Download Dynamics Notes/Labs/HW

PHYSICS UNION
MATHEMATICS
Physics II
Dynamics
Student Edition
Supported by the National Science
Foundation (DRL-0733140).
PUM Physics II
Dynamics
Most of the module activities were adapted from:
A. Van Heuvelen and E. Etkina, Active Learning Guide, !
Addison Wesley, San Francisco, 2006. !
Used with permission.!
!
Some activities area based on FMCE (Thornton and Sokoloff) and on
Ranking Task Exercises in Physics (O’Kuma, Maloney, and Hieggelke).!
!
Contributions of: E. Etkina, M. Blackman, T. Bartiromo, A. Boudreaux, S. Brahmia, J. Chia, C.
D’Amato, J. Flakker, J. Finley, S. Kanim, R. Newman, J. Santonacita, E. Siebenmann, S. Soni,
R. Therkorn, K. Thomas, M. Trinh. !
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 | Dynamics |## #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
Table of Contents
LESSON!1:!FORCE!AS!AN!INTERACTION!
4!
LESSON!1!A!(SUPPLEMENT):!VECTORS!AND!THEIR!COMPONENTS!
14!
LESSON!2:!EXPERIMENTAL!DESIGN!AND!ASSESSMENT!
18!
LESSON!3:!MOTION!DIAGRAMS!&!FORCE!DIAGRAMS!
21!
LESSON!4:!WHAT’S!A!FORCE!TO!DO?!
26!
LESSON!5:!INERTIAL!AND!NONCINERTIAL!REFERENCE!FRAMES!
31!
LESSON!6:!NEWTON’S!SECOND!LAW:!QUALITATIVE!
36!
LESSON!7:!NEWTON’S!SECOND!LAW:!QUANTITATIVE!
41!
LESSON!8:!DESIGN!AN!EXPERIMENT!
48!
LESSON!9:!APPLYING!NEWTON’S!SECOND!LAW!
52!
LESSON!10:!NEWTON’S!THIRD!LAW:!QUALITATIVE!
56!
LESSON!11:!NEWTON’S!THIRD!LAW:!QUANTITATIVE!
58!
LESSON!12:!TWOCBODY!PROBLEMS!
64!
LESSON!13:!DESIGN!AN!EXPERIMENT!
68!
LESSON!14:!FRICTION!
72!
LESSON!15:!PUTTING!IT!ALL!TOGETHER!
78!
LESSON!16:!COMPONENTS!
81!
LESSON!17:!FINDING!THE!COEFFICIENT!
88!
LESSONS!18:!PRACTICE!
91!
LESSON!19:!REVIEW!
97!
PUM | Dynamics | 3#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
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?
4! PUM | Dynamics |##Lesson#1:#Force#as#an#Interaction #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
f) What other objects are interacting with the ball? List each object and the direction of the
push or pull.
1.2 Test Your Reasoning
a) In the previous activity, did you say that air interacts with the ball for part f? How do you
think air interacts with the ball?
b) What experiment can you perform to test your idea about whether the air pushes up or
down on the ball?
c) Use the video experiment on the website or CD if your class does not have the
equipment. Before watching the video or performing the experiment, write a prediction of
what should happen to the bottle based on your hypothesis.
d) Watch the video or perform the experiment: Bottle in Vacuum. (Or on the CD it is on the
List of Videos, Bottle in a vacuum).
e) Summarize what effect the air has on the ball.
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 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?
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.
PUM | Dynamics | Lesson#1:#Force#as#an#Interaction 5#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
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 add to zero (here you need to take the direction
of the force into account), 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 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?
e) Write an expression for the forces exerted on the tennis ball similar to the expression you
wrote for the medicine ball. Is the tennis ball in equilibrium? Explain.
6! PUM | Dynamics |##Lesson#1:#Force#as#an#Interaction #
© Copyright 2014, 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 that being
added to the force that
Earth exerts on the
object make the sum
of the forces zero.
Write a
mathematical
expression for
the forces
exerted on the
object. Specify
your axis.
(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
_______________
(d) You place the object
on a horizontal meter stick
whose ends rest on two
blocks. Record what
happens
_____________________
(e) You place the object
on a thick, foam cushion.
Record what happens
_____________________
PUM | Dynamics | Lesson#1:#Force#as#an#Interaction 7#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
(f) You place the object
on a tabletop.
Record what happens
_____________________
(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 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 (think of different cases, including the case when the supporting object
is tilted).
b) Is this force always equal in magnitude and direction to the force that 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.
8! PUM | Dynamics |##Lesson#1:#Force#as#an#Interaction #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
.
SKETCH
FORCE DIAGRAM
1. Sketch the situation
!
2. Circle the object of
interest
3. Draw a dot
representing the
box
y
FFloor#on#Box#
!!
4. Identify interactions between
the system and other objects;
Earth, floor, rope and surface
FEarth#on#Box#
6. Label the
forces
FRope#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
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 Earth pulls on a

ball, we note the force exerted by 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 Earth is the
external object. If we are interested in the force that the ball exerts on Earth, we will write it as

FBall on Earth .
1.7 Reason
Describe a situation in which a surface exerts ONLY a horizontal force on the object. Draw a
picture of the situation. Then draw a force diagram.
PUM | Dynamics | Lesson#1:#Force#as#an#Interaction 9#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
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 add to
zero? Do the forces in the horizontal direction add to zero?
b) Draw an arrow to indicate the direction of the sum of the
forces. 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
10! PUM | Dynamics |##Lesson#1:#Force#as#an#Interaction #
© Copyright 2014, 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.
#
Now use the rubric below to self-assess your force diagrams. Do not worry about the length of
force arrows yet.
Missing !
An attempt!
Needs improvement!
Acceptable!
No force diagram
is constructed. #
Force diagram is constructed but
contains major errors: missing or
extra forces (not matching with the
interacting objects), incorrect
directions of arrows, or incorrect
relative length of force arrows. #
Force diagram contains
no errors in force arrows
but lacks a key feature
such as labels of forces
with two subscripts or
forces are not drawn
from single point. #
The diagram contains
forces for each interaction
and each force is labeled so
that one can clearly
understand what each force
represents. Relative lengths
of force arrows are correct. #
Homework
1.10 Represent and Reason
a) Draw force diagrams and use them to determine the direction of the sum of the forces
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.
PUM | Dynamics | Lesson#1:#Force#as#an#Interaction 11#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
b) Examine the unlabeled force diagrams and come
up with a real life situation that they might
describe. Then label each force with the
appropriate subscripts.
1.11 Reason You are sitting on a chair reading this
problem. What are the objects that exert forces on the chair? Choose the right answer:
(a) no forces are exerted on the chair; (b) you are exerting a force; (c) you, Earth and
gravity; (d) Earth, floor, and you; (e) floor, you and gravity.
1.12 Represent and reason You are holding a medicine ball steady. Choose the force diagram
that represents the forces exerted on the ball best:
FH on B
FE on B
#
FH on B
#
#
FH on B
FE on B
Fgravity
a)
b)
c)
d)
e)
1.13 Estimate uncertainty
Rob and Tina collected data using a scale that had divisions every newton (N): 0 N, 1 N, 2 N, 3
N, etc. When Tina hung her bag on the spring scale, she wrote the reading of the scale as 2.2 N.
Rob repeated the experiment and wrote the reading as 2.3 N. They used the same bag. Why are
their numbers different? Who do you think is correct? Based on your answer, decide how
precisely you can measure the force with this scale.
1.14 Estimate uncertainty
Find three measuring devices in your house (each one needs to show the quantity that it is
measuring, the units of measurement, and a scale).
a) Write down the experimental uncertainty for each instrument. You may need to recall
how we determined experimental uncertainty in the kinematics module.
b) Now take a measurement with each instrument and write the result so that it incorporates
the experimental uncertainty.
12! PUM | Dynamics |##Lesson#1:#Force#as#an#Interaction #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
1.15 Estimate Uncertainty You drop a tennis ball from the second floor window three times.
Your friend Tina is using a stopwatch to record the time of flight. Here are the results of her
measurements: 1.11 s; 1.14 s; and 1.12 s. What is the average time Tina measured? What is the
random uncertainty of the measurement? Write the result in the form of t = t average ± Δt .
1.16 Significant Figures Two friends are discussing their cash allowance. Jake says that he gets
$15 a month. Robin says that she gets $15.00 a month. Who is more likely to get more per
month?
Reflect: What did you learn about forces in this lesson? You heard the
expression “May the force be with you.” Explain why it is not a possible thing
for a force in physics.
PUM | Dynamics | Lesson#1#A#(Supplement):#Vectors#and#Their#Components 13#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
Lesson 1 A (Supplement): Vectors and Their Components
1a.1 Represent and Reason
(a) In activity 1.5 you used a spring scale and a platform scale. You lowered the object onto a
platform scale so it touched the scale. You recorded the new reading of the spring scale _______
and the reading of the platform scale ________. Represent the situation with a force diagram.
Assign magnitudes to the forces. What is the sum of the forces exerted on the object in this
situation?
(b) Obtain a block (a textbook will work too) and attach two spring scales to it, so that they are
parallel to the desk surface, make a 600 angle with each other and show the same reading.
Observe what happens to the box. Represent the situation with the force diagram. What is the
sum of the forces exerted on the book?
Did You Know?
There are two general types of physical quantities—force is an example of one type and
temperature is an example of the other type. As you know, forces can be exerted in different
directions (direction is very important if you are trying to hammer a nail into the wall), while
there is no direction associated with your body temperature. Physical quantities that contain
information about their magnitude and direction are called vector quantities and are represented
 
by symbols with an arrow on top ( F , v , etc. ). For example, force is a vector quantity. When you
push a door, your push can be represented with a force arrow; the stronger you push, the longer
that arrow must be. The direction of the push is represented by the direction of the arrow on the
force diagram. When you add vector quantities you need to take the magnitude of the arrow into
account and its direction too. Two forces of the same magnitude and opposite direction when
added together, result in a zero force.
Physical quantities that do not contain information about direction are called scalar
quantities and are written using italic symbols ( m, T , etc. ). Mass is a scalar quantity as is
temperature. To manipulate scalar quantities, you use standard arithmetic and algebra rules addition, subtraction, multiplication, division, etc. You add, subtract, multiply, and divide scalars
as though they were ordinary numbers. Vectors are more complicated. For example, you just
pulled on the box exerting equal strength pulls at an angle of 600 with respect to each other. Was
the resultant pull twice as large as the pull exerted by each person, or more or less? In what
direction was the resultant pull? To answer these questions, we need to learn how to perform
mathematical operations on vectors.
14! PUM | Dynamics |##Lesson#1#A#(Supplement):#Vectors#and#Their#Components #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
 
a) Addition A + C
To add two vectors we will use the following graphical
technique to illustrate. Suppose we want to add the two



vectors A and C . To add them graphically, we redraw A and


place the tail of C at the head of A , keeping the length and
the orientation of both vectors the same as before (we can
move vectors parallel to each other; therefore moving vectors


A and C from their original location will not change them.
While we can move vectors from one place to another for
addition, we cannot change the magnitude or direction of a

vector while moving it). Having moved vector C , we draw



another vector, R , from the tail of A to the head of C as in

the figure at the right. This vector R represents the result of


the addition of two vectors A and C . We can write the
  
resultant vector as a mathematical equation: R = A + C . This vector has a magnitude and
direction. As you see in the Figure, the magnitude of the vector is not equal to the sum of the


magnitudes of A and C .
#
 
b) Subtraction A − C
You are familiar with subtraction a little bit already – this is

what we do when we find the Δv vector on motion diagrams.

