1. Sketch the situation
y
2. Circle the object of
interest
FFloor on Box
3. Draw a dot
representing the
box
6. Label the
forces
FRope on Box
4. Identify interactions between
the system and other objects.
Here: Earth, floor, rope and
surface
FEarth on Box
Check for understanding:
What does the length of
an arrow on the diagram
mean?
5. Draw forces to
represent
interactions, watch
the length of
arrows
6 PUM | Dynamics | Lesson 1: Force as an Interaction
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.
Did You Know?
System: A system is the object of interest that we choose to analyze. Make a sketch of the process that you are
analyzing. Then circle the object of interest – your system. Everything outside that system is called the
environment and consists of objects that might interact with and affect the system object’s motion. These are
external interactions.
Force: A force that one object exerts on another characterizes an interaction between the two objects. The
force causes some effect or influence of the one object on the second object. Forces are represented by a
symbol with an arrow above it to show that the force has direction and with two subscripts indicating the
two objects. For example, if the Earth pulls on a ball, we note the force exerted by the Earth on the ball as:
FEarth on Ball .
The arrow above force indicates that force is the physical quantity that both has magnitude and direction.
The symbol also indicated that in this case our system is the ball and the Earth is the external object. If we
are interested in the force that the ball exerts on the Earth, we will write it as FBall on Earth .
1.8 Represent and Reason
A person pushes a box across a very smooth floor.
a) Examine the force diagram to the right. Do the forces in the vertical direction balance? Do the forces in
the horizontal direction balance?
b) Draw an arrow to indicate the direction of the unbalanced force, if there is one. Discuss whether the
result is reasonable.
1.9 Represent and Reason
Read each of the scenarios and then draw a force diagram for the selected object of interest.
1. You are throwing a tennis ball upward.
Consider the moment right before the ball leaves
your hand. The ball is the object of interest.
3. The ball is at the top of the flight. The ball is the
object of interest
PUM | Dynamics | Lesson 1: Force as an Interaction 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.
2. The ball is flying up. The ball is the object of
interest.
4. The ball is being caught by you. Consider the moment
when your hands are stopping the ball. The ball is the
object of interest.
Homework
1.10 Represent and Reason
a) Draw force diagrams and use them to determine the direction of the unbalanced force exerted on the
following objects of interest:
i. A hockey puck moving on ice slows to a stop. The puck is the object of interest.
ii. A box is sliding down an inclined plane. The box is the object of interest.
iii. You start lifting up a heavy suitcase; the suitcase is the object of interest.
iv.
A boat floats in the ocean; the boat is the object of interest.
v. You are pulling a sled on fresh snow at constant speed; the sled is the object of interest.
vi.
You are pushing a lawnmower; the lawnmower is the object of interest.
b) Examine the unlabeled force
diagrams
below and come up with a real
life situation
that they might describe. Then
label each
force with the appropriate
subscripts.
8 PUM | Dynamics |
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: Motion diagrams & Force diagrams
3.1 Observe and Represent
Consider the following experiment: You have a bowling ball and a board (or anything that rolls easily, a
billiard ball or a low friction cart on a track). You place the ball on the floor and push it with the board
continuously trying to exert a constant force.
a) Sketch the situation.
b) Perform the experiment, then describe the motion of the ball in words.
c) List all of the objects interacting with the bowling ball while it is being pushed.
d) Draw a motion diagram for the ball. Indicate the direction of the v arrow.
e) Draw a force diagram for the ball.
3.2 Represent and Reason
a) Look at the force diagram you drew in 3.1 Are there any forces that are balanced? If so, please indicate
which and explain why you think so.
b) Indicate if there is an unbalanced force exerted on the ball. Indicate the direction of the unbalanced
force with an arrow.
c) Indicate the direction of the velocity change arrow ( v ) on the motion diagram.
3.3 Observe and Represent
Consider this new experiment: You push the ball to start it moving. Once it is already rolling, you lightly
push the front of the moving bowling ball continuously with a board in the direction opposite to the direction
of motion.
a) Sketch the situation.
b) Perform the experiment and describe the motion of the ball in words.
c) List all of the objects interacting with the bowling ball while it is being pushed in the direction opposite
to its motion.
d) Draw a motion diagram.
e) Draw a force diagram for the ball.
f) Examine your force diagram. Indicate which forces are balanced and which forces are unbalanced. How
do you know? Draw an arrow to show the direction of the unbalanced force.
g) Indicate the direction of the change in velocity arrow on the motion diagram.
3.4 Represent and Reason
PUM | Dynamics | Lesson 3: Motion diagrams & Force diagrams 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.
