MP02: Motion Diagrams
Velocity and Acceleration of a Power Ball
Learning Goal: To understand the distinction between velocity and acceleration with the use of motion diagrams.
In common usage, velocity and acceleration both can imply having considerable speed. In physics, they are sharply
defined concepts that are not at all synonymous. Distinguishing clearly between them is a prerequisite to
understanding motion. Moreover, an easy way to study motion is to draw a motion diagram, in which the position of
the object in motion is sketched at several equally spaced instants of time, and these sketches (or snapshots) are
combined into one single picture.
In this problem, we make use of these concepts to study the motion of a power ball. This discussion assumes that we
have already agreed on a coordinate system from which to measure the position
of objects as a function of time. Let
and
(also called the position vector)
be velocity and acceleration, respectively.
Harvaran Ghai
Consider the motion of a power ball that is dropped on the floor and bounces back. In the following questions, you
will describe its motion at various points in its fall in terms of its velocity and acceleration.
Part A
You drop a power ball on the floor. The motion diagram of the ball is sketched in the figure
the magnitude of the velocity of the ball is increasing, decreasing, or not changing.
increasing
. Indicate whether
decreasing
not changing
Correct
While the ball is in free fall, the magnitude of its velocity is increasing, so the ball is accelerating.
Part B
Since the length of is directly proportional to the length of
, the vector connecting each dot to the next could
represent velocity vectors as well as position vectors, as shown in the figure here
. Indicate whether the velocity
and acceleration of the ball are, respectively, positive (upward), negative, or zero.
Use P, N, and Z for positive (upward), negative, and zero, respectively. Separate the letters for velocity and
acceleration with a comma.
N,N
Correct
Part C
Now, consider the motion of the power ball once it bounces upward. Its motion diagram is shown in the figure here
. Indicate whether the magnitude of the velocity of the ball is increasing, decreasing, or not changing.
increasing
decreasing
not changing
Correct
Since the magnitude of the velocity of the ball is decreasing, the ball must be accelerating (specifically, slowing
down).
Part D
The next figure
shows the velocity vectors corresponding to the upward motion of the power ball. Indicate
whether its velocity and acceleration, respectively, are positive (upward), negative, or zero.
Use P, N, and Z for positive (upward), negative, and zero, respectively. Separate the letters for velocity and
acceleration with a comma.
P,N
Correct
Part E
The power ball has now reached its highest point above the ground and starts to descend again. The motion diagram
representing the velocity vectors is the same as that after the initial release, as shown in the figure of Part B. Indicate
whether the velocity and acceleration of the ball at its highest point are positive (upward), negative, or zero.
Use P, N, and Z for positive (upward), negative, and zero, respectively. Separate the letters for velocity and
acceleration with a comma.
Z,N
Correct
These examples should show you that the velocity and acceleration can have opposite or similar signs or that one of
them can be zero while the other has either sign. Try hard to think carefully about them as distinct physical
quantities when working with kinematics.
Motion of Two Rockets
Learning Goal: To learn to use images of an object in motion to determine velocity and acceleration.
Two toy rockets are traveling in the same direction (taken to be the x axis). A diagram is shown of a time-exposure
image where a stroboscope has illuminated the rockets at the uniform time intervals indicated.
Harvaran Ghai
Part A
At what time(s) do the rockets have the same velocity?
at time
only
at time
only
at times
and
at some instant in time between
at no time shown in the figure
and
Correct
Part B
At what time(s) do the rockets have the same x position?
at time
only
at time
only
at times
and
at some instant in time between
at no time shown in the figure
Correct
and
Part C
At what time(s) do the two rockets have the same acceleration?
at time
only
at time
only
at times
and
at some instant in time between
at no time shown in the figure
and
Correct
Part D
The motion of the rocket labeled A is an example of motion with uniform (i.e., constant) __________.
and nonzero acceleration
velocity
displacement
time
Correct
Part E
The motion of the rocket labeled B is an example of motion with uniform (i.e., constant) __________.
and nonzero acceleration
velocity
displacement
time
Correct
Part F
At what time(s) is rocket A ahead of rocket B?
before
after
before
only
only
and after
between
and
at no time(s) shown in the figure
Correct
PSS 1.1: (Almost) a Dozen Diagrams
Learning Goal: To practice Problem-Solving Strategy 1.1 for constructing motion diagrams.
A car is traveling with constant velocity along a highway. The driver notices he is late for work so he stomps down
on the gas pedal and the car begins to accelerate. The car has just achieved double its initial velocity when the driver
spots a policeman behind him and applies the brakes. The car then decelerates, coming to rest at a stoplight ahead.
In this problem, you will be asked several questions related to construction of a motion diagram for this situation
and a few others.
Harvaran Ghai
Represent the moving object as a particle. Make simplifying assumptions when interpreting the problem
statement.
MODEL:
VISUALIZE:



A complete motion diagram consists of:
The position of the object in each frame of the film, shown as a dot. Use five or six dots to make the motion
clear but without overcrowding the picture. More complex motions may need more dots.
The average velocity vectors, found by connecting each dot in the motion diagram to the next with a vector
arrow. There is one velocity vector linking each set of two position dots. Label the row of velocity vectors
.
The average acceleration vectors, found using Tactics Box 1.3. There is one acceleration vector linking
each set of two velocity vectors. Each acceleration vector is drawn at the dot between the two velocity
vectors it links. Use
vectors
.
to indicate a point at which the acceleration is zero. Label the row of acceleration
Model
It is appropriate to use the particle model for the car. You should also make some simplifying assumptions.
Part A
Which of the following simplifying assumptions is it reasonable to make in this problem?
A. During each of the three different stages of its motion, the car is moving with constant (possibly zero)
acceleration.
B. During each of the three different stages of its motion, the car is moving with constant (possibly zero)
velocity.
C. The highway is straight (i.e., there are no curves).
D. The highway is level (i.e., there are no hills or valleys).
Enter the letters of all the correct answers in alphabetical order. Do not use commas. For example, if you
think that assumptions C and D are reasonable, enter CD.
ACD
Correct
Visualize
Now draw a motion diagram, including all the elements listed in the problem-solving strategy. Use your diagram to
answer the following questions.
In interpreting the diagrams that follow, assume that the car is moving in a straight line to the right.
Refer to this set of motion diagrams
in answering the following.
Part B
Which of the diagrams best describes the position and the velocity of the car before the driver notices he is late?
A
B
C
Correct
Part C
Which of the diagrams best describes the position and the velocity of the car after the driver hits the gas, but before
he notices the policeman?
A
B
C
Correct
Part D
Which of the diagrams best describes the position and the velocity of the car after the driver notices the policeman?
A
B
C
Correct
Part E
Which of these diagrams
problem introduction?
most accurately depicts the acceleration of the car during the events described in the
Assume that the car is initially moving to the right.
A
B
C
Correct
Now let's use our results for the car and apply them to some other problems. Consider these three situations:



A train has its brakes released and pulls out of the station, slowly picking up speed.
A sled is given a quick push along a horizontal surface; the sled comes to a stop after covering some
distance.
A motorcycle is moving along a straight highway at 105 km/h (the legal speed limit in many states).