 
Basically, to find the vector P , which is equal to A − C , you


need to find the vector that you need to add to C to get A .
c) Practice
Label each vector in the pairs of vectors below and find their
#
A

A#
K#

C#
  
P = A−C
  #
C+P=A

C#
sum and their difference.
#
d) Make up 10 examples of vector addition, in 3 of them the vector s should be in the same
direction, in 3 – in the opposite and in 4 – at an angle. Add the vectors and then act out every
example. You can use any props available, including people.
PUM | Dynamics | Lesson#1#A#(Supplement):#Vectors#and#Their#Components 15#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
Check your Tip to Tail Vector Diagrams:
Scale present (ex. 1 cm = 1 m/s)
Orientation of the diagram remains consistent
Vectors are labeled with magnitude, units and direction
Vectors are done in proper format (tip to tail or coordinate)
Vectors are drawn to scale
Protractors, rulers and straight edge is used when drawing vector diagram
Vector components are solid lines and resultant is dotted (or vice versa)
Supplemental Material for those who can use Phet simulations
Vector Analysis- Tip to Tail Method
Tip to Tail: One vector “tail” starts from the origin and each vector added on after is added to the
arrow “tip” of the previous vector. The vector tips become the new origin for each subsequent
vector arrow. It is very important that you keep your orientation (0º, 90º, 180º, 270º or E, N, W
& S) consistent in each diagram to be as accurate as possible with the direction of the vector. The
magnitude of the vector is represented in the length of the arrow and should be to scale (ex. 1
cm = 1 m/s or 5 cm = 100 N).
You can only add similar vectors with the same units (ex. Velocity and velocity, acceleration and
acceleration, m/s² and m/s²).
PhET Simulation
Go to phet.colorado.edu and click on “Play with Sims” button. From the
side menu bar select “Math” and then select Vector Addition
Simulation. Select the “Run Now” button and a new window should pop
up.
Begin making your tip-to-tail diagrams by selecting a vector arrow from
the bucket. Drag and drop the vector into the space provided. #
16! PUM | Dynamics |##Lesson#1#A#(Supplement):#Vectors#and#Their#Components #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
#
Changing The Vector Arrow
You can change the length and direction of the arrow by clicking on
the tip and dragging it to the direction you choose or extend or
compress it in size. Notice the box on the top has symbols and
numbers. These will change when you change the length or direction
of your arrow. Click on another arrow and it will display its values.
Align the vector arrows so they are tip to tail. To get the resultant,
click on “Show Sum” and a green arrow will appear in the space. Drag
the arrow over to your diagram so that the Resultant (green) arrow’s
tail is at the origin and the tip of the Resultant is at the last arrow tip.
You have just made a tip-to-tail vector diagram.
#
NOTE: When doing this by hand, you will label the vector’s
magnitude (size) and direction next to the vector itself. The Resultant
will be represented by a dotted line instead of a green line and the
components will be solid lines.#
Answer the following questions using the simulation.
1. You take a walk and travel 20 meters in the north direction (90º).
a. Could this arrow represent another type of vector, like “20 m/s North” or “200 N 290º”?
Explain.
b. Next, you turn left and walk 10 meters to the west (180º). Click “show sum” to find the
resultant of these vectors. Arrange the vectors to display a tip-to-tail vector diagram. How
far are you from your initial position? In which direction is the resultant? In what
direction would you have to travel to get from your end point back to your starting
point?
c. Your friend says that you will get different answers if you set up your vectors in the
wrong order, so you need to be extra careful. Do you agree with your friend? Support
your answer with evidence.
2. On the weekend, you decide to run a couple of errands for your parents (the holidays will be
here in no time!) on your way to your friend’s house. You travel East on Rt. 537 driving 15
m/s, then travel North on Rt. 9 moving 21 m/s, and then take the back roads traveling 11 m/s
at 328º to your friends house. What was the resultant velocity for your trip?
For this next problem, select “Style 2” so that you can see the x and y components of a vector at an angle
and use a scale to convert the units associated with the arrow’s length to the force in your force diagram.
3. A boy pulls his sled across newly fallen snow. The weight of the sled is 150 N and the boy
pulls the sled with a constant velocity with a rope that makes a 45º angle with the ground.
The ground exerts an upward force of 100 N and a horizontal force (friction) of 40 N. Draw a
force diagram to represent this situation. What is the force the rope exerts on the sled?
PUM | Dynamics | Lesson#1#A#(Supplement):#Vectors#and#Their#Components 17#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
Lesson 2: Experimental Design and Assessment
2.1 Observe and Find a Pattern
You have the following equipment: a spring scale, a plastic bottle filled with sand, a container
of water, and a ruler (measuring tape).
a) Examine the equipment that you are given. Think of possible questions you can answer
using the equipment. Focus your questions on the magnitudes and directions of forces
that the water exerts on the bottle.
b) As a group, decide what question(s) you are going to investigate and the experiment
you are going to conduct to answer your question(s). Draw a picture of the apparatus.
c) What physical quantities will you measure? What do you think are the dependent and
independent variables? How are you going to record and represent the data?
d) Perform the experiment and collect the data. Represent the interactions of the bottle
with the scale, Earth, and water using a force diagram.
e) What patterns did you find? Summarize your findings for the question(s) you posed.
f) Based on the experiment you performed, does the water push up or down on the
object? Does the push depend on how deeply the object is submerged? If you cannot
answer these questions from you experiment, conduct a second experiment to answer
them.
Homework
2.2 Communicate
Write a report about your investigation so that a person who did not see your experiment can
repeat it and obtain similar results. Use the rubrics on the next page to help write your report. Be
sure to include what you learned from performing the experiment(s)?
2.3 Equation Jeopardy
The following equations are mathematical descriptions of several situations that you might have
encountered in activity 2.1. Try to visualize the equations and describe what situations they could
describe. Choose the upward direction as positive.
18! PUM | Dynamics |##Lesson#2:#Experimental#Design#and#Assessment #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
a) 10.0 N + (-10.0 N) = 0
b) 7.0 N + 3.0 N + (-10.0 N) = 0
c) 4.0 N + 6.0 N + (-10.0 N) = 0
For all three situations draw force vectors and show the total force using vector addition.
PUM | Dynamics | Lesson#2:#Experimental#Design#and#Assessment 19#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
Scientific Ability
Missing
An attempt
Needs some
improvement
Acceptable
Is able to
identify the
question to be
1! investigated
The question is
not mentioned.
An attempt is made
to formulate the
question but it is
described in a
confusing manner,
or is not the
question of interest.
The question is posed
but there are minor
omissions or vague
detail.
The question to be
investigated is clearly
stated.
Is able to
decide what is
to be
measured and
2! identify
independent
and
dependent
variables
The chosen
measurements
will not produce
data that can be
used to achieve
the goals of the
experiment.
The chosen
measurements will
produce data that
can be used at best
to partially achieve
the goals of the
experiment.
The chosen
measurements will
produce data that can
be used to achieve the
goals of the
experiment. However,
independent and
dependent variables are
not clearly
distinguished.
The chosen
measurements will
produce data that can
be used to achieve the
goals of the
experiment.
Independent and
dependent variables are
clearly distinguished.
Is able to use
available
3! equipment to
make
measurements
At least one of
the chosen
measurements
cannot be made
with the available
equipment.
All chosen
measurements can
be made, but no
details are given
about how it is
done.
All chosen
measurements can be
made, but the details of
how it is done are
vague or incomplete.
All chosen
measurements can be
made and all details of
how it is done are
clearly provided.
Is able to
describe what
is observed in
4! words and
with a picture
of the
experimental
setup.
No description is
mentioned.
A description is
mentioned but it is
incomplete. No
picture is present.
A description exists,
but it is mixed up with
explanations or other
elements of the
experiment. A labeled
picture is present.
Clearly describes what
happens in the
experiment both
verbally and by means
of a labeled picture.
Is able to
identify
sources of
5! experimental
uncertainty
No attempt is
made to identify
experimental
uncertainties.
An attempt is made
to identify
experimental
uncertainties, but
most are missing,
described vaguely,
or incorrect.
Most experimental
uncertainties are
correctly identified.
Experimental
uncertainties due to all
instruments are
correctly identified.
Random uncertainty is
considered.
Is able to
evaluate
specifically
how identified
6! experimental
uncertainties
may affect the
data
No attempt is
made to evaluate
experimental
uncertainties.
An attempt is made
to evaluate
experimental
uncertainties, but
most are missing,
described vaguely,
or incorrect. Or
only absolute
uncertainties are
mentioned.
The final result does
take the identified
uncertainties into
account but is not
correctly evaluated.
The final result is
written with the
uncertainty.
20! PUM | Dynamics |##Lesson#2:#Experimental#Design#and#Assessment #
© Copyright 2014, 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 Δv arrow.
e) Draw three force diagrams for the ball as it is rolling on the floor. Place these diagrams
under the motion diagram showing for what clock readings you created them. Does your
picture make sense to you?
3.2 Represent and Reason
a) Look at the force diagrams you drew in 3.1. Are the force arrows changing as the ball
rolls on the floor? Are there any forces that add to zero? If so, please indicate which and
explain why you think so.
b) Indicate the sum of the forces exerted on the ball on each diagram with an arrow.

c) Indicate the direction of the velocity change arrow ( Δv ) on the motion diagram for the
instances matching the force diagrams.
3.3 Observe and Represent
Consider this new experiment: You push the bowling 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. Try to exert a constant force. Make sure you do
not push so hard that the ball stops very quickly. The slower the process occurs, the easier it
will be for you to analyze it.
a) Sketch the situation.
b) Perform the experiment and describe the motion of the ball in words.
PUM | Dynamics | Lesson#3:#Motion#diagrams#&#Force#diagrams 21#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
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 for the entire time that the ball is in motion.
e) Draw three force diagrams for the ball as it is in motion; place those diagrams under the
motion diagram for the relevant clock readings.
f) Examine your force diagrams. Do they change as the ball slows down? Indicate which
forces add to zero and which forces do not. How do you know? Draw an arrow to show
the direction of the sum of all forces exerted on the ball.
g) Indicate the direction of the change in velocity arrow on the motion diagram for each of
the clock readings for which you drew force diagrams.
a) Observe the experiment in Vertical Throw . Draw a motion diagram for the ball, as it is
moving upward and the force diagram for the three instances on the motion diagram.

Compare the direction of the Δv arrow on the motion diagram to the direction of the sum
of the forces on the force diagram.
3.4 Represent and Reason
Place the bowling ball on a smooth floor, push it really hard and after you remove your
hand observe its motion. If the floor is smooth the ball should not slow down.
a) Draw a motion diagram for the ball. What is the direction of the velocity change?
b) Draw force diagrams for the ball for three different clock readings. Are the diagrams the
same or different? What is the direction of the sum of the forces exerted on the ball?
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 sum of the forces that other objects exert on the

ball and in the directions of the v arrows on the motion diagrams for the ball?
b) Is there a pattern in the directions of the sum of the forces forces that other objects exert

on the ball and the directions of the Δv 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.
22! PUM | Dynamics |##Lesson#3:#Motion#diagrams#&#Force#diagrams #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
e) Do you think the forces exerted on an object cause motion or change in motion (consider
pushing an object from rest and pushing an object that is already moving)?
f) Who was the observer recording the velocity changes for the object in the above
activities? 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) Design 2 different experiments whose outcome you can predict using the statement you
formulated in activity 3.5 (c).
Need Some Help?
The statement you are testing is the rule or pattern you noticed between the direction of the sum

of the forces exerted on an object on the force diagram and the direction of the Δv arrow on the
motion diagram for the same object. The experiment you design should try to “rule out” the
pattern, not to “prove” it.
b) Write predictions for the outcome of each experiment based on the pattern you noticed.
c) Perform the experiment and record the outcome. How did your prediction compare to the
outcome? Did you succeed in the disproving the pattern? Explain your judgment.
Homework
3.7 Test the Pattern
You have a medicine ball. When you place it on a bathroom scale, the scale reads 6 pounds (a
unit of force in a 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.
PUM | Dynamics | Lesson#3:#Motion#diagrams#&#Force#diagrams 23#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
d) After you made the prediction, watch the videos. Make sure that in the second clip, you
move frame by frame. Ball Drop then Ball Drop 2.
e) What judgment can you make about the pattern you formulated?
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.
e) A matchbox slides down a steep incline. Draw a motion diagram and a force diagram for
the matchbox as it slides down the incline. Check the consistency of your representations.
3.9 Pose a Problem
Consider the scenario: You are playing ice hockey.
Pose a problem similar to the two activities above. Then solve your problem.
24! PUM | Dynamics |##Lesson#3:#Motion#diagrams#&#Force#diagrams #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
3.10 Reason
Below you have descriptions of processes, unlabeled motion diagrams, and unlabeled force
diagrams. Label the force diagrams and motion diagrams and match the diagrams so all three
describe the same motion.
Words: (a) a book sliding on a table top to a stop; (b) a book dragged on the tabletop at constant
speed; (c) an object thrown upward on its way up; (d) and object thrown upward at the top of its
flight; (e) a car skidding to a stop. Notice that some force diagrams are extra – they do not match
any word descriptions.
Force diagrams
a)
b)
c)
d)
e)
f)
Motion Diagrams
a)
b)
c)
d)
PUM | Dynamics | Lesson#3:#Motion#diagrams#&#Force#diagrams 25#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
Lesson 4: What’s a Force to Do?
4.1 Test an Idea
Aaron has a hypothesis that says “objects always move in the direction of the sum of the forces
exerted on them by other objects.”
a) Design an experiment whose outcome might be consistent with Aaron’s hypothesis.
Carefully describe what you are going to do. Draw a force diagram for the system object
for the experiment you described.
b) Make a prediction about the object’s motion based on Aaron’s hypothesis. Use the
rubrics below to help in your reasoning.
Need Some Help?
A Hypothesis is a general belief, pattern, model, or rule developed from observations.
Hypotheses can be used to develop predictions.
A Prediction is a statement that describes the outcome of a specific experiment (you can only
make a prediction after you design the experiment but you need to make it prior to conducting
the experiment). Predictions must be based on the hypothesis being tested.
We can use our hypothesis and prediction to write an H-D statement, which uses the logic of
hypothetico-deductive reasoning:
If blah-blah is correct (describe my idea/hypothesis/explanation) and I do such and such
(describe your testing experiment), then such and such should happen (describe your predicted
outcome).
But it did not happen (describe the outcome of the experiment), therefore I need to revisit my
hypothesis, examine the experiment, etc.
OR
And it did happen (describe the outcome of the experiment), therefore my hypothesis has not
been disproved yet.
REMEMBER! – When you predict the outcome, the prediction must be based on the
hypothesis under test and NOT your intuition.
c) Now perform the experiment and record the outcome. How did the outcome compare to
the prediction? What can you say about proved Aaron’s hypothesis?
4.2 Test an Idea
a) Think of an experiment you can perform in which you can exert a force on an already
moving object in a direction that is different from the direction of its motion. Describe the
experiment carefully and draw a force diagram.
26! PUM | Dynamics |##Lesson#4:#What’s#a#Force#to#Do? #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
b) Use Aaron’s idea to make a prediction about the object’s motion. Use the rubrics below
to help in your reasoning.
c) Perform the experiment and record the outcome. Compare the outcome to your
prediction. What judgment can you make about Aaron’s HYPOTHESIS?
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
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 to
make a prediction
is made.
A prediction is made
that is distinct from the
hypothesis but is not
based on it.
No judgment is
made about the
hypothesis.
A judgment is made but
is not consistent with
the outcome of the
experiment.
Needs some
improvement
A prediction is made
and is distinct from the
hypothesis but does not
describe the outcome
of the designed
experiment.
A prediction is made
that follows from the
hypothesis but does not
have an if-and-then
structure.
A judgment is made
and is consistent with
the outcome of the
experiment but
assumptions are not
taken into account.
Acceptable
A prediction is
made, is distinct
from the hypothesis,
and describes the
outcome of the
designed experiment.
A prediction is made
that is based on the
hypothesis and has
an if-and-then
structure.
A reasonable
judgment is made
and assumptions are
taken into account.
4.3 Hypothesize
Based on the experiments above and the experiments you performed in lesson 3, formulate the
relationship between the sum of the forces exerted by other objects on the object of interest and
the change in motion of the object of interest.
4.4 Test an Idea
James thinks that when the sum of the forces exerted by other objects on the object of interest is
zero, the object is at rest.
a) Design two experiments to test James’s hypothesis.
b) Write a prediction for each experiment based on James’s hypothesis.
c) Perform the experiments and record the outcomes. Compare the outcomes to your
predictions.
d) What judgment can you make about James’s hypothesis now? Why would James have
such an idea?
PUM | Dynamics | Lesson#4:#What’s#a#Force#to#Do? 27#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
Did You Know?
Relationship between force and motion: If external objects exert the forces on the system
object and these forces do not add to zero, the motion of the system object changes so that the