Imagine that you have a bowling ball that is moving on a very smooth floor (neglect all friction forces). While
the ball is in motion, its velocity does not change.
a) Draw a motion diagram for the ball. What is the direction of the velocity change?
b) Draw a force diagram for the ball. What is the direction of the unbalanced force?
c) If the floor is infinitely long, how long will the ball move before it stops? Should it ever stop?
3.5 Find a Pattern
Consider the experiments you performed in activities 3.1 - 3.4. Examine the force and motion diagrams for
each experiment.
a) Is there a pattern in the directions of the unbalanced forces that other objects exert on the ball and in the
r
directions of the v arrows on the motion diagrams for the ball?
b) Is there a pattern in the directions of the unbalanced forces that other objects exert on the ball and the
directions of the v arrows in the motion diagrams?
c) Use the pattern that you found to formulate a statement relating the force diagram to the motion
diagram.
d) How do you understand the difference between the words “motion” and “change in motion”? Give an
example.
e) Do you think the net force exerted on an object causes motion or change in motion?
f) Who was the observer recording the velocity changes for the ball? Would there be observers for whom
the statement relating the force diagram to the motion diagram would not be true?
3.6 Test the Pattern
a) Go to PhET Forces and Motion Basics Simulation
b) Click on the Tug of War tab and experiment with the simulation.
i. What happens if the force is unbalanced (one side is winning)
ii. What happens if the force starts out unbalanced but then you balance it? Is this consistent
with the rule we have been forming about force and change in motion?
c) Click on the Friction tab and experiment with the simulation (click on speedometer so you can
observe both force and speed).
d) What happens if you reduce the friction to zero and push on the refrigerator until you get to 10
m/s? Does the refrigerator change its motion after this point?
e) Now turn friction up to maximum. Experiment with the simulation again. What changes and
can you explain why?
10 PUM | Dynamics | Lesson 3: Motion diagrams & Force diagrams
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.
f) Now turn friction to a medium level and run the simulation one more time. What changes do
you now notice in the motion of the object?
Homework
3.7 Test the Pattern
You have a medicine ball. When you place it on a bathroom scale, the scale reads 6 pounds (the unit of force in
the British system). Imagine that a friend drops a medicine ball, and it falls straight down on a bathroom scale.
a) Draw a force diagram for the ball when it sits on the scale at rest. Draw a motion diagram for the ball.
b) Draw a motion diagram for the ball when it just touches the scale but is not yet stopped.
c) Draw a force diagram to match the motion diagram.
Assume that the scale reads the force that the scale exerts on the ball. Make a prediction about the
reading of the scale as it stops the falling ball using the pattern between the motion diagram and the
force diagram you formulated and tested during the lesson.
3.8 Represent and Reason
a) Draw a motion diagram for a book sliding on a table coming to a stop. Draw a force diagram for the
book. Are the force diagram and motion diagram consistent with each other? Explain.
b) You are holding a birthday balloon filled with helium. Draw motion and force diagrams for the balloon.
Are the force diagram and motion diagram consistent with each other? Explain.
c) You are holding a birthday balloon filled with helium and then let it go. Draw motion and force
diagrams for the balloon the moment you let it go. Are the diagrams consistent with each other?
Explain.
d) The balloon reaches the ceiling. Draw motion and force diagrams for the balloon the moment the top of
it touches the ceiling. Check the consistency of your representations. Can you represent the balloon as a
particle in this case? Explain.
PUM | Dynamics | Lesson 3: Motion diagrams & Force diagrams 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.
Lesson 5: Inertial and Non-inertial Reference Frames
5.1 Observe and Analyze
You are sitting on a train and place a ping-pong ball on a tray table in front of you. The ping-pong ball is at
rest. All of a sudden, the ball starts rolling towards you. At the same time, your friend who was waiting for
your train to depart, saw the train starting to move in the direction in which you were facing, but she saw the
ball stationary and the train leaving from under it.
a) Describe the motion of the ball when it starts rolling using a motion diagram for each observer: you on
the train and your friend on the platform.
b) Explain the behavior of the ball when it starts rolling using a force diagram for each observer: you on
Newton’s first law of motion: We choose a particular object as the object of interest—the system. If no
other objects interact with the system object or if the sum of all the external forces exerted on the system
object is zero (forces in the y direction are balanced and forces in the x direction are balanced), then the
system object continues moving at constant velocity (including remaining at rest) as seen by observers in
the inertial reference frames.