Part F
Of the three situations described, which object corresponds to the position and velocity diagram shown here?
the train
the sled
the motorcycle
Correct
Note that the diagram shown is not a complete motion diagram; it lacks the vector representing the acceleration of
the object.
Part G
Of the three situations described, which object corresponds to the motion diagram shown here?
the train
the sled
the motorcycle
Correct
Let us now consider another scenario. A car and a truck are moving at the same velocity along a straight highway.
Both drivers apply the brakes at the same moment. The car and truck both come to a stop. The car takes less time to
stop than the truck.
Refer to the motion diagrams shown here
in answering the following.
Assume that both cars are moving to the right.
Part H
Which of the three diagrams shown best describes the motion of the car and truck after the brakes have been
applied?
A
B
C
Correct
Part I
Diagram (B) is incorrect because, according to it:
The car and truck move in different directions.
The car is moving at constant velocity.
The truck is speeding up.
The car and the truck have the same acceleration.
Correct
Part J
Diagram (C) is incorrect because, according to it:
The car and truck move in different directions.
The car is moving at constant velocity.
The truck is speeding up.
The car and the truck have the same acceleration.
Correct
MP04: Using Motion Diagrams
Curved Motion Diagram
The motion diagram shown here
represents a pendulum released from rest at an angle of 45 from the vertical.
The dots in the motion diagram represent the positions of the pendulum bob at eleven moments separated by equal
time intervals. The green arrows represent the average velocity between adjacent dots. Also given is a "compass
rose" of directions in which the different directions are labeled with the letters of the alphabet.
Harvaran Ghai
Part A
What is the direction of the acceleration of the pendulum bob at moment 5?
Enter the letter of the arrow with this direction from the compass rose in the figure. Type Z if the acceleration
vector has zero length.
A
Correct
Part B
What is the direction of the acceleration of the pendulum bob at moments 0 and 10?
Enter the letters of the arrows with these directions from the compass rose in the figure, separated by
commas. Type Z if the acceleration vector has zero length.
directions at moment 0, moment 10 =D,F
Correct
Part C
In which of the following other scenarios could the motion reasonably be represented by the motion diagram in the
introduction?
A.
B.
C.
D.
A weight placed on the rim of a bicycle wheel that is being held off the ground so it can rotate freely
An airplane pulling out of a dive
A race car rounding a turn
A marble released part way up the inside surface of a smoothly rounded bowl
Enter the letters of all possible correct scenarios in alphabetical order. Do not use commas.
AD
Correct
Part D
Assume that the diagram in the problem introduction represents the motion of a ball tied on the end of a string--that
is, a pendulum. Also assume that the interval between each time step in the diagram is 0.10 s. The total time it takes
for this pendulum to swing back and forth, also called the period of the pendulum, is then approximately 2 s.
An interesting fact about the pendulum is that its period is essentially independent of the weight of the ball (or
whatever other object is used). It depends only on the length of the string (with longer period for longer strings) and
the strength of the force of gravity (which is essentially constant over the surface of the earth).
Based on observation or comparison with other real-life pendula, estimate the length
pendulum to have a period of 2 s.
Express
of the string needed for the
in meters. A factor of 3 error is allowed in either direction.
=1.0
Correct
Physics can often seem to be a science of very precise answers. However, having a solid grasp of the fundamental
concepts allows a physicist to make reasonably accurate estimates like this one very quickly. Even if you were
ultimately looking for a more precise answer, an initial estimate gives you a way of checking whether the result of a
long calculation is reasonable.
Average Velocity from a Position vs. Time Graph
Learning Goal: To learn to read and interpolate on a graph of position versus time and to change units.
In this problem you must find the average velocity from a graph of
indicate the average velocity over the time interval from
interval from 1 to 3 s.
to
. Thus
. We will use the notation
to
is the average velocity over the time
Harvaran Ghai
Part A
Find the average velocity over the time interval from 0 to 1 second.
Answer to the nearest integer.
=0
Correct
Part B
Find the average velocity over the time interval from 1 to 3 seconds.
Answer to the nearest integer.
=20
Correct
Part C
Now that you have
and
, find
Give your answer to three significant figures.
.
=13.3
Correct
Note that
is not equal to the simple arithmetic average of
and
for time intervals of different length. You would have to double the weight given to
interval twice as long.
Part D
, because they are averages
because it is for an
Find the average velocity over the time interval from 1 to 5 seconds. You will need to interpolate to find the position
at time
. Do not simply eyeball the position or you will likely not be able to obtain the solution to the desired
accuracy.
Round your answer to two significant figures.
=6.7
Correct
Part E
Obtaining this answer required some interpolation on the graph. Now see if you can express this result in terms of
kilometers per hour.
Express your answer to the nearest integer.
=24
Correct
Part F
Find the average velocity over the time interval from 2.5 to 6.0 seconds.
Express your answer to two significant figures.
=-8.6
Correct
Running and Walking
Tim and Rick both can run at speed
distance
half.
and walk at speed
, with
. They set off together on a journey of
. Rick walks half of the distance and runs the other half. Tim walks half of the time and runs the other
Harvaran Ghai
Part A
How long does it take Rick to cover the distance
Express the time taken by Rick in terms of
?
,
, and
.
=
Correct
Part B
Find Rick's average speed for covering the distance
Express Rick's average speed in terms of
=
Correct
,
.
, and
.
Part C
How long does it take Tim to cover the distance?
Express the time taken by Tim in terms of
,
, and
.
=
Correct
Part D
Who covers the distance more quickly?
Think logically, but without using the detailed answers in the previous parts.
Rick
Tim
Neither. They cover the distance in the same amount of time.
Correct
Part E
In terms of given quantities, by what amount of time,
, does Tim beat Rick?
It will help you check your answer if you simplify it algebraically and check the special case
Express the difference in time,
in terms of
,
, and
.
.
=
Correct
Part F
In the special case that
, what would be Tim's margin of victory
?
0
Correct
Graph of v(t) for a Sports Car
The graph
questions.
shows the velocity of a sports car as a function of time . Use the graph to answer the following
Harvaran Ghai
Part A
Find the maximum velocity
of the car.
Express your answer in meters per second to the nearest integer.
=55
Correct
Part B
During which time interval is the acceleration positive?
Indicate the most complete answer.
to
to
to
to
to
Correct
Part C
Find the maximum acceleration
of the car.
Express your answer in meters per second squared to the nearest integer.
=30
Correct
Part D
Find the minimum magnitude of the acceleration
of the car.
Express your answer in meters per second squared to the nearest integer.
=0
Correct
Part E
Find the distance traveled by the car between 0 and 2 s.
Express your answer in meters to the nearest integer.
=55
Correct
Rearending Drag Racer
To demonstrate the tremendous acceleration of a top fuel drag racer, you attempt to run your car into the back of a
dragster that is "burning out" at the red light before the start of a race. (Burning out means spinning the tires at high
speed to heat the tread and make the rubber sticky.)
You drive at a constant speed of toward the stopped dragster, not slowing down in the face of the imminent
collision. The dragster driver sees you coming but waits until the last instant to put down the hammer, accelerating
from the starting line at constant acceleration, . Let the time at which the dragster starts to accelerate be
.
Harvaran Ghai
Part A
What is
, the longest time after the dragster begins to accelerate that you can possibly run into the back of the
dragster if you continue at your initial velocity?