Δv arrow in a motion diagram describing the motion of this object is in the same direction as the
sum of the forces exerted on the object.
Homework
4.5 Reason
For each part below, identify the system, draw a force diagram, draw a motion diagram, and
determine if the force and motion diagram are consistent.
a) Give an example for an object moving in the direction of the sum of the forces exerted
on it by other objects.
b) Give an example for an object moving in the direction opposite to the sum of the forces
force exerted on it by other objects.
c) Give an example for an object moving at an angle with the sum of the forces exerted on
it by other objects.
d) You throw a small ball upward.
4.6 Represent and Reason
Draw a motion diagram and three force diagrams for each of the following objects (the object
of interest is underlined) once the object is in motion. Make sure that the force diagram and
motion diagram are consistent with each other.
Situation
A ball is dropped and is
falling down.
Force diagrams for
three instances
Motion diagram
#
#
#
28! PUM | Dynamics |##Lesson#4:#What’s#a#Force#to#Do? #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
How do you know if
they are consistent?
#
A ball is thrown
upwards – after it
leaves your hand and
before it reached the
top of the flight
#
#
#
#
#
#
#
#
#
A rabbit sits in its
cage.
#
#
#
A matchbox slides
down a smooth book
cover.
#
#
#
#
#
#
A ball is thrown down.
#
#
A football is landing on
a cushion and the
cushion is being
compressed.
#
#
An air hockey puck
slides across an air
table.
#
PUM | Dynamics | Lesson#4:#What’s#a#Force#to#Do? 29#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
4.7 Reason
Read the statements below and classify each into one of three groups: experimental evidence,
hypothesis, prediction.
a) As the plants grow their mass increases.
b) The mass of the plants increases because you water them.
c) The increase in the mass of the growing plant should be exactly equal to the decrease in
the mass of the potting soil in a pot with a plant. Measure when the soil is dry.
d) The mass of the plants increases because they absorb carbon from the air.
e) The mass of the plants increases because they absorb nutrients from the soil.
f) The increase in the mass of the growing plant should be exactly equal to the mass of
water used to water the plant.
g) Explain how you understand the difference between the terms experimental evidence,
hypothesis, and prediction. Provide your own example for each.
Reflect: Is the following statement true or false: Objects on Earth always
move in the direction of the sum!of!the!forces exerted on them by other
objects. If you disagree, how can you convince others in your opinion?
30! PUM | Dynamics |##Lesson#4:#What’s#a#Force#to#Do? #
© Copyright 2014, 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 pingpong 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 the train and your friend on the platform.
Observer
You on a train that is
starting to move
Your friend on the platform
Motion diagram for the ball
(description of motion)
Force diagram for the ball
(explanation of motion)
Are the diagrams consistent?
Explain.
c) What can you say about the relationship between the sum of the forces exerted on the ball
and change in motion for the observer on the train that starts to move?
d) How will you rewrite the relationship between force and change in motion to include the
role of the observer?
e) Why do you think your friend on the platform did not see the ball starting to move (i.e.
change its motion) when the train started to move?
PUM | Dynamics | Lesson#5:#Inertial#and#NonKinertial#Reference#Frames 31#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
Did You Know?
Inertial reference frame: Inertia is the phenomenon when an object continues moving at
constant velocity if no other objects interact with it or if the sum of all these interactions is zero.
Reference frames in which we can observe this phenomenon are called inertial reference frames.
If the sum of all forces exerted on the object is zero, then in an inertial reference frame, the
object’s velocity remains constant.
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 add to zero and forces in the x
direction add to zero), then the system object continues moving at constant velocity (including
remaining at rest) as seen by observers in the inertial reference frames.
5.2 Reason
Consider the following idea: The relationship between the sum of the forces and the change in
motion depends on the observer.
Apply this idea using all of the videos at: Learning Cycle.
For some observers, THE RELATIONSHIP between the direction of the sum of the forces

exerted on the object of interest (the system) and the direction of the Δv arrow DOES NOT
WORK. Identify such observers in every experiment.
Need Some Help?
Recall in the kinematics module that motion is relative. For each video, describe the motion of
the object from multiple reference frames.
Recall that when we draw force diagrams, we only consider the forces exerted on the system
object(s).
5.3 Reason
a) You are a passenger in a car. All of a sudden, your head jerks backwards. Explain this
experiment from the reference frame of the car. Explain this experiment from the
reference frame of the pavement.
b) Describe what an observer on the ground sees when you stumble on a rock or slip on a
banana peel (focus on the motion of your feet and your head, assuming that the head is
only loosely attached to the body). Then describe what you observe.
32! PUM | Dynamics |##Lesson#5:#Inertial#and#NonKinertial#Reference#Frames #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
c) Use Newton’s first law to explain the observations of the Earth-based observer and your
observations for the situation described in b. Who is in the inertial reference frame? How
do you know?
d) Imagine that you have an infinitely long, smooth table covered in sand. A bowling ball is
hit once so that it starts rolling on the table but stops after 2 m. After removing some of
the sand and repeating the experiment, the ball stops rolling after 5 m. How far do you
think the ball will roll if ALL the sand is removed?
e) If you take a ball whose mass is half of the mass of the bowling ball, how will the
outcome of the last experiment change?
Homework
5.4 Explain
a) Provide two examples when the forces exerted on an object of interest add to zero.
b) Choose an observer who will see this object moving with a constant velocity.
c) Now choose an observer who will see this object moving with a changing velocity.
d) Draw pictures of the same situation as seen by the two different observers. What is
different about the reference frames of these two observers?
Need Some Help?
Example: A pendulum with a pendulum bob is attached to the ceiling of a car. For an observer
sitting in the car, the bob instantly starts moving towards him. For an observer on the ground, the
bob remains at rest but the car accelerates forward.
5.5 Represent and Reason
An elevator starts at rest on the ground floor of a building and stops at the top floor. The
elevator then returns to the bottom floor. The elevator is our object of interest. The observer is
on the ground.
Complete the table that follows to determine how the force the supporting cable exerts on the
elevator compares to the force Earth exerts on the elevator. The motion diagram and the force
diagram should be consistent with each other and with the rule relating motion and forces
developed in lesson 3.
PUM | Dynamics | Lesson#5:#Inertial#and#NonKinertial#Reference#Frames 33#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
Experiment
Sketch a motion
diagram.
Draw three force
diagrams for 3
consecutive clock
readings on the
motion diagram.
Check the consistency of
the diagrams.
(a) Elevator hangs at rest
at the ground floor.
(b) Elevator starts
moving upward with
increasing speed.
(c) Elevator moves at a
constant upward speed.
(d) Elevator slows as it
approaches the top floor.
(e) Elevator starts
moving down with
increasing speed.
(f) Elevator moves down
at constant speed.
34! PUM | Dynamics |##Lesson#5:#Inertial#and#NonKinertial#Reference#Frames #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
(g) Elevator slows to a
stop on ground floor.
Summarize what you learned about how the magnitude of the force that the supporting cable
exerts on the elevator compares to the force that Earth exerts on the elevator when the
elevator moves at constant speed and changing speed.
Reflect: Explain to your friends/parents/siblings the meaning of the words
“inertial/non inertial reference frame” and give 2 examples of inertial frames
from your everyday life and 2 examples of non-inertial reference frames.
PUM | Dynamics | Lesson#5:#Inertial#and#NonKinertial#Reference#Frames 35#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
Lesson 6: Newton’s Second Law: Qualitative
6.1 Observe and Find a Pattern
Student A is on rollerblades and stands in front of a
detector. The motion detector produces the velocityversus-time graphs shown. Student B (not on
rollerblades) stands behind Student A and pushes her
forward. Student A starts moving. The surface is very
(linoleum floor).
motion
smooth
Describe any patterns you see on the graph.
a) Draw a motion diagram and a force diagram for student A (1) when student B is pushing
and (2) when she is not pushing. Are the diagrams consistent with each other for each
time interval?
b) What is the meaning of the slope of the graph? Write a mathematical function that
describes each part of one line on the graph. Write a second mathematical function that
describes each part of a second line on the graph. What is the difference in the functions?
c) What can you say about the relationship between the sum of the forces exerted on an
object and its acceleration?
6.2 Reason
Refer to the activity 6.1.
a) Why do you think student A does not stop moving when student B stops pushing?
b) Break each motion down into two parts: when student B is pushing and when student B is
not pushing. What is different about the motion of student A for the three cases? What is
the same?
c) Imagine that while student A is moving at a constant speed, student B starts pulling her in
the direction opposite to her motion, exerting a constant force. Draw a motion diagram
and a force diagram for student A and extend the existing velocity versus time graphs to
represent the experiment.
d) Repeat c but imagine that student B pulls even harder in the direction opposite to student
A’s motion.
36! PUM | Dynamics |##Lesson#6:#Newton’s#Second#Law:#Qualitative #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
6.3 Test the pattern
Download on your computer PHET simulation “Forces in one dimension”. Turn off “Friction”.
Click on “graph Applied Force” and “Graph velocity.” Use the file cabinet as your object.
Predict the shapes of the graphs (sketch them) if you first exert a force of 200N on the cabinet for
6 seconds and then turn the force off but continue playing the simulation. After you sketched the
predicted graphs, play the simulation and reconcile your predictions with the simulation graphs.
Repeat the same procedure for the force of 400 N.
6.4 Observe and Find a Pattern
Student A is still on rollerblades, but this time she is wearing a backpack filled with textbooks.
Student B pushes Student A several times; each time, student A adds three more books to the
backpack. Student B pushes exerting the same force each time.
a) Use the graph to find a qualitative pattern between the change in Student A’s velocity
and the amount of stuff in her backpack.
Did You Know?
Mass: Mass m characterizes the amount of matter in an object and the ability of the object to
change velocity in response to interactions with other objects. The unit of mass is called a
kilogram (kg). Mass is a scalar quantity, and masses add as scalars.
b) What can you say about the velocity of Student A after Student B stops pushing her?
c) Does the mass of an object affect the velocity of an object or the change of velocity
when there is a force exerted on it?
d) How does changing the mass of the object affect the acceleration of an object if the force
exerted on it is the same?
6.5 Test the pattern (Phet simulations)
Use the same simulation as before. This time predict the shapes of the applied force and velocity
graphs you use the same magnitude force first exerted on the file cabinet (200 kg) and then on
the refrigerator (400 kg). After you made the predictions, run the simulation. Did your prediction
match the graphs on the simulation? If not, how can you reconcile?
6.6 Find a Relationship
Summarize how the change in the velocity of the object depends on the forces exerted on it
by other objects. Then summarize how the change in the velocity of the object depends on
PUM | Dynamics | Lesson#6:#Newton’s#Second#Law:#Qualitative 37#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
the mass. Use the words: more, less, and constant. Now combine these two relationships into
one.
6.7 Test the Relationship
a) Use the relationship you formulated in activity 6.4 to predict the shape of the velocity
versus time graph for an object that is dropped
b) Use the relationship you formulated in activity 6.4 to predict the shape of the velocity
versus time graph for an object that is thrown downward.
c) Explain the shape of the graphs in terms of the relationship you are testing.
d) Conduct the experiment using a motion detector. If there is no motion detector in
your classroom, use the graphs provided by your teacher to compare the prediction
with the actual outcome.
e) Revise the relationship if the prediction does not match the outcome.
Did You Know?
Two new words are used in physics to describe the processes and objects in activity 6.3. The first
is “inertia”. It describes the motion of an object that does not interact with any other objects –
AND – inertia also describes the motion of an object the forces exerted on which add to zero.
Only observers in “inertial reference frames” observe inertia.
The second word is “inertness”. Inertness is a property of objects that describes how hard it is
to change their velocity by exerting forces on them.
6.8 Reason
Examine the graphs in activity 6.4. Discuss which parts of the graph relate to the word
“inertia” and which parts relate to the word “inertness”. What is a familiar physical quantity
that is a quantitative measure of inertness?
Homework
6.9 Represent and reason
Draw a velocity versus time graph for the following scenario: A bowling ball is rolling on the
floor. Then a person starts pushing it very lightly in the direction opposite to the direction of
motion and continues pushing for a while. Finally the person stops pushing the ball.
38! PUM | Dynamics |##Lesson#6:#Newton’s#Second#Law:#Qualitative #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
How many possible graphs can you draw for this scenario? Examine the effects of
assumptions on your answer.
6.10 Reason
When you talk about the change in velocity of an object and the sum of the forces exerted on
that object, what is the cause and what is the effect? How do you know?
6.11 Equation Jeopardy
The following system of equations describes forces exerted on an object in the vertical and in the
horizontal direction. Describe two different situations that this system can describe in which the
object of interest is moving in two different directions:
x direction: 12.0 N +(- 9.0 N) = 3.0 N
y direction: 20.0 N +(-20.0 N) = 0
6.12 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
PUM | Dynamics | Lesson#6:#Newton’s#Second#Law:#Qualitative 39#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
6.13 Reason
You throw a ball up. Draw a motion diagram and two force diagrams for the ball on its way up
and another set of diagrams (one motion and two force) for the ball on its way down: (a) first
ignoring air resistance, and then (b) taking air resistance into account. When does the ball have