Homework
5.4 Explain
A pendulum with a pendulum bob is attached to the ceiling of a car. When the car accelerates forward,
describe what the motion of the pendulum will be for an observer sitting in the car vs. an observer watching
from the curb. For which observer is Newton’s 1st Law valid?
6.10 Ranking Tasks
Examine the forces exerted on each object and the mass of each object. Rank the magnitude of the accelerations
of the objects from largest to smallest. Each arrow represents a force exerted by some other object on the object
of interest. Be sure to explain the reasoning behind your ranking.
A
B
C
1,000
g
400 g
200 g
D
E
F
1,000
g
500 g
200 g
12 PUM | Dynamics | Lesson 3: Motion diagrams & Force diagrams
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: Newton’s Second Law: Quantitative
7.1 Observe and Find a Pattern
Imagine an experiment in which one or more identical springs pull one or more identical carts in the same
direction on a smooth horizontal track. (The springs are stretched the same amount so that each spring exerts
the same force on the cart.)
Experiment number
1
2
3
4
5
6
7
8
9
Number of springs
0
1
2
3
4
1
1
2
2
Number of carts
1
1
1
1
1
2
3
2
3
Acceleration of carts
0
1.03 m/s2
1.98 m/s2
3.03 m/s2
3.95 m/s2
0.51 m/s2
0.32 m/s2
1.02 m/s2
0.66 m/s2
b) Use the data in the table above to devise a relationship that shows how the carts’ acceleration depends
on the carts’ mass and on the sum of the forces exerted on the carts by the springs, the Earth, and the
track.
7.2 Observe and Find a Pattern
Imagine springs are attached to both ends of a cart. The springs can pull the cart left or right. Each spring
pulls with the same strength, but the number of springs on either side of the cart can vary.
a) Examine the data in the table that follows.
Experiment
Number of springs
pulling to the right
Number of springs
pulling to the left
1
2
3
4
5
3
1
3
4
2
3
2
1
1
6
Acceleration
of the cart
0
–1.03 m/s2
1.98 m/s2
3.03 m/s2
–3.95 m/s2
7.3 Explain
In the two previous activities, you analyzed experiments in which the motion of an object was affected by
other objects.
PUM | Dynamics | Lesson 7: Newton’s Second Law: Quantitative 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.
a) Mathematically represent the relationship between the object’s acceleration, the unbalanced force
exerted on it by other objects, and its mass. Make sure that you write the relationship as a cause-effect
relationship.
Did You Know?
In the previous lessons, you have developed and tested Newton’s Second Law of Motion.
Newton’s Second Law of Motion: We choose a particular object, or group of objects, as our system object.
The acceleration a of the system is directly proportional to the unbalanced (net) force
r
r
r
r
Fnet  F1 on S  F2 on S  ...  Fn on S  Fn on S exerted by other objects on the system object and inversely
proportional to the mass m of the system object:
r
r

Fnet Fn on S
a

mS
mS
Homework
7.5 Reason
When you studied kinematics, you learned that all objects fall with the same acceleration: 9.8 m/s2. Use
this observational evidence and Newton’s Second Law to write a mathematical expression for the force that
the Earth exerts on any object.
7.6 Reason
a) Two forces are exerted on an object in the vertical direction: a 20 N force downward and a 10 N force
upward. The mass of the object is 25 kg. (1) What do you know about the motion of this object? (2)
Represent the motion of the object with a force diagram and a motion diagram.
b) You pull a 20-kg sled, exerting an unbalanced, horizontal force of 30 N on it for 10 seconds. (1) What is
the acceleration of the sled? (2) What is the speed of the sled after 3 seconds? (3) What force do you
need to exert on the sled if you wish to keep it going at that constant velocity?
c) You hang a picture using two ropes, each at an angle of 30 with the vertical. (1) Draw a sketch of the
situation. (2) Draw a force diagram for the picture. (3) If the mass of the picture is 5 kg, what is the force
that each rope must exert on the picture to keep it stable? (4) How can you use trigonometry to solve the
problem?
14 PUM | Dynamics | Lesson 7: Newton’s Second Law: Quantitative
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: Applying Newton’s Second Law
9.2 Regular Problem
In a grocery store, you push a 14.5 kg shopping cart. It is initially rolling at a constant speed of 2 m/s. You push
on it in the direction opposite to its motion exerting a force of 12 N.
a) Draw a force diagram and a motion diagram for the cart when you start pushing in the direction opposite
to its motion.
b) Assuming you push the cart exerting constant force for a while, how far will it travel in 3 seconds?
(Ignore friction for all parts of this problem.) Use the problem-solving strategy steps illustrated above.