=
Correct
Part B
Assuming that the dragster has started at the last instant possible (so your front bumper almost hits the rear of the
dragster at
), find your distance from the dragster when he started. If you calculate positions on the way to
this solution, choose coordinates so that the position of the drag car is 0 at
. Remember that you are solving for
a distance (which is a magnitude, and can never be negative), not a position (which can be negative).
=
Correct
Part C
Find numerical values for
and
in seconds and meters for the (reasonable) values
(26.8 m/s)
and
.
Separate your two numerical answers by commas, and give your answer to two significant figures.
,
Correct
=0.54,7.2 s, m
The blue curve shows how the car, initially at
accelerating drag car (red) at
, continues at constant velocity (blue) and just barely touches the
.
Motion of a Shadow
A small source of light
is located at a distance
moves toward the wall with constant velocity
.
from a vertical wall. An opaque object with a height of
of magnitude . At time
, the object is located at the source
Harvaran Ghai
Part A
Find an expression for
, the magnitude of the velocity
of the top of the object's shadow, at time .
Express the speed of the top of the object's shadow in terms of , ,
, and
.
=
Correct
Rocket Height
A rocket, initially at rest on the ground, accelerates straight upward from rest with constant net acceleration , until
time
, when the fuel is exhausted.
Harvaran Ghai
Part A
Find the maximum height
that the rocket reaches (neglecting air resistance).
Express the maximum height in terms of , , and/or . Note that in this problem, is a positive number
equal to the magnitude of the acceleration due to gravity.
= a*t1*((a*t1)/g)-(.5g((a*t1)/g)2)+(.5(t1)2*a)
Correct
Part B
If the rocket's net acceleration is
for
Express your answer numerically in meters, using
, what is the maximum height the rocket will reach?
.
=1470 m
Correct
A Flower Pot Falling Past a Window
As you look out of your dorm window, a flower pot suddenly falls past. The pot is visible for a time , and the
vertical length of your window is
. Take down to be the positive direction, so that downward velocities are
positive and the acceleration due to gravity is the positive quantity .
Assume that the flower pot was dropped by someone on the floor above you (rather than thrown downward).
Harvaran Ghai
Part A
From what height
above the bottom of your window was the flower pot dropped?
Express your answer in terms of
, , and .
=
Correct
Part B
If the bottom of your window is a height above the ground, what is the velocity
of the pot as it hits the
ground? You may introduce the new variable , the speed at the bottom of the window, defined by
.
Express your answer in terms of some or all of the variables
,
=
Correct
Resolving Vector Components with Trigonometry
, ,
, and .
Often a vector is specified by a magnitude and a direction; for example, a rope with tension
exerts a force of
magnitude in a direction 35 degrees north of east. This is a good way to think of vectors; however, to calculate
results with vectors, it is best to select a coordinate system and manipulate the components of the vectors in that
coordinate system.
Harvaran Ghai
Part A
Find the components of the vector with length and angle with respect to the x axis as shown, named . Don't
forget that when multiplying two factors, you must include a multiplication symbol; also, the cos and sin functions
must have parentheses around their arguments. For example, a vector might take the form p*sin(Q),m*cos(N).
Write the components in the form x,y.
=
Correct
Part B
Find the components of the vector with length and angle
Write the components in the form x,y.
with respect to the x axis as shown, named
.
=
Correct
Notice that vectors
Part C
and
have the same form despite their placement with respect to the y axis on the drawing.
Find the components of the vector with length and angle
Express your answer in terms of and
as shown, named
.
. Write the components in the form x,y.
=
Correct
Tracking a Plane
A radar station, located at the origin of xz plane, as shown in the figure
, detects an airplane coming straight at
the station from the east. At first observation (point A), the position of the airplane relative to the origin is
position vector
has a magnitude of 360
tracked for another 123
and is located at exactly 40
. The
above the horizon. The airplane is
in the vertical east-west plane for 5.0 , until it has passed directly over the station
and reached point B. The position of point B relative to the origin is
(the magnitude of
is 880 ). The
contact points are shown in the diagram, where the x axis represents the ground and the positive z direction is
upward.
Harvaran Ghai
Part A
Define the displacement of the airplane while the radar was tracking it:
components of
?
. What are the
Express
in meters as an ordered pair, separating the x and z components with a comma, to two
significant figures.
=-1100,26
Correct
Two Forces Acting at a Point
Two forces,
and
, act at a point.
negative x axis in the second quadrant.
negative x axis in the third quadrant.
has a magnitude of 9.80
has a magnitude of 6.40
and is directed at an angle of 60.0 above the
and is directed at an angle of 53.2 below the
Harvaran Ghai
Part A
What is the x component of the resultant force?
Express your answer in newtons.
-8.73
Correct
Part B
What is the y component of the resultant force?
Express your answer in newtons.
3.36
Correct
Part C
What is the magnitude of the resultant force?
Express your answer in newtons.
9.36
Correct
A Push or a Pull?
Learning Goal: To understand the concept of force as a push or a pull and to become familiar with everyday forces.
A force can be simply defined as a push or a pull exerted by one object upon another.
Although such a definition may not sound too scientific, it does capture three essential properties of forces:



Each force is created by some object.
Each force acts upon some other object.
The action of a force can be visualized as a push or a pull.
Since each force is created by one object and acts upon another, forces must be described as interactions. The proper
words describing the force interaction between objects A and B may be any of the following:

"Object A acts upon object B with force

"Object A exerts force

"Force
is applied to object B by object A."

"Force
due to object A is acting upon object B."
."
upon object B."
One of the biggest mistakes you may make is to think of a force as "something an object has." In fact, at least two
objects are always required for a force to exist.
Each force has a direction: Forces are vectors. The main result of such interactions is that the objects involved
change their velocities: Forces cause acceleration. However, in this problem, we will not concern ourselves with
acceleration--not yet.
Harvaran Ghai
Some common types of forces that you will be dealing with include the gravitational force (weight), the force of
tension, the force of friction, and the normal force.
It is sometimes convenient to classify forces as either contact forces between two objects that are touching or as
long-range forces between two objects that are some distance apart. Contact forces include tension, friction, and the
normal force. Long-range forces include gravity and electromagnetic forces. Note that such a distinction is useful
but not really fundamental: For instance, on a microscopic scale the force of friction is really an electromagnetic
force.
In this problem, you will identify the types of forces acting on objects in various situations.
First, consider a book resting on a horizontal table.
Part A
Which object exerts a downward force on the book?
the book itself
the earth
the surface of the table
Correct
Part B
The downward force acting on the book is __________.
a contact force
a long-range force
Correct
Part C
What is the downward force acting on the book called?
tension
normal force
weight
friction
Correct
Part D
Which object exerts an upward force on the book?
the book itself
the earth
the surface of the table
Correct
Part E
The upward force acting on the book is __________.
a contact force
a long-range force
Correct
Part F
What is the upward force acting on the book called?
tension
normal force
weight
friction
Correct
Now consider a different situation. A string is attached to a heavy block. The string is used to pull the block to the
right along a rough horizontal table.