the largest Δv ? (c) Represent the motion of the ball for the case of no air resistance with the
position versus time graph and velocity versus time graph.
6.14 Reason
A string pulls horizontally on a cart so that it moves faster and faster along a smooth frictionless
horizontal surface. When the cart is moving medium-fast, the pulling is stopped abruptly. (a)
Describe carefully in words what happens to the cart’s motion when the pulling stops. (b)
Illustrate your description with motion diagrams, force diagrams and position and velocityversus-time graphs. Indicate on the graphs when the pulling stopped. What assumptions did you
make?
6.15 Reason
Your friend solving the previous problem says that after the string stops pulling, the cart starts
slowing down. (a) Give a reason why he would think this way. (b) Do you agree with his
opinion? Explain your opinion. (c) Explain carefully how you can design an experiment to test
his idea.
Reflect: What did you learn about Newton’s laws in this lesson? How did you
learn it? If you were to explain in simple words to a person who never took
physics what Newton’s second law is about, what would you say? Why do you
think Newton’s second law is important?
40! PUM | Dynamics |##Lesson#6:#Newton’s#Second#Law:#Qualitative #
© Copyright 2014, 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
a) Use the data table above. Draw pictures and force diagrams for the cart in Experiments 1
– 4.
Picture and Force Diagram
for Experiment 1
Picture and Force Diagram for
Experiment 2
Force Diagram for
Experiment 3
Force Diagram for
Experiment 4
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, Earth, and the track.
Need Some Help?
When doing such analysis, it is possible to devise a relationship between the dependent variable
and each independent variable, one at a time. Then combine these relationships to get a final
relationship.
PUM | Dynamics | Lesson#7:#Newton’s#Second#Law:#Quantitative 41#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
For example, use some of the data to see how the acceleration depends on the number of springs.
Then use other parts of the data to see how the acceleration depends on the number of carts.
You can also collect your own data using the video experiments if you prefer (optional): Glider
on air-track and Glider with varying masses
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
b) Draw a force diagram for the cart in each experiment. Show the horizontal forces only;
the upward force exerted by the surface on the cart’s wheels and the downward force
exerted by Earth on the cart balance.
Force Diagram for Experiment 1
Force Diagram for Experiment 2
Force Diagram for Experiment
4
Force Diagram for Experiment
5
Force Diagram for Experiment 3
42! PUM | Dynamics |##Lesson#7:#Newton’s#Second#Law:#Quantitative #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
c) Explain why there are negative signs in the acceleration column of the data table.
d) Use the data in the table above to devise a relationship between the cart’s acceleration
and the sum of the forces exerted by the springs, Earth, and the track on the cart.
e) Is this relationship consistent with the relationship you came up with in the previous
activity? Explain.
7.3 Explain
In the two previous activities, you analyzed experiments in which the motion of an object was
affected by other objects.
a) Mathematically represent the relationship between the object’s acceleration, the sum of
the forces exerted on it by other objects, and its mass. Make sure that you write the
relationship as a cause-effect relationship.
Need Some Help?
Think of cause-effect relationships you encounter in everyday life. For example, you set your
alarm early (cause), you get to use the bathroom first (effect). Another example would be, you
stub your toe on a rock (cause), your toe starts aching (effect).
Determine what variables in part a can be causes (independent variables) and what variable can
be the effect (dependent variable).
b) Represent the relationship graphically. What is your dependent variable and what are
your independent variables? How many graphs do you need in order to represent the
relationship?
c) How does acceleration of an object depend on the sum of the forces exerted on it? How
does the acceleration depend on the mass of the object?
d) What variable(s) did you hold constant in answering the questions above? Why was this
necessary?
PUM | Dynamics | Lesson#7:#Newton’s#Second#Law:#Quantitative 43#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
e) What does it mean if one quantity is directly proportional to another quantity? What does
it mean if it is inversely proportional? Give integer examples and draw graphs to explain
your reasoning.
7.4 Test Your Idea
a) Design an experiment whose outcome you can predict using the idea that you invented in
7.3.
b) Make a list of the equipment will you need. Describe the experiment in words and with a
picture.
Here’s An Idea!
Use the motion of objects that you are familiar with. One possibility would be a ball thrown
upward.
c) Make a prediction about the outcome of the experiment using the idea you invented in
7.3.
d) Perform the experiment and record the outcome. How does your prediction compare to
the outcome? What judgment can you make about your idea?
7.5 Test Your Idea
You can now test your idea with the Phet simulations. Use Forces in 1 dimension again. Friction
should be off. Turn on force and acceleration graphs. Choose the object you wish to push and the
force you wish to exert. Do not play the simulation yet. Sketch the graphs of force and
acceleration with the scales on them labeled carefully. Then play the simulation – do you need to
revise your predicted graphs? After you do this, sketch the graphs position and velocity versus
time and then play the simulation again. If you have time, repeat for a different force, make sure
you vary both magnitude and direction!
Did You Know?
In the previous activities, 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 sum of the
forces exerted on it by other objects F1 on S + F2 on S + ... + Fn on S = ΣFn on S and inversely
proportional to the mass m of the system object:

 ΣFn on S
aS =
mS
44! PUM | Dynamics |##Lesson#7:#Newton’s#Second#Law:#Quantitative #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
Newton’s second law is written for the vectors. You cannot use them to make calculations,
thus you need to use it in component form. To do this, use the force diagram. If
the forces are pointed in the positive direction along x or y axes, they enter the
y
equation with positive signs, if in the opposite direction – with negative signs.
F2 on o#
For example: 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 2 kg.
How do we write Newton’s second law in component form for this situation?
Consider the upward to be the positive direction. The force diagram for the