Lesson 10: Newton’s Third Law: Qualitative
10.1 Observe and Explain
Student A and Student B both wear rollerblades or are on chairs with wheels. Student B pushes Student A
abruptly.
a) Observe what happens during the instant of the push to both students, and describe your observations in
words.
b) Draw motion diagrams and force diagrams for each student for the instant when B pushes A. Use the
diagrams to explain the observations.
10.2 Test your Idea
Use Newton’s Second Law and the explanation that you devised in the previous activity to predict what will
happen if Students A and B, both on rollerblades, start throwing a heavy medicine ball back and forth to each
other. If you have the equipment, perform the experiment and then check whether your prediction matches the
outcome. You can also watch the video of the experiment at:
http://paer.rutgers.edu/pt3/experiment.php?topicid=3&exptid=30
a) What was your hypothesis?
c) What was your prediction?
d) Did the outcome of the experiment prove the hypothesis to be right or fail to disprove it?
10.3. Apply
Examine a fan cart on your desk. Turn on the fan and observe its motion. Draw a motion diagram for the cart
and decide what object exerts the force to accelerate the cart. Use the sail on your desk to test your answer.
Record all your observations and the testing experiment very carefully.
PUM | Dynamics | Lesson 9: Applying Newton’s Second Law 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.
Homework
10.4 Represent and Reason
Fill in the empty spaces and draw pictures representing the situations. Place force arrows on the pictures;
remember to think about the lengths of the arrows.
a) When the Earth exerts a force on the book, the book exerts a force on __________
b) When a table exerts a force on the book, the book exerts a force on _________
c) When a tennis racket exerts a force on the ball, the ball exerts a force on_________
d) When car tires push back on the Earth’s surface, the Earth’s surface _________
10.5 Relate
List 5 everyday experiences that support the idea that when object B exerts a force on object A, object A will
simultaneously exert a force on object B. Discuss whether you can always observe the effects of these forces
that the interacting objects exert on each other. List possible reasons.
16 PUM | Dynamics | Lesson 9: Applying Newton’s Second Law
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: Newton’s Third Law: Quantitative
Did You Know?
Newton’s Third Law of Motion: When two objects interact, object 1 exerts a force on object 2. Object 2 in
turn exerts an equal-magnitude, oppositely directed force on object 1:
r
F1 on 2   F2 on 1 . Each force above is caused by one object and is exerted on another object. Since these two
forces are exerted on two different objects, they cannot be added to find a net force.
Homework
11.4 Reason
a) A book sits on the tabletop. What is the Newton’s Third Law pair for the force that the Earth exerts on
the book?
b) If the Earth exerts a 5 N force on the book, what is the force that the book exerts on the Earth?
c) What is the acceleration of the book if the Earth is the only object exerting a force on it?
d) Why does the book fall onto the Earth but the Earth does not fall onto the book? The mass of the Earth is
6.00 x 1024 kg.
Need Some Help?
Use the Earth’s mass to calculate the Earth’s acceleration. What does this tell you about the motion of the
Earth?
e) The Sun’s mass is 2.00 x 1030 kg. It pulls on the Earth, exerting a force of about 1020 N. What is the
force that the Earth exerts on the Sun?
11.5 Reason
Two students sit on office chairs with wheels. Student A pushes student B away from him. Student B does
nothing. Does student B exert the force on A? How do you know?
11.6 Reason
a) You hit a stationary puck with a hockey stick. The stick exerts a 100 N horizontal force on the puck.
What is the force exerted by the puck on the stick? How do you know?
b) A truck rear ends a small sports car that is moving in the same direction as the truck. The collision
makes the truck slow down and the sports car is propelled forward. What object exerts a larger force on
the other object: the truck on the car or the car on the truck. Explain how your answer reconciles with
Newton’s third law and with the fact that the sports car is damaged more than the truck.
PUM | Dynamics | Lesson 11: Newton’s Third Law: Quantitative 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.
c) The Earth pulls on an apple exerting a 1.0 N force on it. What is the force that the apple exerts on the
Earth? Why does the apple fall towards the Earth but the Earth does not move towards the apple?
d) The tree branch exerts a 1.0 N force holding the apple. What is the force that the apple exerts on the tree
branch?
11.7 Reason
Use Newton’s third law to predict what will happen if you try to open a door wearing rollerblades. Draw a force
diagram for yourself to help make the prediction.