Part G
Which object exerts a force on the block that is directed toward the right?
the block itself
the earth
the surface of the table
the string
Correct
Part H
The force acting on the block and directed to the right is __________.
a contact force
a long-range force
Correct
To exert a tension force, the string must be connected to (i.e., touching) the block.
Part I
What is the force acting on the block and directed to the right called?
tension
normal force
weight
friction
Correct
Part J
Which object exerts a force on the block that is directed toward the left?
the block itself
the earth
the surface of the table
the string
Correct
Part K
The force acting on the block and directed to the left is __________.
a contact force
a long-range force
Correct
Part L
What is the force acting on the block and directed to the left called?
tension
normal force
weight
friction
Correct
Now consider a slightly different situation. The same block is placed on the same rough table. However, this time,
the string is disconnected and the block is given a quick push to the right. The block slides to the right and
eventually stops. The following questions refer to the motion of the block after it is pushed but before it stops.
Part M
How many forces are acting on the block in the horizontal direction?
0
1
2
3
Correct
Once the push has commenced, there is no force acting to the right: The block is moving to the right because it was
given a velocity in this direction by some force that is no longer applied to the block (probably, the normal force
exerted by a student's hand or some spring launcher).
Once the contact with the launching object has been lost, the only horizontal force acting on the block is directed to
the left--which is why the block eventually stops.
Part N
What is the force acting on the block that is directed to the left called?
tension
normal force
weight
friction
Correct
The force of friction does not disappear as long as the block is moving. Once the block stops, fricion becomes zero
(assuming the table is perfectly horizontal).
Free-Body Diagrams: Introduction
Learning Goal: To learn to draw free-body diagrams for various real-life situations.
Imagine that you are given a description of a real-life situation and are asked to analyze the motion of the objects
involved. Frequently, that analysis would involve finding the acceleration of the objects. That, in turn, requires that
you find the net force.
To find the net force, you must first identify all of the forces involved and then add them as vectors. Such a
procedure is not always trivial. It is helpful to replace the sketch of the situation by the drawing of the object
(represented as a particle) and all the forces applied to it. Such a drawing is called a free-body diagram. This
problem will walk you through several examples of free-body diagrams and will demonstrate some of the possible
pitfalls.
Here is the general strategy for drawing free-body diagrams:



Identify the object of interest. This may not always be easy: A sketch of the situation may contain many
objects, each of which has a different set of forces acting on it. Including forces acting on different
objects in the same diagram will lead to confusion and a wrong solution.
Draw the object as a dot. Draw and clearly label all the forces acting on the object of interest. The
forces should be shown as vectors originating from the dot representing the object of interest. There are
two possible difficulties here: omitting some forces and drawing the forces that either don't exist at all
or are applied to other objects. To avoid these two pitfalls, remember that every force must be applied
to the object of interest by some other object--or, as some like to say, "every force must have a source."
Once all of the forces are drawn, draw the coordinate system. The origin should coincide with the dot
representing the object of interest and the axes should be chosen so that the subsequent calculations of
vector components of the forces will be relatively simple. That is, as many forces as possible must be
either parallel or perpendicular to one of the axes.
Harvaran Ghai
It should come as good news that, even though real life can present us with a wide variety of situations, we will be
mostly dealing with a very small number of forces. Here are the principal ones of interest:




Weight, or the force due to gravity. Weight acts on every object and is directed straight down unless we are
considering a problem involving the nonflat earth (e.g., satellites).
Normal force. The normal force exists between two surfaces that are pressed against each other; it is always
perpendicular to the surfaces.
Force of tension. Tension exists in strings, springs, and other objects of finite length. It is directed along the
string or a spring. Keep in mind that a spring can be either compressed or stretched whereas a string can
only be stretched.
Force of friction. A friction force exists between two surfaces that either move or have a tendency to move
relative to each other. Sometimes, the force of air drag, similar in some ways to the force of friction, may
come into play. These forces are directed so that they resist the relative motion of the surfaces. Keep in
mind that to simplify problems you often assume friction is negligible on smooth surfaces. In addition, the
word friction commonly refers to resistive forces other than air drag that are caused by contact between
surfaces so you can ignore air drag in problems unless you are told to consider its effects.
The following examples should help you learn to draw free-body diagrams. We will start with relatively simple
situations in which the object of interest is either explicitly suggested or fairly obvious.
Part A
A hockey puck slides along a horizontal, smooth icy surface at a constant velocity as shown.
diagram for the puck. Which of the following forces are acting on the puck?
A.
B.
C.
D.
E.
F.
G.
Draw a free-body
weight
friction
force of velocity
force of push
normal force
air drag
acceleration
Type the letters corresponding to all the correct answers in alphabetical order. Do not use commas. For
instance, if you think that only answers C and D are correct, type CD.
AE
Correct
There is no such thing as "the force of velocity." If the puck is not being pushed, there are no horizontal forces
acting on it. Of course, some horizontal force must have acted on it before, to impart the velocity--however, in the
situation described, no such "force of push" exists. Also, the air drag in such cases is assumed to be negligible.
Finally, the word "smooth" usually implies negligible surface friction. Your free-body diagram should look like the
one shown here.
Part B
Consider a block pulled by a horizontal rope along a horizontal surface at a constant velocity as shown.
The
tension in the rope is nonzero. Draw a free-body diagram for the block. Which of the following forces are acting on
the block?
A.
B.
C.
D.
E.
F.
G.
weight
friction
force of velocity
force of tension
normal force
air drag
acceleration
Type the letters corresponding to all the correct answers in alphabetical order. Do not use commas. For
instance, if you think that only answers C and D are correct, type CD.
ABDE
Correct
Because the velocity is constant, there must be a force of friction opposing the force of tension. Since the block is
moving, it is kinetic friction. Your free-body diagram should look like that shown here.
Consider the following situation in parts C - F.
A block is resting on a slope as shown.
Part C
Which of the following forces are acting on the block?
A.
B.
C.
D.
E.
weight
kinetic friction
static friction
force of push
normal force
Type the letters corresponding to all the correct answers in alphabetical order. Do not use commas. For
instance, if you think that only answers C and D are correct, type CD.
ACE
Correct
Part D
What is the direction of the force due to gravity acting on the block?
vertically upward
vertically downward
perpendicular to the slope
upward along the slope
downward along the slope
Correct
Part E
What is the direction of the normal force acting on the block?
vertically upward
vertically downward
perpendicular to the slope
upward along the slope
downward along the slope
Correct
Part F
Draw the free-body diagram for the block. What is the direction of the force of friction acting on the block?
vertically upward
vertically downward
perpendicular to the slope
upward along the slope
downward along the slope
Correct
Without friction, the block would slide down the slope; so the force of static friction must oppose such a motion and
be directed upward along the slope. Your free-body diagram should look like that shown here.
Now consider a block sliding up a rough slope after having been given a quick push as shown.
Part G
Which of the following forces are acting on the block?
A. weight
B. kinetic friction
C. static friction
D. force of push
E. normal force
F. the force of velocity
Type the letters corresponding to all the correct answers in alphabetical order. Do not use commas. For
instance, if you think that only answers C and D are correct, type CD.
ABE
Correct
The word "rough" implies the presence of friction. Since the block is in motion, it is kinetic friction. Once again,
there is no such thing as "the force of velocity." However, it seems a tempting choice to some students since the
block is going up.