 ΣFn on O F1 on O + F2 on O
=
situation is on the right. Newton’s second law is a =
.
mO
mO
F1 on o#
Component of force 1 in the y direction is negative and the component of force 2 is positive.
Newton’s second law in component form for the situation becomes:
ay =
F1 on O y + F2 on O y
mO
=
(−F1 on O ) + (F2 on O ) (−20 N) + (10 N)
=
= −5 N/kg = −5 m/s 2 . Follow
mO
2 kg
the steps in the above equation carefully to see how you can find the acceleration. The sign of
acceleration is negative. It means that the acceleration points down. The object is either sloping
down on its way up or speeding up on its way down.
Homework
7.6 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 Earth exerts on any object.
7.7 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. (3) Write Newton’s second law in component form.
b) You pull a 20-kg sled on a horizontal surface, exerting a 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?
PUM | Dynamics | Lesson#7:#Newton’s#Second#Law:#Quantitative 45#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
7.8 Represent and Reason
Complete the table that follows. The system object is underlined.
Write a word
description of
the situation.
Sketch the situation
and circle the
system object.
Draw a force
diagram; do not
forget the axes.
Label the
forces, if
needed. Draw a
motion
diagram.
Draw the
direction of the
acceleration
and of the sum
of the forces.
Are they
consistent?
An elevator
pulled by a
rope is slowing
down on its
way up.
Write Newton’s Second Law in
component form.
TR on O + (−FE on O )
m
m 800 N+(-1000 N)
−2.0 2 =
s
10 kg
ay =
30o#
Did You Know?
Gravitational force: The magnitude of the gravitational force that Earth exerts on any object
near its surface equals the product of the object’s mass m and the gravitational constant g:
FE on O = m g
where g = 9.8 m/s2 = 9.8 N/kg on or near the earth’s surface. The force points toward the center
of Earth.
7.9 Reason
Two small balls of the same material, one of mass m and the other of mass 2m , are dropped
simultaneously from the Leaning Tower of Pisa. Earth exerts a bigger force:
(a) On the 2 m ball.
(b) On the m ball.
(c) Earth exerts same force on both balls because they fall with the same acceleration.
46! PUM | Dynamics |##Lesson#7:#Newton’s#Second#Law:#Quantitative #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
Reflect: What did you learn about Newton’s laws in this lesson? How did you
learn it? If you were to explain in simple words to a person who never took
physics what Newton’s second law is about, what would you say? Why do you
think Newton’s second law is important?
PUM | Dynamics | Lesson#7:#Newton’s#Second#Law:#Quantitative 47#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
Lesson 8: Design an Experiment
8.1 Design an Experiment
You have a spring and a set of objects of known masses. Your goal is to create a mathematical
model for the spring constant, which relates the forces exerted by the spring on an object that
stretches it to the stretch of the spring.
a) Design an experiment to determine the relationship between how much the spring
stretches from the original unstretched position as it pulls on an object and the force that
it exerts on that object.
b) Sketch the setup for your experiment and outline the plan. What instruments are you
going to use?
c) What are you going to measure? What are the dependent and independent variables?
d) Perform the experiment, collect the data, and decide how you can best represent it both
graphically and mathematically.
e) Represent the data and decide how you will represent the uncertainties in the
measurements on your graph.
f) What pattern do you notice? How can you express the pattern mathematically? Examine
the units of the slope on the graph. What information do the units give you about the
spring?
Did You Know?
A spring constant is a quantity that determines the force that the spring must exert on an object
in order to stretch it by 1 m from its original length. The unit of the spring constant, k, is 1 N/m.
g) What is the spring constant of your spring? How certain are you in your answer?
h) Finish your lab report. Make sure that data are easy to understand and the experiments is
described clearly enough for anyone to repeat it and obtain similar results.
8.2 Reason
A 0.5 kg object attached to a 0.5 m spring stretches it by 0.1 m. Draw a force diagram for this
situation and determine the spring constant of the spring.
48! PUM | Dynamics |##Lesson#8:#Design#an#Experiment #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
8.3. Design an Experiment
Attach a 100 g object to a spring scale and lift the scale slowly. Notice the reading on the
scale. Can you get it to read 1 N when the object is moving? Can you get it to read more?
Can you get it to read less? How were you able to achieve this?
8.4 Test an Idea
Ritesh says: “If you hang an object on a spring scale, the scale always reads how much force
Earth exerts on the object”.
Erin says: “The scale always reads the sum of the forces exerted on the object”.
a) Think of what you can do to test their opinions. Describe the experiment.
b) Make a prediction based on each of the hypotheses .
c) Perform the experiment or watch the video and record the outcome: Spring Scale
d) Make a judgment about each person’s hypothesis. What can you say the scale measures?
Did You Know?
Elastic force: An object that stretches or compresses like a spring exerts an elastic force on
some other object that is causing it to stretch or compress. The elastic force exerted by the
spring on that object points in the direction opposite to the stretch (or compression). The
magnitude of the force is directly proportional to the distance x that the elastic object (spring)
is stretched or compressed from its equilibrium position (at x = 0):
FS on O= k x
The force constant k in units N/m is a measure of the stiffness of the spring (the ratio of the force
in N needed to stretch the spring 1 m).
Homework
8.5 Reflect
Make a list of things you learned from the lab activity above with the springs. How did you learn
each?
PUM | Dynamics | Lesson#8:#Design#an#Experiment 49#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
8.6 Reason
Calculate the relative uncertainties of your scale measurements. When was your knowledge
about the force more accurate?
Need Some Help?
Recall from kinematics that when you measure a quantity and the measurement is small, the
uncertainty is a big percentage of the measurement. When the measurement is large, the
uncertainty is a smaller percentage of the measurement.
The relative uncertainty is the fraction of the measurement due to the uncertainty.
For example, when the measurement is 2 N and the uncertainty is 0.5 N, the relative
uncertainty becomes 0.5/2 = 0.4 or 40%. When the measurement is 15 N and the
uncertainty is 0.5 N, the relative uncertainty is 0.5/15 = 0.3 or 30%.
Therefore for the accuracy of measurement with a particular instrument, we want to measure
quantities that are large.
8.7 Reason
What is the uncertainty in your parents’ car speedometer? When is the measurement of the speed
more accurate: driving in the city or on a highway? Explain.
8.8 Reason
If you are to stand on a scale in a very powerful elevator in a very tall building, what might
happen to the scale reading as the elevator takes you from the 1st floor to the 42nd floor? Use
force diagrams, motion diagrams, and Newton’s Second Law to explain your answer.
8.9 Regular problem
You hang a 5-pound bag of groceries on a spring scale that has a visible spring. The scale
stretches from being 20 cm long to being 26 cm long. What is the spring constant? (Hint: make
sure you use SI units for all quantities. If you do not know how to convert pounds to Newton’s
use the textbook or the internet for help.)
8.10. Regular problem
You probably know from experience that door springs are difficult to stretch. (a) What maximum
force do you need to exert on a relaxed spring with a 1.2 ×104 -N/m spring constant to stretch it
6.0 cm from its equilibrium position? (b) What average force do you need to exert?
50! PUM | Dynamics |##Lesson#8:#Design#an#Experiment #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
8.11. Reason
Why is it harder to stretch the spring that is already stretched than an unstretched spring? Explain
your answer using force diagrams, graphs, etc.
8.12 Reason
A person stands on a bathroom scale in a spaceship that is drifting at constant velocity in outer
space far from the Solar system. What is the reading of the bathroom scale compared to its
reading when on Earth?
(a) The same as on Earth since the mass of the person did not change.
(b) Zero since the person is not pressing on the scale at all.
(c) About half the reading on Earth.
8.13 Reason
A man stands on a scale and holds a heavy object in his hands. What happens to the scale reading
if the man quickly lifts the object upward and then stops lifting it?
(a) The reading increases, returns briefly to the reading when standing stationary, then decreases.
(b) The reading decreases, returns briefly to the reading when standing stationary, then increases.
(c) Nothing since the mass of the person with the book remains the same. Thus the reading does
not change.
PUM | Dynamics | Lesson#8:#Design#an#Experiment 51#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
Lesson 9: Applying Newton’s Second Law
Problem-Solving Strategy for Dynamics Problems
Sketch and Translate:
Read the text of the problem at least three times to make sure you understand what the problem
is saying. Visualize the problem, make sure you see what is happening.
• Sketch the situation described in the problem; include all known information.
• Choose a system object and make a list of objects that interact with the system.
• Indicate the direction of acceleration, if you know it.
Simplify and Diagram:
• Consider the system as a particle.
• Decide if you can ignore any interactions of the environment with the system object.
• Draw a force diagram for the system. Label the forces with two subscripts. Make sure the
diagram is consistent with the acceleration of the system object (if known). Include
perpendicular x- and y-coordinate axes.
• Draw a motion diagram and make sure the force and the motion diagrams match.
Represent Mathematically:
• Apply Newton’s Second Law in component form to the situation you represented in the force
diagram.
• Add kinematics equations if necessary.
Solve and Evaluate:
• Solve the equations for an unknown quantity and evaluate the results to see if they are
reasonable (the magnitude and the sign of the answer, its units, and how the result changes if one
of the quantities becomes zero – it the result what your common sense tells you? Make sure you
go back to the force and motion diagrams to make sure your answer is consistent with both.
#
Need Some Help?
Here is an example that applies the strategy shown above: A 5-kg object
(Earth exerts a 50 N force on it) is lifted by a cable that exerts a 70 N
force on it. Calculate the acceleration of the object.
Translate: The object is our system; Earth and the cable interact with the
object. The acceleration is up.
FC on O
y
FE on O
Simplify and Diagram: In this case there are two forces exerted on the
object – one exerted by the cable and one exerted by Earth. We choose the positive axis to be#
down.
Represent Mathematically: Newton’s second law in component form: aO y =
FC on O y + FE on O y
mO
Solve and Evaluate: The component of the force exerted by the cable is negative as the force
points in the negative direction, the component of the force exerted by Earth is positive. Thus
(−70 N) + 50 N
aO y =
= (−4 N/kg) = (-4 m/s 2 ) . The negative sign of acceleration means that it
5 kg
52! PUM | Dynamics |##Lesson#9:#Applying#Newton’s#Second#Law #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
is pointed upward – the elevator is accelerating in the upward direct (not necessarily moving in
the upward direction, it can be slowing down).
9.1 Represent and Reason
Description
of the
object of
interest is
underlined
1) A 2.2 kg
bucket of
clams sits at
rest on a
desk.
A
Sketch the
situation.
Circle the
object of
interest.
Draw the
direction of
the
acceleration,
if known.
B
Translate
the givens
into
physical
quantities.
C
Draw a force
diagram for the
object of interest.
D
Can you
evaluate
any of
the
forces in
the force
diagram?
E
Write Newton’s Second Law in
component form.
Fill in anything you know and
solve for anything you do not
know.
Which
are
negative
and
which
are
positive?
Given in
description:
a = 0 (sits )
m = 2.2kg
Ftable&on&
bucket
# Bucket
FE on B y =
-mg=-22
N
ay =
0=
FEarth on Bucket y + FTable on Bucket y
m
(−22N) + FTable on Bucket
2.2kg
FEarth&on&
bucket
2) A 5kg
bucket of
clams hangs
motionless
from a
spring that
stretches 40
cm.
3) A man
pulls a 40kg
refrigerator
up an
elevator
shaft with a
rope at a
constant
speed.
Come up
with your
own for an
object in
equilibrium
with 3 or
more other
objects
interacting
with it.
PUM | Dynamics | Lesson#9:#Applying#Newton’s#Second#Law 53#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
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
2.0 m/s. You push on it in the direction opposite to its motion exerting a force of 12.0 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) Write Newton’s second law in component form for the process.
c) 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.
9.3 Regular problem
An astronaut, while pushing a beam into place on the International Space Station, exerted a
150-N force on the beam. The beam accelerates at 0.15 m/s 2 . Determine the mass of the beam.
9.4 Design an Experiment
a) Describe an experiment that you can design to test Newton’s Second Law. Decide on
what aspect of the law you can test.
b) Brainstorm different possibilities and decide which ones are the best. Think of the criteria
for choosing the best experiment.
d) What equipment will you need? What data will you collect?
a) What is your prediction about the relationships in the data based on Newton’s Second
Law?
b) What is the difference between Newton’s Second Law and the predictions?
9.5 Reason
An elevator is pulled upward so it moves with increasing upward speed—the force exerted by
the cable on the elevator is constant and greater than the downward gravitational force exerted by
Earth. When the elevator is moving up medium fast, the force exerted by the cable on the
elevator changes abruptly to just balance the downward gravitational force exerted by Earth—the
54! PUM | Dynamics |##Lesson#9:#Applying#Newton’s#Second#Law #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
sum of the forces that the cable and Earth exert on the elevator is now zero. Now what happens
to the elevator? Explain. Represent your answer with a position and velocity-versus-time graphs.
What assumptions did you make?
Homework
9.6 Practice
Fill in the table below. The system object is underlined.
Description
of the object
of interest is
underlined
A
Sketch the
situation.
Circle the
object of
interest.
Draw a
motion
diagram
and the
direction of
the
acceleration,
if known
B
Translate
the givens
into
physical
quantities.
C
Draw a
force
diagram
for the
object of
interest.
D
Can you
evaluate any of
the force
components in
the force
diagram?
Which are
negative and
which are
positive? What
if you changed
the direction of
the axes?
E
Write Newton’s Second Law in
component form.
What can you determine using
the information in the problem?
1) A 72 kg
crate on a
freight
elevator
accelerates
upwards at a
rate of 0.2
m/s2 while
moving
down.
2) A 172.0
kg crate on a
freight
elevator
accelerates
downwards
at a rate of
0.4 m/s2
while
moving up.
3) A physics
teacher of
mass m is
holding onto
a rope
attached to a
hot air
balloon and
is
accelerating
upwards at a
m/s2.
PUM | Dynamics | Lesson#9:#Applying#Newton’s#Second#Law 55#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
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. If you are not observing the real-life experiment, watch the video
experiments: Pull and Push
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.
c) Why didn’t Student B’s velocity change in activities 6.1 and 6.3? Explain using force
diagrams.
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: Passing the Ball
a) What was your hypothesis?
b) Where did the hypothesis come from?
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.
56! PUM | Dynamics |##Lesson#10:#Newton’s#Third#Law:#Qualitative #
© Copyright 2014, 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 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 Earth’s surface, 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.
PUM | Dynamics | Lesson#10:#Newton’s#Third#Law:#Qualitative 57#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
Lesson 11: Newton’s Third Law: Quantitative
11.1 Observe and Find a Pattern
The goal of this experiment is to determine a mathematical relationship between the force that
object A exerts on object B and the force that object B exerts on object A when they are
interacting with each other.
Available Equipment: Force probe sensors with hooks on the end and computers.
Note on equipment: You will be using a new sensor for this experiment; it is called a force
probe. A force probe is a sensor that sends a signal to the computer indicating the force exerted
on its tip. The software interface helps you plot force as a function of time for two different force
probes.
a) Take one of the probes, connect it to the computer, and gently pull or push on it. Examine
the graph on the screen and make sure it makes sense to you.
Need Some Help?
The force probe is an object. You exerted a force on this object, and this force was recorded as a
function of time.
b) Design and perform enough experiments to find a pattern in the readings of the two
probes when they record forces that two interacting objects exert on each other.
Here’s An Idea!
Place one force probe on the table and tap it gently with the second probe. The probes are very
delicate, and if you use them to tap each other, you must do it lightly. Even a light tap will
register on the graph.
c) Write a short report, summarizing the experiments and your findings. For your report, be
sure to include the following:
1) For each experiment, describe the setup in words and sketch the graphs that you
see on the computer.
2) Find a pattern in the pairs of graphs representing the force-versus-time functions
recorded by each probe during an interaction.
58! PUM | Dynamics |##Lesson#11:#Newton’s#Third#Law:#Quantitative #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
3) Formulate a tentative rule relating the force that object A exerts on object B to the
force that the object B exerts on the object A.
4) Use the observational experiment rubric and data analysis rubric to write your
report about this experiment.
11.2 Test Your Hypothesis
Available Equipment: Force probe sensors with hooks on the end, a track, carts, objects of
different masses to put on the carts, cushions, elastic bands, and computers.
a) State the rule you are going to test which you devised in 11.1.
b) Brainstorm a list of possible experiments whose outcome can be predicted with the help
of the rule. Choose two experiments from your list.
c) Briefly describe your chosen design. Include a labeled sketch.
d) Use the tentative rule to make a prediction about the outcome of each experiment.
e) Perform each experiment. Record the outcome.
f) Does the outcome support the prediction? Explain.
g) Based on the prediction and the outcome of the experiments, what is your judgment about
the rule?
h) Think of the assumptions that you used to make the predictions for the testing
experiments. How could these assumptions have affected your judgment?
i) Talk to your classmates in other lab groups and find out about their results. Are they
consistent with yours?
j) Use the testing experiment rubric to write your report.
11.3 Reason
Use the rule that you devised and tested to decide who exerts a larger force for the following
situations:
PUM | Dynamics | Lesson#11:#Newton’s#Third#Law:#Quantitative 59#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
1. A mosquito on a car’s windshield or a car’s windshield when a mosquito smashes into it;
2. A reflex hammer (Taylor hammer) on your knee or your knee on the hammer when a
doctor taps your knee with it.
Reconcile your answers with your observations of these phenomena.
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:


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 the sum of
the forces that we use in the Newton’s second law to find the acceleration of an object.
11.4 Reason
A horse is pulling a carriage. According to Newton’s third law the force exerted by the horse on
the carriage is the same in magnitude but points in the opposite direction of the force exerted by
the carriage on the horse. How are the horse and carriage able to move forward?
a) The horse is stronger.
b) The total force exerted on the carriage is the sum of the forces exerted by the horse and
the ground’s surface in the horizontal direction.
c) The sum of the forces exerted on the horse is the sum of the static friction force exerted
on its hooves by the surface and the force exerted on it by the carriage.
d) Both b and c are correct.
Homework
11.5 Reason
a) A book sits on the tabletop. What is the Newton’s Third Law pair for the force that Earth
exerts on the book?
b) If Earth exerts a 5 N force on the book, what is the force that the book exerts on Earth?
c) What is the acceleration of the book if Earth is the only object exerting a force on it?
60! PUM | Dynamics |##Lesson#11:#Newton’s#Third#Law:#Quantitative #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
d) Why does the book fall onto Earth but Earth does not fall onto the book? The mass of
Earth is 6.00 x 1024 kg.
Need Some Help?
Use Earth’s mass to calculate Earth’s acceleration. What does this tell you about the motion of
Earth?
e) The Sun’s mass is 2.00 x 1030 kg. It pulls on Earth, exerting a force of about 1020 N.
What is the force that Earth exerts on the Sun?
11.6 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?
11.7 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.
c) Earth pulls on an apple exerting a 1.0 N force on it. What is the force that the apple exerts
on Earth? Why does the apple fall towards Earth but 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?
11.8 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.
11.9 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
PUM | Dynamics | Lesson#11:#Newton’s#Third#Law:#Quantitative 61#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
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?
11.10 Reason
The Moon orbits Earth because Earth exerts a force on it. The Moon, therefore, has to exert a
force on Earth. What is the visible result of this force?
11.11 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?
Earth on the person or the floor on the person?
11.12 Reason
Your young sister jumps off a chair holding a 0.50-kg container of ice cream. What is the force
with which the container pushes on her hand when she is in the air? Explain.
(a) Almost 5 N .
(b) The same as she exerts on the container.
(c) 0 N .
(d) Both b and c are correct.
62! PUM | Dynamics |##Lesson#11:#Newton’s#Third#Law:#Quantitative #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
11.13 Reason
A spaceship moves in outer space. What happens to its motion if there are no external forces
exerted on it? If there is a constant force exerted on it in the direction of its motion? If something
exerts a force opposite its motion?
a) It just keeps moving; it speeds up with constant acceleration; it slows down with constant
acceleration.
b) It slows down; it moves with constant velocity; it slows down.
c) It slows down; moves with constant velocity; it stops instantly.
11.14 Reason
You stand on a bathroom scale in a moving elevator. What happens to the scale reading if all of
a sudden the cable holding the elevator breaks?
(a) The reading will increase. (b) The reading will not change.
(c) The reading will decrease a little. (d) The reading will drop to 0 instantly.
11.15 Estimate
Imagine that all people on Earth decided to jump simultaneously. What will happen to Earth?
Reflect: What did you learn about Newton’s third law so far? How is it
different from the second law? Give 3 examples of Newton’s third law from
everyday life.
PUM | Dynamics | Lesson#11:#Newton’s#Third#Law:#Quantitative 63#
© Copyright 2014, 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.
2#
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) Is your answer to (c) consistent with Newton’s Third Law?
e) 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.
64! PUM | Dynamics |##Lesson#12:#TwoKBody#Problems #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
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.
12.3. Test your ideas
a) Examine the experimental setup in the video experiment at: Atwood's Machine
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
A person pulls on a rope, which in turn pulls a crate across a horizontal smooth surface, as shown
below.
Three motion diagrams for the crate are shown below (with v arrows only). In the table that
follows, construct a force diagram for the crate. Check the consistency of your motion diagrams
and force diagrams.
(a)
(b)
(c)
PUM | Dynamics | Lesson#12:#TwoKBody#Problems 65#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
(a)
(b)
(c)
#
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.
12.7 Regular problem
You pull a rope oriented horizontally. The other end of the rope is attached to the front of first of
two wagons that have the same 20-kg mass. The rope exerts a force of magnitude T1 on the first
wagon. The wagons are connected together by a second horizontal rope that exerts a force of
magnitude T2 on the second wagon. Determine the magnitudes of T1 and T2 if the acceleration of
the wagons is 2.0 m/s 2 .
12.8 Regular problem
Rope 1 pulls horizontally, exerting a force of 45 N on an 18-kg wagon attached by a second
horizontal rope to a second 12-kg wagon. Make a list of physical quantities you can determine
using this information, and solve for three of them (one has to be a kinematics quantity).
12.9 Regular problem
A person holds a 250-g block that is connected to a 300-g block with a string going over a light
pulley with no friction in the bearing (similar to a system in the video experiment). After the
person releases the 200-g block, it starts moving upward and the heavier block descends.
(a) Choose each block as a system and draw force and motion diagrams for them.
66! PUM | Dynamics |##Lesson#12:#TwoKBody#Problems #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
(b) What is the acceleration of each block? (c) What is the force that the string exerts on each
block? (d) How long will it take each block to traverse 1.0 m?
12.10 Reason
In the pulley-2 block system in the problem above, a 300-g block was replaced with the block of
the unknown mass. The first block is hanging 0.8 m above the second block. The system is at
rest. Nobody is holding any of the block. What is the mass of the unknown block?
PUM | Dynamics | Lesson#12:#TwoKBody#Problems 67#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
Lesson 13: Design an Experiment
13.1 Design an Experiment
You have helium balloons and air balloons at your disposal. You can attach different objects to
the balloons.
a) Decide what questions you can investigate using one or both balloons. Brainstorm a list
of questions.
b) Decide which one you and your group can perform with the equipment available.
c) Decide what you need to know more about in order to pursue the question.
d) Make a sketch of the experimental design and list the physical quantities that you plan to
measure.
e) Are you going to conduct an observational experiment or a testing experiment? If it is a
testing experiment, make sure that you outline the hypothesis being tested and the
prediction of the outcome of the experiment based on the hypothesis. If you cannot make
a prediction, you should consider whether you are actually doing a testing experiment.
f) Perform the experiment, record your data, represent the data, and perform necessary
calculations. Do not forget about the uncertainties when you are presenting the final
results.
g) Write a report about your investigation. Include sketches of the equipment, motion
diagrams and force diagrams if necessary, tables of data, graphs, and your findings,
including the uncertainties. Make sure that the report is clear enough that a person who
has not seen the experiment can repeat it.
h) Use the rubrics to improve your report.
68! PUM | Dynamics |##Lesson#13:#Design#an#Experiment #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
Ability
Absent
An attempt
Needs some
improvement
Acceptable
Is able to
formulate the
question to be
investigated
The question to be
investigated is not
mentioned.
The question is posed
but it is not clear.
The question is posed
but it involves more
than one variable.
The question is
posed and it
involves only one
variable.
Is able to design
an experiment to
answer the
question
The experiment
does not answer
the question.
The experiment is
related to the question
but will not help
answer it.
The experiment
investigates the
question but might
not produce the data
to find a pattern.
The experiment
investigates the
question and might
produce the data to
find a pattern.
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.
It is clear what will
be measured, and
independent and
dependent variables
are identified but the
choice is not
explained.
It is clear what will
be measured and
independent and
dependent variables
are identified and
the choice is
explained.
Is able to use
available
equipment to
make
measurements
At least one of the
chosen
measurements
cannot be made
with the available
equipment.
All chosen
measurements can be
made, but no details
are given about how it
is done.
All chosen
measurements can be
made, but the details
of how it is done are
vague or incomplete.
All chosen
measurements can
be made and all
details of how it is
done are clearly
provided.
Is able to
describe what is
observed in
words, pictures
and diagrams.
There is no
description of what
was observed.
A description is
mentioned but it is
incomplete. No
picture is present.
A description exists,
but it is mixed up
with explanations or
other elements of the
experiment. A
labeled picture is
present.
Clearly describes
what happens in the
experiments both
verbally and by
means of a labeled
picture.
Is able to
construct a
mathematical (if
applicable)
relationship that
represents a
trend in data
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.
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.
The relationship
represents the trend
accurately and
completely and an
analysis of how
well it agrees with
the data is included
(if applicable).
PUM | Dynamics | Lesson#13:#Design#an#Experiment 69#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
Homework
13.2 Finish Your Report
In the next class, you will need to evaluate one of your classmate’s report and your classmate
will evaluate yours. Make a list of necessary elements that you will look for in your classmate’s
report.
13.3 Reason (adapted from FMCE)
Questions below refer to a toy car which can move to the right or left on a horizontal surface
along a straight line (the + position axis). The positive direction is to the right.
0
+
Different motions of the car are described below. Choose the letter (A to G) of the accelerationtime graph which corresponds to the motion of the car described in each statement.
You may use a choice more than once or not at all. If you think that none is correct, answer
choice J.
_____1.
The car moves toward the right (away from the origin), speeding up at a constant acceleration.
_____2.
The car moves toward the right, slowing down at a constant acceleration.
70! PUM | Dynamics |##Lesson#13:#Design#an#Experiment #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
_____3.
The car moves toward the left (toward the origin) at a constant velocity.
_____4.
The car moves toward the left, speeding up at a constant acceleration.
_____5.
The car moves toward the right at a constant velocity.
Questions below refer to a coin which is tossed straight up. After it is released it moves upward,
reaches its highest point and falls back down again. Use one of the following choices (A through
G) to indicate the acceleration of the coin during each of the stages of the coin's motion
described below. Take up to be the positive direction. Answer choice J if you think that none is
correct.
A.
The acceleration is in the negative direction and constant.
B.
The acceleration is in the negative direction and increasing
C.
The acceleration is in the negative direction and decreasing
D.
The acceleration is zero.
E.
The acceleration is in the positive direction and constant.
F.
The acceleration is in the positive direction and increasing
G.
The acceleration is in the positive direction and decreasing
___6.
The coin is moving upward after it is released.
___7.
The coin is at its highest point.
___8.
The coin is moving downward.
Reflect: Among the multiple choice questions find those that relate to forces
changing in time. So far we mostly studied constant forces. What happens to
Newton’s second law if the sum of the forces exerted on the object changes
with time?
PUM | Dynamics | Lesson#13:#Design#an#Experiment 71#
© Copyright 2014, 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) Fill in the table that follows by constructing a force diagram for the block (the system) for
these five situations.
The block sits on The spring
the table with no pulls on the
scale pulling it.
block, which
does not start
moving.
The spring pulls
harder, but the
block still does
not move.
The spring pulls
on the block, and
the block is just
about to start
moving.
The spring pulls
the block at a slow,
constant velocity.
b) 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.
c) 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?
d) What object is exerting this friction force for the scenarios given above?
e) 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
72! PUM | Dynamics |##Lesson#14:#Friction #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
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.
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
PUM | Dynamics | Lesson#14:#Friction 73#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
b) Express mathematically a relationship between the normal force and the maximum static
friction force.
14.4. Test your idea
Design an experiment to test that the magnitude of the maximum static friction force is equal
to fs surface on object = µ N surface on object . Describe what you will do, what data you will collect and
what the predicted outcome should be if the expression is correct. Then perform the
experiment and make a judgment about the hypothesis.
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 that Earth exerts 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 add to zero. 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?
When two objects touch each other, they exert a force on each other. In physics it is customary to
break the force that each exerts on the other into two forces – a force perpendicular to the
surfaces and the force parallel to the surfaces.
Normal force: This is the perpendicular component of the total force that one object exerts on
the other object. it points perpendicular to the surfaces that are 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: The parallel component of the force that two objects exert on each other
is called a friction force. The friction force that one object exerts 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 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
74! PUM | Dynamics |##Lesson#14:#Friction #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
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 = µ 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 = µ k N
14.6 Design an Experiment
Use Phet simulations apply your understanding of friction forces. Go to Forces in 1 Dimension.
This time do not turn off Friction. Choose three objects and design experiments that will allow
you to determine what coefficient of static and kinetic friction the programmers who write the
simulation used. Write a report in which you will describe the experiments and explain how you
used the results to determine the coefficients. Do your results make sense? Did the programmers
use the knowledge of Newton’s laws and friction forces to write the program?
Homework
14.7 Observe and Represent
Imagine that you could watch yourself walk in slow motion. Analyze your steps in terms of
the force of friction that the floor exerts on your foot and in terms of Newton’s Second and
Third Laws. In order to do this, break the step into two parts: (1) when you put the foot down
to finish up the previous step, and (2) when you are pushing off the floor to start a new step.
Draw force diagrams to represent your reasoning.
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?
14.9 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 sum of the forces exerted on the desk is 27 N.
PUM | Dynamics | Lesson#14:#Friction 75#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
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?
14.10 Regular problem
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 kinetic or static friction? Explain.
14.11 Regular Problem
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 is the acceleration of the box?
14.12 Reason
A car is moving to the right increasingly faster. A leaf is on the back vertical side of the car and
does not slide down. Explain how this can be.
14.13 Regular problem
The Ford P2000 fuel cell car has a mass of 1520-kg. While it is traveling at 20 m/s, the driver
applies the brakes to stop the car on a wet surface with a 0.40 coefficient of friction. (a) How far
does the car travel before stopping? (b) If a different car with the mass 1.5 times as much as the
mass of the Ford P2000 is on the road traveling at the same speed and the coefficient of friction
between the road and the tires is the same, what will its stopping distance be? Does the answer
make sense to you?
76! PUM | Dynamics |##Lesson#14:#Friction #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
14.14 Reason
Compare the ease of pulling a lawn mower and pushing it. In particular, in which case is the
friction force that the grass exerts on the mower greater?
(a) The same. (b) Pulling is easier. (c) Pushing is easier.
(d) Not enough information to answer.
PUM | Dynamics | Lesson#14:#Friction 77#
© Copyright 2014, 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?
15.2 Reason
Describe in words a problem for which the following equation is a solution and draw the force
diagram that is consistent with the equation (specify the direction of the axis):
(30 kg)(–1.0 m/s 2 ) = 100 N – fS on O
15.3 Reason
Explain the whiplash phenomenon from the point of view of an observer on the ground and an
observer in the car.
15.4 Reason
You push a bowling ball along a bowling alley. Draw force diagrams for the ball: (a) just before
you let it go; (b) when the ball is rolling along the alley; (c) as the ball is hitting a pin. (d) For
each force exerted on the ball in parts (a)-(c), draw to the side the Newton’s third law pair force
and indicate the object on which these third law forces are exerted.
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?
78! PUM | Dynamics |##Lesson#15:#Putting#It#All#Together #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
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.
15.9 Regular Problem
A Navy Seal of mass 80 kg parachuted into an enemy harbor. At one point while he was falling,
the resistive force that air exerted on him was 520 N . What can you determine about his
motion?
15.10 Regular Problem
Karen Nyberg, 60-kg astronaut, sits on a bathroom scale in a rocket that is taking off vertically
with an acceleration of 3 g . What does the scale read?
15.11 Regular Problem
A 0.10-kg apple falls off a tree branch that is 2.0 m above the grass below. The apple sinks
0.060 m into the grass while stopping. Determine the force that the grass exerts on the apple
while stopping it. Indicate any assumptions you made. [This is a two-part problem.]
15.12 Regular Problem
An 80-kg fireman slides 5.0 m down a fire pole. He holds the pole and exerts a 500-N steady
force that resists his downward slide so that he is not going too fast when he reaches the bottom.
He stops in 0.40 m by bending his knees. What can you determine using this information? [This
is a two-part problem.]
PUM | Dynamics | Lesson#15:#Putting#It#All#Together 79#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
15.13 Reason
Earth exerts a 1.0-N gravitational force on a tennis ball as it falls toward the ground. (a) What
force does the ball exert on Earth? (b) Compare their accelerations due to these forces. The mass
of the ball is about 100 g and the mass of Earth is about 6 × 10 24 kg .
15.14 Reason
You push a bowling ball along a bowling alley. Draw force diagrams for the ball: (a) just before
you let it go; (b) when the ball is rolling along the alley (for two clock readings); (c) as the ball is
hitting a bowling pin. (d) For each force exerted on the ball in parts (a)–(c), draw to the side the
Newton’s third law force, and indicate the object on which these third law forces are exerted.
80! PUM | Dynamics |##Lesson#15:#Putting#It#All#Together #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
Lesson 16: Components
16. 1 Reason
The x- and y-components of several unknown forces are listed next ( Fx , Fy ). For each force,
draw on an x, y coordinate system the components of the force vectors and determine the
magnitude and direction of each force: (a) (+100 N, –100 N), (b) (–300 N, –400 N), and (c) (–
400 N, +300 N).
16.2 Reason
The x- and y-components of several unknown forces are listed next ( Fx , Fy ). For each force,
draw on an x, y coordinate system the components of the force vectors, and determine the
magnitude and direction of each force: (a) (–200 N, +100 N), (b) (+300 N, +400 N), and (c)
(+400 N, –300 N).
Need Some Help?
Scalar components of a vector
#
When working with well-defined x - and y -axes, we do not