11.8 Represent and Reason
Your friend says that if Newton’s third law is correct, no object would ever start moving. Here is his argument:
“You pull a sled exerting a 50 N force on it. According to Newton’s third law the sled exerts the force of 50 N
on you in the opposite direction. The total force is zero, thus the sled should never start moving. But it does.
Thus Newton’s third law is wrong.”
What is your opinion about this answer? How can you convince your friend of your opinion?
11.9 Reason
The Moon orbits the Earth because the Earth exerts a force on it. The Moon, therefore, has to exert a force on
the Earth. What is the visible result of this force?
11.10 Regular Problem
A person of mass m is standing on the floor of an elevator that starts from the first floor and reaches the 21st
floor.
a) Make two kinematics models for the motion of the elevator. Describe them in words. What is the same
about the two models? What is different?
b) Now describe the same models with motions diagrams, with position, velocity and acceleration versus
time graphs, and with algebraic functions.
c) Choose one of the models and draw force diagrams for the person for three different parts of the trip.
d) Write a mathematical expression that will help you determine the magnitude of the force that the person
exerts on the floor when the acceleration of the elevator is upward and again when the acceleration is
downward. What is a reasonable magnitude for the elevator’s acceleration?
e) Who is pushing harder – the elevator’s floor on the person or the person on the floor? The Earth on the
person or the floor on the person?
18 PUM | Dynamics | Lesson 11: Newton’s Third Law: Quantitative
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: Two-Body Problems
12.1 Represent and Reason
1
2
You push two crates that have different masses on a smooth surface. Fill in the table that follows for the two
situations. Use the rule that relates the forces that two objects exert on each other while interacting (devised in
lesson 11).
Situation 1: You push crate 1, which pushes against the smaller crate 2.
Situation 2: You now reverse the positions of the crates and push crate 2, which pushes on larger crate 1.
(a) You push crate 1.
Show the force that 2 exerts on 1.
Show the force that 1 exerts on 2.
1
(b) You push crate 2. Show the force that 1 exerts on 2.
2
Show the force that 2 exerts on 1.
1
c) Based on the diagrams in (a) and (b), should it be easier to push the crates in one situation than the
other? Explain.
d) Calculate the sum of the forces from part (a), the force crate 1 exerts on crate 2 and the force crate 2
exerts on crate 1. How does this compare to the sum of these forces for part (b)? What does this imply
about the magnitude of the force of one crate on the other and vice versa?
12.2 Represent and Reason
This time, instead of pushing two crates, you connect them with a rope and attach another rope to crate 1.
You pull this second rope horizontally, exerting a force Fyou on crate1. The masses of the crates are m1 and m2.
a) Find the acceleration of the crates. Decide what assumptions you need to make to solve this problem.
Follow the problem-solving strategy.
b) Find the force that the rope connecting the two crates exerts on crate 2. Again, be sure to follow the
problem-solving strategy.
PUM | Dynamics | Lesson 12: Two-Body Problems 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.
1
k
g
12.3. Test your ideas
a) Examine the experimental setup in the video experiment at:
http://paer.rutgers.edu/pt3/experiment.php?topicid=3&exptid=105
b) Before watching the video predict how much each object in the video will move in 1 second. (Based on
Newton’s 2nd Law) Your prediction should contain an uncertainty value with it. Make sure you follow
the problem-solving strategy closely and list all of your assumptions.
c) How might the result be different from your prediction if the assumptions are not valid?
Homework
12.3 Continued
Perform the experiment (watch the video) and record the outcome.
Clearly describe of how you found whether the prediction matched the outcome of the experiment.
What can you say about Newton’s Laws based on this experiment?
12.4 Represent and reason
S
p
1r
i
kn
gg
p
u
s
h
e
s
d
o
w
n
12.5 Represent and Reason
Examine the system on the right. Jon says that the force the rope
exerts on the cart is always equal to m1 g. Why would Jon have such
an opinion? Do you agree or disagree? Explain your answer.
m1
12.6 Reason
You use the setup described in activity 12.5. You first hold the cart with your hand so the system is at rest.
Then you abruptly push the cart to the left and let it go. Describe the motion of the cart in words after you let
it go. Explain the motion using force diagrams for both the cart and the hanging object. Then sketch the
acceleration versus time graph for the cart.
20 PUM | Dynamics | Lesson 12: Two-Body Problems
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: Friction
14.1 Observe and Find a Pattern
Perform the following experiment: Rest a wooden block (or some other object, like your shoe) on a table.