Part H
Draw the free-body diagram for the block. What is the direction of the force of friction acting on the block?
vertically upward
vertically downward
perpendicular to the slope
upward along the slope
downward along the slope
Correct
The force of kinetic friction opposes the upward motion of the block. Your free-body diagram should look like the
one shown here.
Part I
Now consider a block being pushed up a smooth slope. The force pushing the block is parallel to the slope.
Which of the following forces are acting on the block?
A. weight
B. kinetic friction
C. static friction
D. force of push
E. normal force
Type the letters corresponding to all the correct answers in alphabetical order. Do not use commas. For
instance, if you think that only answers C and D are correct, type CD.
ADE
Correct
Your free-body diagram should look like the one shown here.
The force of push is the normal force exerted, possibly,
by the palm of the hand of the person pushing the block.
In all the previous situations just described, the object of interest was explicitly given. Let us consider a situation
where choosing the objects for which to draw the free-body diagrams is up to you.
Two blocks of masses
and
are connected by a light string that goes over a light frictionless pulley. The block
of mass
is sliding to the right on a rough horizontal surface of a lab table.
Part J
To solve for the acceleration of the blocks, you will have to draw the free-body diagrams for which objects?
A.
B.
C.
D.
E.
F.
the block of mass
the block of mass
the connecting string
the pulley
the table
the earth
Type the letters corresponding to all the correct answers in alphabetical order. Do not use commas. For
instance, if you think that only answers C and D are correct, type CD.
AB
Correct
Part K
Draw the free-body diagram for the block of mass
none
. How many forces are exerted on this block?
one
two
three
four
Correct
Your free-body diagram should look like that shown here.
Part L
Draw the free-body diagram for the block of mass
none
one
two
three
four
Correct
. How many forces are exerted on this block?
Your free-body diagram should look like that shown here.
Understanding Newton's Laws
Harvaran Ghai
Part A
An object cannot remain at rest unless which of the following holds?
The net force acting on it is zero.
The net force acting on it is constant and nonzero.
There are no forces at all acting on it.
There is only one force acting on it.
Correct
If there is a net force acting on a body, regardless of whether it is a constant force, the body accelerates. If the body
is at rest and the net force acting on it is zero, then it will remain at rest. The net force could be zero either because
there are no forces acting on the body at all or because several forces are acting on the body but they all cancel out.
Part B
If a block is moving to the left at a constant velocity, what can one conclude?
There is exactly one force applied to the block.
The net force applied to the block is directed to the left.
The net force applied to the block is zero.
There must be no forces at all applied to the block.
Correct
If there is a net force acting on a body, regardless of whether the body is already moving, the body accelerates. If a
body is moving with constant velocity, then it is not accelerating and the net force acting on it is zero. The net force
could be zero either because there are no forces acting on the body at all or because several forces are acting on the
body but they all cancel out.
Part C
A block of mass
is acted upon by two forces:
you say about the block's motion?
It must be moving to the left.
(directed to the left) and
(directed to the right). What can
It must be moving to the right.
It must be at rest.
It could be moving to the left, moving to the right, or be instantaneously at rest.
Correct
The acceleration of an object tells you nothing about its velocity--the direction and speed at which it is moving. In
this case, the net force on (and therefore the acceleration of) the block is to the right, but the block could be moving
left, right, or in any other direction.
Part D
A massive block is being pulled along a horizontal frictionless surface by a constant horizontal force. The block
must be __________.
continuously changing direction
moving at constant velocity
moving with a constant nonzero acceleration
moving with continuously increasing acceleration
Correct
Since there is a net force acting, the body does not move at a constant velocity, but it accelerates instead. However,
the force acting on the body is constant. Hence, according to Newton's 2nd law of motion, the acceleration of the
body is also constant.
Part E
Two forces, of magnitude
and
, are applied to an object. The relative direction of the forces is unknown.
The net force acting on the object __________.
A. cannot be equal to
B. cannot be equal to
C. cannot be directed the same way as the force of
D. must be greater than
Enter the letters of all the correct answers in alphabetical order. Do not use commas. For example, if you
think only the last option is correct, enter D.
A
Correct
Conceptual Questions on Newton's 1st and 2nd Laws
Learning Goal: To understand the meaning and the basic applications of Newton's 1st and 2nd laws.
In this problem, you are given a diagram
representing the motion of an object--a motion diagram. The dots
represent the object's position at moments separated by equal intervals of time. The dots are connected by arrows
representing the object's average velocity during the corresponding time interval.
Your goal is to use this motion diagram to determine the direction of the net force acting on the object. You will
then determine which force diagrams and which situations may correspond to such a motion.
Harvaran Ghai
Part A
What is the direction of the net force acting on the object at position A?
upward
downward
to the left
to the right
The net force is zero.
Correct
The velocity vectors connecting position A to the adjacent positions appear to have the same magnitude and
direction. Therefore, the acceleration is zero--and so is the net force.
Part B
What is the direction of the net force acting on the object at position B?
upward
downward
to the left
to the right
The net force is zero.
Correct
The velocity is directed to the right; however, it is decreasing. Therefore, the acceleration is directed to the left--and
so is the net force.
Part C
What is the direction of the net force acting on the object at position C?
upward
downward
to the left
to the right
The net force is zero.
Correct
The horizontal component of the velocity does not change. The vertical component of the velocity increases.
Therefore, the acceleration--and the net force--are directed straight downward.
The next four questions are related to the force diagrams numbered 1 to 6.
These diagrams represent the forces
acting on a moving object. The number next to each arrow represents the magnitude of the force in newtons.
Part D
Which of these diagrams may possibly correspond to the situation at point A on the motion diagram?
Type, in increasing order, the numbers corresponding to the correct diagrams. Do not use commas. For
instance, if you think that only diagrams 3 and 4 are correct, type 34.
6
Correct
Part E
Which of these diagrams may possibly correspond to the situation at point B on the motion diagram?
Type, in increasing order, the numbers corresponding to the correct diagrams. Do not use commas. For
instance, if you think that only diagrams 3 and 4 are correct, type 34.
35
Correct
Part F
Which of these diagrams may possibly correspond to the situation at point C on the motion diagram?
Type, in increasing order, the numbers corresponding to the correct diagrams. Do not use commas. For
instance, if you think that only diagrams 3 and 4 are correct, type 34.
24
Correct
Part G
Which of these diagrams correspond to a situation where the moving object (not necessarily the one shown in the
motion diagram) is changing its velocity?
Type, in increasing order, the numbers corresponding to the correct diagrams. Do not use commas. For
instance, if you think that only diagrams 3 and 4 are correct, type 34.
12345
Correct
Consider the following situations:
A.
B.
C.
D.
E.
F.
G.
H.
I.
J.
A car is moving along a straight road at a constant speed.
A car is moving along a straight road while slowing down.
A car is moving along a straight road while speeding up.
A hockey puck slides along a smooth (i.e., frictionless) icy surface.
A hockey puck slides along a rough concrete surface.
A cockroach is speeding up from rest.
A rock is thrown horizontally; air resistance is negligible.
A rock is thrown horizontally; air resistance is substantial.
A rock is dropped vertically; air resistance is negligible.
A rock is dropped vertically; air resistance is substantial.