need to use the vector components of F , but can instead provide the
same information about the vector by specifying what are called the

scalar components Fx and Fy of the force F . For example, in the

Figure on the right, we see that F is 5 N and points 37 0 above the

negative x -axis. The x vector component of force F is 4 N long
y
F=5N
370
1N
x

and points in the negative x direction. Thus, the scalar x component of F is Fx = –4 N .

Similarly, the y vector component of F is 3 N in the positive y direction. Thus, we can specify

the y effect of F as Fy = +3 N . Thus, the x and y components Fx = –4 N and Fy = +3 N tell

us everything we need to know about the force F . Note that since the scalar components of a
vector are scalars, they are not written with a vector symbol above them.
Finding the magnitude and direction of a force from its scalar components
The scalar components of a vector contain all the information needed to reconstruct the
original vector. To do this, first place the x -vector component of the vector so that its tail is at
the origin (Figure on the next page). The x -scalar component indicates the direction and
PUM | Dynamics | Lesson#16:#Components 81#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
magnitude of the x -vector component along the x -axis. Then place the y -vector component so
it its tail is at the head of the x -vector component. We also know the direction and magnitude of
the y -vector component relative to the y -axis from the sign and magnitude of the y -scalar
component. The force vector points from the tail of the x -vector component to the head of the
y -vector component. Together, the original vector and its two vector components form a right
triangle, and the lengths of the sides of the triangle are the magnitudes of the respective vectors.
Since the magnitudes of the vector components are equal to the magnitudes of the scalar
components, we can use the Pythagorean theorem to determine the magnitude F of the force
(how strong the force is, in newtons):
F=
Fx 2 + Fy 2 .
We can also use this triangle, along with some trigonometry to determine the angle θ
(3.1)
( 90 0 or
less) that the force makes relative to the positive or negative x -axis:
tan θ =
Fy
Fx
,
or
θ = tan–1
Fy
Fx
.
Thus, if we know the components Fx and Fy , we can determine the magnitude and direction of

the force F .

Finally,#if#we#know#the#magnitude#of# F ,#we#can#find#its#scalar#components:#
Fx = F cosθ ; Fy = F sin θ .
16.3 Reason
Finding components of a vector
a) Represent each vector below as the sum of two vectors, one that is parallel to the x-axis
and one that is parallel to the y-axis. Assign some integer values to the length of the

vector a and the angle ϑ and find the length of the components in terms of your chosen
values. Show your work.

b) Now express the length of each component in terms of the length of the vector a and the
angle ϑ. Do not forget the sign!
82! PUM | Dynamics |##Lesson#16:#Components #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
16. 4 Represent and reason
You are pulling a 20-kg crate on a rug exerting a 300-N force at a 300 angle with the horizontal.
The maximum coefficient of static friction between the rug and the crate is 0.5. Represent the
situation with a force diagram, motion diagram and mathematically. Make sure your vertical
forces and force components balance.
16.5 Represent and reason
Inclined planes
Draw a force diagram and show the direction of the acceleration for an object sliding down an
inclined plane tilted at an angle ϑ with the horizontal.
On the force diagram, consider the direction of the following forces exerted on the object
described above.
a) Consider the frictional force (which is parallel to the surface) exerted by the surface of
the incline on the object and the normal force exerted by the incline on the object (which
points perpendicular to the surface). How do they compare to each other?
b) Consider the force exerted by Earth on the object. How does the direction of the force
that Earth exerts on the object compare to those mentioned in part (a)? How does the
magnitude of this force compare to the magnitude of the normal force?
c) Which forces are perpendicular to each other? Which of the forces that are exerted on the
object are not perpendicular?
PUM | Dynamics | Lesson#16:#Components 83#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
16.6 Evaluate
Examine the two diagrams below, which represent an object of mass m accelerating down an
inclined plane. The incline has a rough surface. One diagram has an axis that is upright. The
other diagram has an axis that is tilted. For each situation, draw a force diagram and write
ΣFy ⎞
⎛
ΣF ⎞
⎛
Newton’s Second Law in variable/component form ⎜ ax = x ⎟ and ⎜ ay =
for each
⎝
⎝
m ⎠
m ⎟⎠
situation. Which coordinate axis would you prefer to use for evaluating objects on inclines and
why?
Scenario 1: Upright Axis
!

a#
y!
x!
θ!
Scenario 2: Tilted Axis
y!
x!
a#
θ!
Did You Know?
Components of a force: The force that one object exerts on another has the same effect on
that object as the perpendicular components Fx and Fy of the force. The values of the
components are:
Fx = ± F cos θ
Fy = ± F sin θ
where F is the magnitude of the force and q is the angle (90o or less) between the positive or
negative x-axis and the direction of the force. The component is positive if it projects in the
positive direction of the x- or y-axis and negative if it projects in the negative direction of the
axes.
84! PUM | Dynamics |##Lesson#16:#Components #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
Newton’s Second Law in component form:
F
+ F2 on S x + ... + Fn on S x Fnet x
ax = 1 on S x
=
mS
mS
ay =
F1 on S y + F2 on S y + ... + Fn on S y
mS
=
Fnet y
mS
16.7 Regular Problem
You are trying to pull a sled with two children on it up a hill that makes a 20° angle with the
horizontal. The combined mass of the sled and children is 80 kg. The coefficient of kinetic
friction is 0.2.
How hard should you pull parallel to the hill’s surface if you:
a) want the sled to move at a constant speed?
b) want the sled to accelerate at 1 m/s2 ?
c) If you let the sled sit on the hill, will it slide down or stay in place? The maximum coefficient
of static friction is 0.25.
d) If you wish to use a pulley system to pull the kids up, what should be the mass of the object
attached to the other side of the pulley to pull the sled up at constant speed?
PUM | Dynamics | Lesson#16:#Components 85#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
Homework
16.8 Represent and Reason
Complete#the#table#that#follows.#!
Write a
description
of the
situation in
words.
Sketch the situation
and circle the system
object.
Draw a force diagram with Draw the
perpendicular axes. Label direction of
the forces if needed.
acceleration and
of the sum of the
forces. Are they
consistent?
1. An
elevator is
slowing
down on its
way up.
y!
T
#
R on O
x!
F
#
E on O
2.
y!
# 
NS on O #

TR on O #
30o#
30o#

FE on O #
3.
x!
y!

NS on O #
60o#
30o#
30o#
4.

FE on O #

TR on 1 #
y!

TR on 2
#
1#
x!

FE on 1 #
2#
y!
86! PUM | Dynamics |##Lesson#16:#Components #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
x!
x!