Attach a large spring scale to a string attached to the front of the block. Pull the scale harder and harder. Notice
what happens to the scale reading while the block does not move. Notice the reading right before the block
starts moving and right after. Keep the block moving but not accelerating.
a) Describe in words how the magnitude of the force that the table’s surface exerts on the block varies with
the force exerted by the spring pulling on the block.
b) Compare the magnitude of the force just before the block starts moving to the magnitude when it is
moving at a constant velocity. What do you observe?
c) What object is exerting this friction force for the scenarios given above?
d) Summarize your findings for the friction force exerted on an object at rest and on the same object
moving at a constant velocity.
Did You Know?
The friction force is a resistive force exerted by the surface on an object. There are two kinds of friction forces
you observed in the experiments above. The static friction force is variable. As you saw, once the maximum
static friction force is overcome, the object will start to move. The kinetic friction force is the resistive force
exerted on a moving object.
14.2 Observe and Find a Pattern
Instead of the block in the previous activity, you have rectangular blocks with different surface areas and
different types of surfaces on which the block slides horizontally.
The force that the string exerts on the block (as measured by the spring scale reading) when the block just starts
to slide is recorded in the table that follows. This force is equal in magnitude to the maximum static friction
force (as we discovered in the previous activity). Examine the data in the table that follows.
Mass of the block
1 kg
1 kg
1 kg
1 kg
1 kg
1 kg
Surface area
0.1 m2
0.2 m2
0.3 m2
0.1 m2
0.1 m2
0.1 m2
Quality of
surfaces
Medium smooth
Medium smooth
Medium smooth
A little rougher
Even rougher
Roughest
Maximum
static friction force
3.1 N
3.0 N
3.1 N
4.2 N
5.1 N
7.0 N
Now decide how the maximum static friction force that the surface exerts on the block depends on the surface
area of the block and on the roughness of the two surfaces.
PUM | Dynamics | Lesson 14: Friction 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.
14.3 Observe and Find a Pattern
A spring scale pulls a 1 kg block over a medium smooth surface. The reading of the scale can be used to
determine the magnitude of the maximum static friction force—in this instance, the force when the block starts
to slide. In some experiments, a compressible spring also pushes vertically down on the block (see the second
block).
Use the data in the table to draw a graph of the maximum static friction force versus the normal force the
surface exerts on the block.
Extra downward
force exerted on the
1-kg block
0N
5N
10 N
20 N
Normal force
exerted by the
board on the block
10 N
15 N
20 N
30 N
Maximum
static friction
force
3N
4.5 N
6N
9N
b) Express mathematically a relationship between the normal force and the maximum static friction
force.
14.5 Reason
a) Take a textbook and drag it with your pinky finger. Repeat but this time have your neighbor push down
lightly on the book. Repeat 3 more times with your neighbor pushing down successively harder. Draw a
force diagram for each case. What can you say about the maximum static friction force?
b) Consider the previous activity. Why would we consider the normal force exerted on the object rather
than the force of the Earth exerted on the object?
c) A person is holding a book against a vertical wall, pushing on it horizontally. The book is at rest. Draw a
force diagram for the book. Check if all forces balance. Which force prevents the book from falling
down? Why, if you do not push on the book hard enough, does the book start falling?
Did You Know?
Normal force: When two objects touch each other, they exert a normal force on each other. The force of the
one object on the other object points perpendicular to the surface of contact. Often one symbol N is used to
denote this force (do not confuse with the Newton, N). There is no equation for calculating the normal force. Its
magnitude must be determined for each situation by some other method.
Static friction force: When two objects touch each other, they exert a friction force on each other. The
friction force of the one object on the other object points parallel to the surfaces of contact. If the objects are
not moving with respect to each other, the friction force that they exert on each other is static. The static
22 PUM | Dynamics | Lesson 14: Friction
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.
friction force between two surfaces opposes the tendency of one surface to move across the other and provides
flexible resistance (as much as is needed) to prevent motion—up to some maximum value. This maximum
static friction force depends on the relative roughness of the surfaces (on the coefficient of static friction µs
between the surfaces) and on the magnitude of the normal force N between the surfaces. The magnitude of
the static friction force is always less than or equal to the product of these two quantities:
Fs surface on object  s N
Kinetic friction force: The kinetic friction force between two surfaces is exerted parallel to the surfaces and
opposes the motion of one surface relative to the other surface. The kinetic friction force depends on the
relative roughness of the surfaces (on the coefficient of kinetic friction µk) and on the magnitude of the
normal force N between the surfaces:
Fk surface on object  k N
Homework
14.7 Evaluate
Jamie says that the force of friction is something that we should reduce in order to make the cars go faster.