Part H
Which of these situations describe the motion shown in the motion diagram at point A?
Type the letters corresponding to all the right answers in alphabetical order. Do not use commas. For
instance, if you think that only situations C and D are correct, type CD.
AD
Correct
Part I
Which of these situations describe the motion shown in the motion diagram at point B?
Type the letters corresponding to all the right answersin alphabetical order. Do not use commas. For
instance, if you think that only situations C and D are correct, type CD.
BE
Correct
Part J
Which of these situations describe the motion shown in the motion diagram at point C?
Type the letters corresponding to all the right answers in alphabetical order. Do not use commas. For
instance, if you think that only situations C and D are correct, type CD.
G
Correct
A World-Class Sprinter
World-class sprinters can accelerate out of the starting blocks with an acceleration that is nearly horizontal and has
magnitude
.
Harvaran Ghai
Part A
How much horizontal force must a sprinter of mass 54.0
acceleration?
Express your answer in newtons.
exert on the starting blocks to produce this
=810
Correct
Part B
Which body exerts the force that propels the sprinter, the blocks or the sprinter?
the blocks
the sprinter
Correct
To start moving forward, sprinters push backward on the starting blocks with their feet. As a reaction, the blocks
push forward on their feet with a force of the same magnitude. This external force accelerates the sprinter forward.
Block on an Incline
A block lies on a plane raised an angle from the horizontal. Three forces act upon the block:
gravity;
, the normal force; and
block from sliding
.
, the force of
, the force of friction. The coefficient of friction is large enough to prevent the
Harvaran Ghai
Part A
Consider coordinate system a, with the x axis along the plane. Which forces lie along the axes?
only
only
only
and
and
and
and
Correct
and
Part B
Which forces lie along the axes of the coordinate system b, in which the y axis is vertical?
only
only
only
and
and
and
and
Correct
and
Now you are going to ignore the general rule (actually, a strong suggestion) that you should pick the coordinate
system with the most vectors, especially unknown ones, along the coordinate axes. You will find the normal force,
, using vertical coordinate system b. In these coordinates you will find the magnitude
and y equations, each multiplied by a trigonometric function.
appearing in both the x
Part C
Because the block is not moving, the sum of the y components of the forces acting on the block must be zero. Find
an expression for the sum of the y components of the forces acting on the block, using coordinate system b.
Express your answer in terms of some or all of the variables
,
,
, and .
Correct
Part D
Because the block is not moving, the sum of the x components of the forces acting on the block must be zero. Find
an expression for the sum of the x components of the forces acting on the block, using coordinate system b.
Express your answer in terms of some or all of the variables
,
,
, and .
Correct
Part E
To find the magnitude of the normal force, you must express
in terms of
equations you found in the two previous parts, find an expression for
since
involving
is an unknown. Using the
and but not
.
=
Correct
Congratulations on working this through. Now realize that in coordinate system a, which is aligned with the plane,
the y-coordinate equation is
for
, which leads immediately to the result obtained here
.
CONCLUSION: A thoughtful examination of which coordinate system to choose can save a lot of algebra.
Hanging Chandelier
A chandelier with mass is attached to the ceiling of a large concert hall by two cables. Because the ceiling is
covered with intricate architectural decorations (not indicated in the figure, which uses a humbler depiction), the
workers who hung the chandelier couldn't attach the cables to the ceiling directly above the chandelier. Instead, they
attached the cables to the ceiling near the walls. Cable 1 has tension
Cable 2 has tension
and makes an angle of
and makes an angle of
with the ceiling.
with the ceiling.
Harvaran Ghai
Part A
Find an expression for
, the tension in cable 1, that does not depend on
Express your answer in terms of some or all of the variables
acceleration due to gravity .
,
, and
.
, as well as the magnitude of the
=
Correct
Problem 5.11
Harvaran Ghai
Part A
An astronaut's weight on earth is 805
309 N
Correct
. What is his weight on Mars, where
Problem 5.12
A woman has a mass of
.
Harvaran Ghai
Part A
What is her weight on earth?
539 N
Correct
Part B
What is her mass on the moon, where
55.0 kg
Correct
Part C
What is her weight on the moon?
89.1 N
Correct
Problem 5.14
The figure shows the velocity graph of a
passenger in an elevator.
Harvaran Ghai
Part A
What is the passenger's apparent weight at
1040 N
Correct
?
Part B
At t =
?
735 N
Correct
Part C
At t =
?
585 N
Correct
Pushing a Chair along the Floor
A chair of weight 120
of
= 35.0
lies atop a horizontal floor; the floor is not frictionless. You push on the chair with a force
directed at an angle of 41.0 below the horizontal and the chair slides along the floor.
Harvaran Ghai
Part A
Using Newton's laws, calculate , the magnitude of the normal force that the floor exerts on the chair.
Express your answer in newtons.
=143
Correct
Board Pulled Out from under a Box
A small box of mass
is sitting on a board of mass
and length
. The board rests on a frictionless
horizontal surface. The coefficient of static friction between the board and the box is . The coefficient of kinetic
friction between the board and the box is, as usual, less than .
Throughout the problem, use for the magnitude of the acceleration due to gravity. In the hints, use
magnitude of the friction force between the board and the box.
for the
Harvaran Ghai
Part A
Find
, the constant force with the least magnitude that must be applied to the board in order to pull the board
out from under the the box (which will then fall off of the opposite end of the board).
Express your answer in terms of some or all of the variables
answer.
,
,
, , and
. Do not include
in your
=
Correct
Friction Force on a Dancer on a Drawbridge
A dancer is standing on one leg on a drawbridge that is about to open. The coefficients of static and kinetic friction
between the drawbridge and the dancer's foot are
and
, respectively.
represents the normal force exerted on
the dancer by the bridge, and represents the gravitational force exerted on the dancer, as shown in the drawing
. For all the questions, we can assume that the bridge is a perfectly flat surface and lacks the curvature
characteristic of most bridges.
Harvaran Ghai
Part A
Before the drawbridge starts to open, it is perfectly level with the ground. The dancer is standing still on one leg.
What is the x component of the friction force,
?
Express your answer in terms of some or all of the variables
,
, and/or
.
=0
Correct
This shows a very important point. When you are not told that an object is slipping or on the verge of slipping, then
the friction force is determined using Newton's laws of motion in conjunction with the observed motion and the other
forces on the object. Under these circumstances the friction force is limited by or but is otherwise not
necessarily related to or .
Part B
The drawbridge then starts to rise and the dancer continues to stand on one leg. The drawbridge stops just at the
point where the dancer is on the verge of slipping. What is the magnitude
Express your answer in terms of some or all of the variables
in your answer.
,
of the frictional force now?
, and/or
. The angle should not appear
=
Correct
Part C
Then, because the bridge is old and poorly designed, it falls a little bit and then jerks. This causes the person to start
to slide down the bridge at a constant speed. What is the magnitude
of the frictional force now?
Express your answer in terms of some or all of the variables
in your answer.
,
, and/or
. The angle should not appear
=
Correct
Part D
The bridge starts to come back down again. The dancer stops sliding. However, again because of the age and design
of the bridge it never makes it all the way down; rather it stops half a meter short. This half a meter corresponds to
an angle
degree (see the diagram, which has the angle exaggerated). What is the force of friction
Express your answer in terms of some or all of the variables ,
,
,
, and/or
now?