FE on 2 #
Write Newton’s
Second Law in
component form.
Reflect: How can you explain to someone who does not study physics what a
component of a force is? Why is it important to understand components?
Give an example from everyday life when understanding force components
helps you move heavy objects.
PUM | Dynamics | Lesson#16:#Components 87#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
Lesson 17: Finding the coefficient
17.1 Design an Experiment
You are to determine the maximum coefficient of static friction between a shoe and a board in
two different ways. You have the following equipment: the shoe, a spring scale, the board, a
meter stick, a pulley, objects of different masses, and a protractor (you do ot need o use all fo
them).
a) Devise a method using the spring scale.
b) Describe the experiment you will perform and the mathematical procedure that you will
use to solve the problem.
c) What quantities will you measure and what quantities will you calculate?
d) What are you assuming to be true in your procedure?
e) Perform the experiment and calculate the coefficient of static friction. Do not forget that
you cannot obtain an exact value. How do you know if the result is reasonable?
f) Devise a second method using the block and board or a pulley and objects of different
masses but not using the spring scale. Repeat steps (b) – (e).
g) Compare the outcome of the two methods. Do they agree within expected uncertainties?
Explain.
h) Write a complete report about your experiment.
88! PUM | Dynamics |##Lesson#17:#Finding#the#coefficient #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
Use the rubrics to improve your report.
Ability to design and conduct an application experiment
Scientific Ability
Missing
An attempt
Needs some
improvement
Acceptable
1
Is able to identify
the problem to be
solved
No mention is
made of the
problem to be
solved.
An attempt is made to
identify the problem
to be solved but it is
described in a
confusing manner.
The problem to be
solved is described
but there are minor
omissions or vague
details.
The problem to be
solved is clearly
stated.
The experiment
does not solve
the problem.
2
Is able to design a
reliable
experiment that
solves the
problem
The experiment
attempts to solve the
problem but due to
the nature of the
design the data will
not lead to a reliable
solution.
The experiment
attempts to solve
the problem but due
to the nature of the
design there is a
moderate chance
the data will not
lead to a reliable
solution.
The experiment
solves the problem
and has a high
likelihood of
producing data that
will lead to a reliable
solution.
Is able to use
available
equipment to
make
measurements
At least one of
the chosen
measurements
cannot be made
with the
available
equipment.
All of the chosen
measurements can be
made, but no details
are given about how it
is done.
All of the chosen
measurements can
be made, but the
details about how
they are done are
vague or
incomplete.
All of the chosen
measurements can be
made and all details
about how they are
done are provided
and clear.
Is able to evaluate
the results by
means of an
independent
method
No attempt is
made to evaluate
the consistency
of the result
using an
independent
method.
A second independent
method is used to
evaluate the results.
However there is little
or no discussion about
the differences in the
results due to the two
methods.
A second
independent
method is used to
evaluate the results.
The results of the
two methods are
compared using
experimental
uncertainties. But
there is little or no
discussion of the
possible reasons for
the differences
when the results are
different.
A second
independent method
is used to evaluate
the results and the
evaluation is done
with the experimental
uncertainties. The
discrepancy between
the results of the two
methods, and
possible reasons are
discussed.
Is able to identify
the shortcomings
in an
experimental
design and
suggest specific
improvements
No attempt is
made to identify
any
shortcomings of
the experimental
design.
An attempt is made to
identify
shortcomings, but
they are described
vaguely and no
specific suggestions
for improvements are
made.
Some shortcomings
are identified and
some improvements
are suggested, but
not all aspects of
the design are
considered.
All major
shortcomings of the
experiment are
identified and
specific suggestions
for improvement are
made.
3
4
5
PUM | Dynamics | Lesson#17:#Finding#the#coefficient 89#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
Is able to choose a
productive
mathematical
procedure for
solving the
experimental
problem
Mathematical
procedure is
either missing,
or the equations
written down are
irrelevant to the
design.
A mathematical
procedure is
described, but is
incorrect or
incomplete, due to
which the final
answer cannot be
calculated.
Correct and
complete
mathematical
procedure is
described but an
error is made in the
calculations.
Mathematical
procedure is fully
consistent with the
design. All quantities
are calculated
correctly. Final
answer is meaningful.
7
Is able to identify
the assumptions
made in using the
mathematical
procedure
No attempt is
made to identify
any
assumptions.
8
No attempt is
made to
determine the
effects of
assumptions.
Relevant
assumptions are
identified but are
not significant for
solving the
problem.
The effects of
assumptions are
determined, but no
attempt is made to
validate them.
All relevant
assumptions are
correctly identified.
Is able to
determine
specifically the
way in which
assumptions
might affect the
results
An attempt is made to
identify assumptions,
but the assumptions
are irrelevant or
incorrect for the
situation.
The effects of
assumptions are
mentioned but are
described vaguely.
6
90! PUM | Dynamics |##Lesson#17:#Finding#the#coefficient #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
The effects of the
assumptions are
determined and the
assumptions are
validated.
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.
PUM | Dynamics | Lessons#18:#Practice 91#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
Sketch the situation described in the problem;
provide all known information.
Choose a system object in the sketch from the
first cell of this table and make a list of objects
that interact with the system object. Simplify
objects and interactions if necessary.
Draw a force diagram for the system object.
Label the forces. Make sure the diagram is
consistent with the motion of the system.
Include perpendicular x and y axes.
Apply Newton’s Second Law in component
form (x and y axes) to the situation shown in
the force diagram.
Combine the results from the above force
analysis with kinematics to determine the
unknown quantity. Evaluate the result to see
if it is reasonable (unit, magnitude, and value
for limiting situations).
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?
92! PUM | Dynamics |##Lessons#18:#Practice #
© Copyright 2014, 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?
Sketch the situation described in the problem;
provide all known information.
Choose a system object in the sketch from the
first cell of this table and list objects that interact
with the system.
Draw a force diagram for the system object.
Label the forces. Make sure the diagram is
consistent with the motion of the system object.
Include perpendicular x and y axes.
Apply Newton’s Second Law in component form
(x and y axes) to the situation shown in the force
diagram you drew.
Solve the equations for the unknown quantities.
Evaluate the results to see if they are reasonable
(units, magnitudes, and values for limiting
situations).
PUM | Dynamics | Lessons#18:#Practice 93#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
18.6 Evaluate the Solution
Identify any errors in the solution to the following problem and provide a corrected
solution if there are errors.
The problem: A 1000 kg elevator is moving down at 6.0 m/s. It slows to a stop in 3.0
m as it approaches the ground floor. Determine the force that the cable supporting the
elevator exerts on the elevator as the elevator stops. Assume that g = 10 N/kg.
,#
Proposed solution: The elevator at the right is the object of interest. It is considered a
particle, and the forces that other objects exert on the elevator are shown in the force
diagram. The acceleration of the elevator is:
a = v02/2d = (6.0 m/s)2/2(3.0 m) = 6.0 m/s2.
The force of the cable on the elevator while stopping is:
T = ma = (1000 kg)(6.0 m/s2) = 6000 N.
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.
94! PUM | Dynamics |##Lessons#18:#Practice #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
18.8 Design an Experiment
Design a balloon racer. You are given 2 balloons, straws, paper,
and tape. The racer should be designed using your understanding
of “forces”. You will race this balloon racer against other
students in the 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
d) What assumptions did you make when doing these calculations? How do your
assumptions affect your calculated estimate?
e) What is your relative uncertainty in each value? What could you have done to reduce
uncertainty?
v#
18.9 Reason
The graph for velocity of book being pushed by Jenny across a
table shown on the right. Draw 5 force digraphs for the book for
selected clock readings (marked on the graph).
t1#
t2#
t4#
t5#
18.10 Reason
Go to the Phet simulation Forces in 1 Dimension. Turn friction off. You goal is to choose how
to manipulate the force so the graph acceleration versus time looks as follows:
18.11 Test your ideas
Go to http://paer.rutgers.edu/pt3/experimentindex.php?topicid=3&cycleid=2
You will find several testing experiments listed below. For each experiment, follow instructions
and answer all of the questions on the website. Submit your answers to your teacher.
a) Cart on Track: Qualitative Testing Experiment
PUM | Dynamics | Lessons#18:#Practice 95#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
b) Atwood's Machine with a spring scale Part 1: Qualitative Testing Experiment
c) Atwood's Machine with Spring Scale Part 2: Qualitative Testing Experiment
d) Spring scale up and down: Qualitative Testing Experiment
Reflect: If you could write a note to your past self about solving problems
involving Newton’s laws, what would you say?
96! PUM | Dynamics |##Lessons#18:#Practice #
© Copyright 2014, 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) ΣF 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 +(– 0.4 Ffloor 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.
PUM | Dynamics | Lesson#19:#Review 97#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
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 sum of the forces exerted on
the object in each case if the mass of the
object is 250 kg.
Velocity)(m/s))
19.3 Graph Jeopardy
40!
30!
20!
10!
0!
"10!
"20!
"30!
"40!
0!
10!
20!
30!
40!
50!
60!
70!
Time)(s))
19.4 Reason
Earth exerts a 5-N force on an apple, what is the force that the apple exerts on Earth?
19.5 Reason
You are pulling two boxes (10 kg and 15 kg) connected with a rope on a horizontal surface. You
exert a 250 N force at an angle of 270 with the horizontal by pulling the second rope attached to a
15-kg box. Represent the situation with the force and motion diagram. Write Newton’s second
law in component form for each box. Consider two cases: the surface is a smooth floor and the
boxes are made of laminated cardboard and the floor is a carpet and the boxes are made of
regular cardboard.
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
98! PUM | Dynamics |##Lesson#19:#Review #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
80!
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?
19.14 Reason
Questions below refer to collisions between a car and trucks. For each description of a collision
below, choose the one answer from the possibilities 1 though 6 that best describes the forces
between the car and the truck.
1
The truck exerts a greater amount of force on the car than the car exerts on the truck.
2
The car exerts a greater amount of force on the truck than the truck exerts on the car.
3
Neither exerts a force on the other; the car gets smashed simply because it is in the way of
the truck.
4
The truck exerts a force on the car but the car doesn't exert a force on the truck.
5
The truck exerts the same amount of force on the car as the car exerts on the truck.
PUM | Dynamics | Lesson#19:#Review 99#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
In questions a-d the truck is
much more massive than
the car.
a.
They are both moving at the same speed right before they collide. Which choice
describes the forces?
b.
The truck is moving and the car is standing still right before they collide. Which
choice describes the forces?
c.
The car is moving much faster than the truck right before they collide. Which choice
describes the forces?
d.
The heavier truck is standing still when the car hits it. Which choice describes the
forces?
In questions e and f the
truck is a small pickup and
has the same mass as the
car.
e.
Both the truck and the car are moving at the same speed right before they collide.
Which choice describes the forces?
f.
The car is standing still when the car hits it. Which choice describes the forces?
Questions below refer to a large
truck which breaks down out on the
road and receives a push back to
town by a small compact car.
Pick one of the choices 1 through 5 below which correctly describes the forces that the car exerts
on the truck and the truck exerts on the car for each of the descriptions (35-38).
1
The force exerted by of the car pushing against the truck is equal to that exerted by the truck
pushing back against the car.
100! PUM | Dynamics |##Lesson#19:#Review #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
2
The force exerted by the car pushing against the truck is less than that exerted by the truck
pushing back against the car.
3
The force exerted by the car pushing against the truck is greater than that exerted by the truck
pushing back against the car.
4
The car's engine is running so it exerts a force as it pushes against the truck, but the truck's
engine isn't running so it can't exert a force back against the car.
5
Neither the car nor the truck exert any force on each other. The truck is pushed forward simply
because it is in the way of the car.
a.
The car is pushing on the truck, but not hard enough to make the truck move.
b.
The car, still pushing the truck, is speeding up to get to cruising speed.
c.
The car, still pushing the truck, is at cruising speed and continues to travel at the
same speed.
d.
The car, still pushing the truck, is at cruising speed when the truck puts on its brakes
and causes the car to slow down.
19.15 Represent and reason
Choose 4 of the situations in the exercise above and draw force diagrams for both the car and the
truck. Do these force diagrams explain the situations? Are the forces that the car and truck exert
on each other always equal in magnitude? Why? If yes, why don’t they cancel each other?
19.16. Reason Heather thinks that the engine is the object that pushes the car, Tara think is it is
the ground. Who do you think is correct? How can you convince your opponent in your opinion?
Additional conceptual exercises and problems
19.17 Represent and Reason
A person pulls a rope, which in turn pulls a crate across a horizontal, rough surface, as shown
below.
Three motion diagrams are shown below for the crate (with velocity arrows only). In the
table that follows, construct a force diagram for the crate and make the horizontal arrows the
correct relative lengths.
(a)
(b)
(c)
PUM | Dynamics | Lesson#19:#Review 101#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#
(a)
(b)
(c)
19.18 Represent and Reason
You are pulling up on a suitcase, exerting an upward force of 200 N on it. The mass of the
suitcase is 15 kg. Draw a force diagram for the suitcase; make sure the lengths of the arrows
represent the relative magnitudes of the forces. Use the force diagram to draw the motion
diagram for the suitcase. If the suitcase is accelerating, what is its acceleration? (Hint: think
about all the forces exerted on the suitcase)
19.19 Represent and Reason
In the previous problem, what is the force that the suitcase exerts on your hand? Why can
you lift the suitcase but the suitcase does not lift you?
19.20 Represent and Reason
You continue lifting the suitcase, but now you are exerting an upward force of 147 N on it.
Draw a force diagram for the suitcase; make sure the lengths of the arrows represent the
relative magnitudes of the forces. Use the force diagram to draw the motion diagram for the
suitcase. If the suitcase is accelerating, what is its acceleration?
19.21 Represent and Reason
You continue lifting the suitcase, but now you are exerting an upward force of 100 N on it.
Draw a force diagram for the suitcase; make sure the lengths of the arrows represent the
relative magnitudes of the forces. Use the force diagram to draw the motion diagram for the
suitcase. If the suitcase is accelerating, what is its acceleration? In which direction is the
suitcase moving?
15!
10!
The speed of an object changes, as shown in the graph
below. The mass of the object is 5 kg.
a) Describe the object’s motion in words, as fully as
possible.
102! PUM | Dynamics |##Lesson#19:#Review #
© Copyright 2014, Rutgers, The State University of New Jersey.#
#
Velocity)(m/s))
19.22 Represent and Reason
5!
0!
"5!
"10!
0!
2!
4!
Time)(s))
6!
8!
b) Determine the sum of the forces exerted on the object for each segment of the graph.
c) Write a mathematical function describing the first segment of the graph.
d) Devise a story about the object’s motion.
19.23 Represent and Reason
You throw a tennis ball upward.
Draw a motion diagram for the ball when (a) it is still in contact with your hand, about to leave
it, (b) when it is moving up, (c) when it is at the top of its flight, (d) when it is moving down, and
(e) when you catch it.
Then draw a force diagram for (a) – (e).
Estimate the ball’s acceleration for (a) – (e).
19.24 Explain
Give an example for (a) when an object is moving to the right but the sum of the forces exerted
on it is pointing to the left, (b) when an object is moving down but the sum of the forces exerted
on it is up, and (c) when an object is moving down but the sum of the forces exerted on it is zero.
19.25 Argue
Jake says that cars move because the engines push them. Do you agree with Jake? Explain your
opinion; use force diagrams. How can you convince Jake of your opinion?
Reflect: Congratulations! You completed the Dynamics module. If you were
to make a list of three most important ideas you learned in it, what would
those be?
After you make up the list, write a short description of each idea with an
everyday example. The best descriptions will become a part of the module for
the students next year!
PUM | Dynamics | Lesson#19:#Review 103#
© Copyright 2014, Rutgers, The State University of New Jersey. #
#