What friction force could she mean? Do you agree or disagree with her opinion? If you agree, how would
you argue for it? If you disagree, how would you argue against it?
14.8 Represent and Reason
Some students are trying to move a heavy desk across the room. Diana pushes it across the floor at the same
time that Omar and Jeff pull on it. Omar pulls on the desk, exerting a (-150) N force, and Jeff pulls exerting a
(-125) N force. There is also a (-200) N friction force exerted by the floor on the desk. The net force exerted
on the desk is 27 N.
a) Make a sketch of the situation.
b) Draw a force diagram for the desk. Draw a motion diagram.
c) Write an algebraic statement that describes the force diagram you drew.
d) How hard is Diana pushing?
e) Is the desk moving with a constant velocity or is it speeding up? How do you know?
f) What would happen if, after a few seconds, the boys stopped pulling?
PUM | Dynamics | Lesson 14: Friction 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.
Lesson 15: Putting It All Together
15.1 Reason
You stand on a bathroom scale that reads 712 N (160 lb) (You can use your own weight in newtons in this
problem if you wish). You place the scale on an elevator floor and stand on the scale.
a) What does it read at the beginning of the ride when the elevator accelerates up at 2.0 m/s2?
b) What does the scale read when the elevator continues to move up at a constant speed of 4.0 m/s?
c) What does it read at the end of the ride when the elevator slows down at a rate of magnitude 2.0 m/s2?
15.5 Reason
James Steward, 2002 Motocross/Supercross Rookie of the Year, is leading the race when he runs out of gas near
the finish line. He is moving at 16 m/s when he enters a section of the course covered with sand where the
effective coefficient of friction is 0.90. Will he be able to coast through this 15-m long section to the finish line
at the end? If yes, what is his speed at the finish line? What assumptions did you have to make to solve this
problem?
15.6 Reason
According to Auto Week magazine, a Chevrolet Blazer traveling at 60 mph (97 km/h) can stop in 48 m on a
level road. Determine the coefficient of friction between the tires and the road. Do you think this is coefficient
of kinetic or static friction? Explain.
15.7 Reason
A 50-kg box rests on the floor. The coefficients of static and kinetic friction between the bottom of the box and
the floor are 0.70 and 0.50, respectively. (a) What is the minimum force a person needs to exert on it to start the
box sliding? (b) After the box starts sliding, the person continues to push it exerting the same force. What will
happen to the box? Answer this question quantitatively.
15.8 Reason
A wagon is moving to the right faster and faster. A book is pressed against the back vertical side of the wagon
and does not slide down. Explain how this can be.
24 PUM | Dynamics | Lesson 15: Putting It All Together
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.
Lessons 18: Practice
18.1 Represent and Reason
A book rests on a table.
a) Draw a sketch of the situation and identify objects that interact with the book.
b) Draw forces representing these interactions (a force diagram for the book).
c) If the book is stationary, these forces are equal in magnitude and opposite in direction. Can we say that
they represent Newton’s Third Law pair forces? If not, why not?
d) Draw the Newton’s Third Law force pairs for each force shown in the force diagram from part (b).
Identify the cause of each of these forces and the objects on which each of these forces is exerted.
18.2 Regular Problem
A large plane with a mass of 3.5 x 105 kg lands on a runway at a speed of 27 m/s. If the frictional force exerted
by the road and the air on the plane is 4.3 x 105 N
a) How long does it take the plane to stop?
b) How far does the plane travel in this time?
c) What is the effective coefficient of friction?
d) What is the force that the plane exerts on the runway?
18.3 Regular Problem
The driver of a 1560-kg Toyota Avalon, traveling at 24 m/s on a level, paved road, hits the brakes to stop for a
red light. Determine the distance needed to stop the car if the coefficient of kinetic friction between the car tires
and the road is 0.80.
18.4. Regular Problem
To give a 17 kg child a ride on a 3.4 kg sled, two teenagers pull at 35° angles to the direction of the sled’s
motion (see picture). The unpacked snow exerts a frictional force of 57 N. If both teenagers pull, each exerting
a force of 55 N, what is the acceleration of the sled?
PUM | Dynamics | Lessons 18: Practice 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.
18.5 Regular Problem
Two of your neighbor’s children (40 kg together) sit on a sled. You push on the back child, exerting a 50 N
force on him directed 37o below the horizontal. The sled slides forward with a constant velocity. Complete the
table below to answer the question: What is the coefficient of kinetic friction between the snow and the sled?
18.7 Evaluate the solution
Identify any errors in the solution to the following problem and provide a
corrected solution if there are errors.