.
=
Correct
Skydiving
A sky diver of mass 80.0
(including parachute) jumps off a plane and begins her descent.
Throughout this problem use 9.80
for the magnitude of the acceleration due to gravity.
Harvaran Ghai
Part A
At the beginning of her fall, does the sky diver have an acceleration?
No; the sky diver falls at constant speed.
Yes and her acceleration is directed upward.
Yes and her acceleration is directed downward.
Correct
This Error! Hyperlink reference not valid. shows the sky diver (not to scale) with her position, speed, and
acceleration graphed as functions of time. You can see how her acceleration drops to zero over time, giving constant
speed after a long time.
Part B
At some point during her free fall, the sky diver reaches her terminal speed. What is the magnitude of the drag force
due to air resistance that acts on the sky diver when she has reached terminal speed?
Express your answer in newtons.
=784
Correct
Part C
For an object falling through air at a high speed , the drag force acting on it due to air resistance can be expressed
as
,
where the coefficient
depends on the shape and size of the falling object and on the density of air. For a human
body, the numerical value for
Using this value for
is about 0.250
, what is the terminal speed
.
of the sky diver?
Express you answer in meters per second.
=56.0
Correct
Recreational sky divers can control their terminal speed to some extent by changing their body posture. When
oriented in a headfirst dive, a sky diver can reach speeds of about 54 meters per second (120 miles per hour). For
maximum drag and stability, sky divers often will orient themselves "belly-first." In this position, their terminal
speed is typically around 45 meters per second (100 miles per hour).
Part D
When the sky diver descends to a certain height from the ground, she deploys her parachute to ensure a safe landing.
(Usually the parachute is deployed when the sky diver reaches an altitude of about 900
after deploying the parachute, does the skydiver have a nonzero acceleration?
No; the sky diver keeps falling at constant speed.
--3000
.) Immediately
Yes and her acceleration is directed downward.
Yes and her acceleration is directed upward.
Correct
Part E
When the parachute is fully open, the effective drag coefficient of the sky diver plus parachute increases to
60.0
. What is the drag force
Express your answer in newtons.
=1.88×105
Correct
Part F
acting on the sky diver immediately after she has opened the parachute?
What is the terminal speed
of the sky diver when the parachute is opened?
Express your answer in meters per second.
=3.61
Correct
A typical "student" parachute for recreational skydiving has a drag coefficient that gives a terminal speed for landing
of about 2 meters per second (5 miles per hour). If this seems slow based on video or real-life sky divers you have
seen, that may be because the sky divers you saw were using high-performance parachutes; these offer the sky
divers more maneuverability in the air but increase the terminal speed up to 4 meters per second (10 miles per hour).
An Object Accelerating on a Ramp
Learning Goal: Understand that the acceleration vector is in the direction of the change of the velocity vector.
In one dimensional (straight line) motion, acceleration is accompanied by a change in speed, and the acceleration is
always parallel (or antiparallel) to the velocity.
When motion can occur in two dimensions (e.g. is confined to a tabletop but can lie anywhere in the x-y plane), the
definition of acceleration is
in the limit
.
In picturing this vector derivative you can think of the derivative of a vector as an instantaneous quantity by thinking
of the velocity of the tip of the arrow as the vector changes in time. Alternatively, you can (for small
approximate the acceleration as
)
.
Obviously the difference between
in the same direction,
and
will be parallel to
but is in a slightly different direction, then
in both magnitude and direction, hence
is another vector that can lie in any direction. If it is longer but
. On the other hand, if
will be perpendicular to
has the same magnitude as
. In general,
can have any direction relative to
can differ from
.
This problem contains several examples of this.Consider an object sliding on a frictionless ramp as depicted here.
The object is already moving along the ramp toward position 2 when it is at position 1. The following questions
concern the direction of the object's acceleration vector, . In this problem, you should find the direction of the
acceleration vector by drawing the velocity vector at two points near to the position you are asked about. Note that
since the object moves along the track, its velocity vector at a point will be tangent to the track at that point. The
acceleration vector will point in the same direction as the vector difference of the two velocities. (This is a result of
the equation
given above.)
Harvaran Ghai
Part A
Which direction best approximates the direction of
straight up
when the object is at position 1?
downward to the left
downward to the right
straight down
Correct
Part B
Which direction best approximates the direction of
straight up
when the object is at position 2?
upward to the right
straight down
downward to the left
Correct
Even though the acceleration is directed straight up, this does not mean that the object is moving straight up.
Part C
Which direction best approximates the direction of
when the object is at position 3?
upward to the right
to the right
straight down
downward to the right
Correct
Problem 6.3
A particle's trajectory is described by
and
, where is in s.
Harvaran Ghai
Part A
What is the particle's speed at t
2.00 m/s
Correct
Part B
What is the particle's speed at
12.6 m/s
Correct
= 4.50 ?
Part C
What is the particle's direction of motion, measured from the x-axis, at
0?
270 counterclockwise from the +x axis
Correct
Part D
What is the particle's direction of motion, measured from the x-axis, at
11.4 counterclockwise from the +x axis
Correct
Projectile Motion Tutorial
= 4.50 ?
Learning Goal: Understand how to apply the equations for 1-dimensional motion to the y and x directions
separately in order to derive standard formulae for the range and height of a projectile.
A projectile is fired from ground level at time
, at an angle with respect to the horizontal. It has an initial
speed . In this problem we are assuming that the ground is level.
Harvaran Ghai
Part A
Find the time
Express
it takes the projectile to reach its maximum height.
in terms of
, , and (the magnitude of the acceleration due to gravity).
=
Correct
Part B
Find
, the time at which the projectile hits the ground.
Express the time in terms of
, , and .
=
Correct
Part C
Find
, the maximum height attained by the projectile.
Express the maximum height in terms of
=
, , and .
Correct
Part D
Find the total distance
lands.
(often called the range) traveled in the x direction; in other words, find where the projectile
Express the range in terms of
, , and .
=
Correct
The actual formula for
1.
2.
3.
is less important than how it is obtained:
Consider the x and y motion separately.
Find the time of flight from the y-motion
Find the x-position at the end of the flight - this is the range.
If you remember these steps, you can deal with many variants of the basic problem, such as: a cannon on a hill that
fires horizontally (i.e. the second half of the trajectory), a projectile that lands on a hill, or a projectile that must hit a
moving target.
Horizontal Cannon on a Cliff
A cannonball is fired horizontally from the top of a cliff. The cannon is at height
ball is fired with initial horizontal speed .
above ground level, and the
Harvaran Ghai
Part A
Assume that the cannon is fired at
position of the cannonball at the time
and that the cannonball hits the ground at time
. What is the y
?
Express the y position of the cannonball in terms of
answer.
. The quantities
and should not appear in your
=
Correct
Part B
Given that the projectile lands a distance
Express the initial speed in terms of
from the cliff, as shown, find the initial speed of the projectile,
, , and
.
.
=
Correct
Part C
What is the y position of the cannonball when it is a distance
Express the position of the cannonball in terms of
from the hill?
only.
=
Correct
Problem 6.13
A boat takes 3.70
to travel 15.0
down a river, then 5.30
to return.