The problem: You push a 20-kg lawn mower, exerting a 100-N force on it. You
push 37o below the horizontal. The effective coefficient of kinetic friction
between the grass and mower is 0.60. Determine the acceleration of the lawn
mower. Assume that g = 10 m/s2.
Proposed solution: The situation is pictured at the right. The mower is the
object of interest and is considered a particle. The forces that other objects
exert on the mower are shown in the force diagram. The magnitude of the
kinetic friction force is:
,,
fk = µk N = 0.60(20 kg)(10 m/s2) = 120 N.
The acceleration of the mower is:
a = (F – fk)/m = (100 N – 120 N)/(20 kg) = –1.0 m/s2.
18.8 Design an Experiment
Design a balloon racer. You are given 2 balloons, straws, paper,
The racer should be designed using your understanding of
You will race this balloon racer against other students in the
and tape.
“forces”.
class.
a) Design a method in which you can determine the time it
takes to travel a given distance when relative uncertainty is taken into account.
b) Design a method to determine the average acceleration during this time when relative uncertainty is
taken into account.
c) Design a method in which to determine the average force the air pushing its way out of the balloon
exerts on the balloon itself when relative uncertainty is taken into account. Be sure to include force
diagrams
26 PUM | Dynamics | Lessons 18: Practice
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 19: Review
19.1 Equation Jeopardy
Several mathematical statements are listed below. For each statement, describe a problem for which this
statement could be a solution. Then represent the statement using a force diagram and a motion diagram. For
one of the forces involved in the situation find Newton’s third law pair.
a) Funbalanced on object = (9.8 N/kg) x (3 kg)
b) (-7 m/s) + (2 m/s) = (3 s) x a
c) (-35 N) + (9.8 N) = (1 kg) x a
d) Frope on sled – FJake on sled = (35 kg) x (0 m/s2)
e) a 
m1  m2
g
m1  m2
f) (70 N) cos 300 – 0.4Ffloor on crate = (5 kg) x a
19.2 Diagram Jeopardy
Six force diagrams are shown below. Describe a situation for each diagram; be sure each diagram can represent
the situation created for it. For each situation, in what direction is the object moving? How many answers can
you have? Draw a matching motion diagram and write Newton’s Second Law in component form for each
scenario.
19.3 Graph Jeopardy
Three lines on the graph below describe three motions of an
object. Tell a story about each motion. Draw a motion
diagram and a force diagram. How many answers can you
have? Determine the unbalanced force in each case if the
mass of the object is 250 kg.
19.4 Reason
The Earth exerts a 5-N force on an apple, what is the force that the apple exerts on the Earth?
PUM | Dynamics | Lesson 19: Review 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.
19.6 Reason
A horse is pulling a cart. According to Newton’s third law the force that the cart exerts on the horse is always
the same in magnitude and opposite in direction to the force that the horse exerts on the cart. How does the
horse ever manage to get the cart moving?
19.7 Reason
A woman pushes a 60 kg couch along a rough surface. The couch accelerates at a rate of 0.5 m/s2. Coefficient
of kinetic friction between the couch and the floor μk = 0.13. Make a list of physical quantities you can
determine using this information and determine 2 of those quantities.
19.8 Reason
A football player exerts a force of 1800 N to push a 40 kg blocking sled with an acceleration of 10 m/s2 over a
very rough surface. Make a list of physical quantities you can determine using this information and determine 2
of those quantities.
19.9 Reason
A car locks its brakes and skids to a stop with an acceleration of 4 m/s2. For tires on the road, μk = 0.25.
Assume the car has a mass of 2000 kg
19.10 Reason
Mr. T. pulls a 400 kg walk-in refrigerator behind his car as he drives. The road exerts a 3000 N force on the car
but the car does not accelerate. Explain why. Make a list of quantities you can determine using this
information. Determine 2 of them.
19.11 Reason
A football player exerts a force of 1800 N to push a 40 kg blocking sled on a rough surface. The μk between the
surface and sled is 0.5. Determine everything you can using this information.
19.12 Reason
A car slows to a stop with an acceleration of 8 m/s2. Assume the force of friction exerted by the air and the road
on the car is 15000 N. Pose a question about this situation that you can answer and provide additional
information if necessary.
19.13 Reason
Two objects of masses m1 and m2 are connected with a light rope going over a light pulley. Draw a picture
representing is situation. Then determine the accelerations of the object when the system is let go and the force
that the rope exerts on both objects. How many different scenarios can you come up with? How are the
acceleration and the force difference depending on the scenario?
28 PUM | Dynamics | Lesson 19: 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.