Harvaran Ghai
Part A
How fast is the river flowing?
0.612 km/h
Correct
Crossing a River
A swimmer wants to cross a river, from point A to point B. The distance
distance
(from C to B) is 150
swimmer makes an angle of
the figure.
, and the speed
(0.785
(from A to C) is 200
of the current in the river is 5
, the
. Suppose that the
) with respect to the line from A to C, as indicated in
Harvaran Ghai
Part A
To swim directly from A to B, what speed , relative to the water, should the swimmer have?
Express the swimmer's speed numerically, to three significant figures, in units of kilometers per hour.
=4.04
Correct
Another way to do this problem, without using any kinematics, would be to add the swimmer's and river's velocities
vectorially, and set the angle that this vector makes with AC or the river bank equal to that which AB makes with
the same.
Speed of a Bullet
A bullet is shot through two cardboard disks attached a distance
, as shown.
apart to a shaft turning with a rotational period
Harvaran Ghai
Part A
Derive a formula for the bullet speed in terms of , , and a measured angle between the position of the hole in
the first disk and that of the hole in the second. If required, use , not its numeric equivalent. Both of the holes lie at
the same radial distance from the shaft. measures the angular displacement between the two holes; for instance,
means that the holes are in a line and
means that when one hole is up, the other is down. Assume that
the bullet must travel through the set of disks within a single revolution.
=
Correct
Direction of Acceleration of Pendulum
Learning Goal: To understand that the direction of acceleration is in the direction of the change of the velocity,
which is unrelated to the direction of the velocity.
The pendulum shown makes a full swing from
to
. Ignore friction and assume that the string is
massless. The eight labeled arrows represent directions to be referred to when answering the following questions.
Harvaran Ghai
Part A
Which of the following is a true statement about the acceleration of the pendulum bob,
.
is equal to the acceleration due to gravity.
is equal to the instantaneous rate of change in velocity.
is perpendicular to the bob's trajectory.
is tangent to the bob's trajectory.
Correct
Part B
What is the direction of
when the pendulum is at position 1?
Enter the letter of the arrow parallel to
H
Correct
.
Part C
What is the direction of
at the moment the pendulum passes position 2?
Enter the letter of the arrow that best approximates the direction of
C
Correct
.
We know that for the object to be traveling in a circle, some component of its acceleration must be pointing radially
inward.
Part D
What is the direction of
when the pendulum reaches position 3?
Give the letter of the arrow that best approximates the direction of
F
Correct
.
Part E
As the pendulum approaches or recedes from which position(s) is the acceleration vector
almost parallel to the
velocity vector .
position 2 only
positions 1 and 2
positions 2 and 3
positions 1 and 3
Correct
Banked Frictionless Curve, and Flat Curve with Friction
A car of mass
traveling at speed enters a banked turn covered with ice. The road is banked at an angle , and
there is no friction between the road and the car's tires. A cross section of the curve is shown in the diagram.
Harvaran Ghai
Part A
What is the radius of the turn (assuming the car continues in uniform circular motion around the turn)?
Express your answer in terms of some or all of the variables
acceleration due to gravity .
, , , as well as the magnitude of the
=
Correct
Part B
Now, suppose that the curve is level (
) and that the ice has melted, so that there is a coefficient of static
friction between the road and the car's tires. What is
, the minimum value of the coefficient of static friction
between the tires and the road required to prevent the car from slipping? Assume that the car's speed is still and
that the radius of the curve is given by .
Express your answer in terms of some or all of the variables , ,
acceleration due to gravity .
, as well as the magnitude of the
=
Correct
At the Test Track
You want to test the grip of the tires on your new race car. You decide to take the race car to a small test track to
experimentally determine the coefficient of friction. The racetrack consists of a flat, circular road with a radius of 45
. Error! Hyperlink reference not valid. shows the result of driving the car around the track at various speeds.
Harvaran Ghai
Part A
What is , the coefficient of static friction between the tires and the track?
Express your answer to two significant figures.
=0.95
Correct
Conical Pendulum I
A bob of mass
is suspended from a fixed point with a massless string of length
are to investigate the motion in which the string moves in a cone with half-angle .
(i.e., it is a pendulum). You
Harvaran Ghai
Part A
What tangential speed, , must the bob have so that it moves in a horizontal circle with the string always making an
angle from the vertical?
Express your answer in terms of some or all of the variables
gravity .
,
, and , as well as the acceleration due to
=
Correct
Part B
How long does it take the bob to make one full revolution (one complete trip around the circle)?
Express your answer in terms of some or all of the variables
gravity .
,
, and , as well as the acceleration due to
Correct
Problem 7.18
Harvaran Ghai
Part A
What is the acceleration due to gravity of the sun at the distance of the earth's orbit?
6.00×10−3
Correct
Problem 7.19
The passengers in a roller coaster car feel 50% heavier than their true weight as the car goes through a dip with a
40.0
radius of curvature.
Harvaran Ghai
Part A
What is the car's speed at the bottom of the dip?
14.0
Correct
Problem 7.23
A car speeds up as it turns from traveling due south to heading due east. When exactly halfway around the curve, the
car's acceleration is 3.40
, 40.0 north of east.
Harvaran Ghai
Part A
What is the radial component of the acceleration at that point?
3.39
Correct
Part B
What is the tangential component of the acceleration at that point?
0.296
Correct
A Satellite in Orbit
A satellite used in a cellular telephone network has a mass of 1850
700
and is in a circular orbit at a height of
above the surface of the earth.
Harvaran Ghai
Part A
What is the gravitational force
on the satellite?
Take the gravitational constant to be = 6.67×10−11
and the radius of the Earth to be = 6.38×106 .
, the mass of the earth to be
= 5.97×1024
,
Express your answer in newtons.
=1.47×104
Correct
Part B
What fraction is this of the satellite's weight at the surface of the earth?
Take the free-fall acceleration at the surface of the earth to be
= 9.80
.
0.811
Correct
Although it is easy to find the weight of the satellite using the constant acceleration due to gravity, it is instructional
to consider the weight calculated using the law of gravitation:
. Dividing the gravitational force on
the satellite
by , we find that the ratio of the forces due to the earth's gravity is simply
the square of the ratio of the earth's radius to the sum of the earth's radius and the height of the orbit of the satellite
above the earth,
. This will also be the fraction of the weight of, say, an astronaut in an orbit at the
same altitude. Notice that an astronaut's weight is never zero. When people speak of "weightlessness" in space, what
they really mean is "free fall."
Gravitational Acceleration inside a Planet
Consider a spherical planet of uniform density . The distance from the planet's center to its surface (i.e., the
planet's radius) is
. An object is located a distance
located inside of the planet.)
from the center of the planet, where
. (The object is
Harvaran Ghai
Part A
Find an expression for the magnitude of the acceleration due to gravity,
Express the acceleration due to gravity in terms of ,
=
Correct
,
, and
, inside the planet.
, the universal gravitational constant.
Part B
Rewrite your result for
function of R.
in terms of
Express your answer in terms of
,
, the gravitational acceleration at the surface of the planet, times a
, and
.
=
Correct
Notice that increases linearly with , rather than being proportional to
center of the planet, as required by symmetry.
. This assures that it is zero at the
Part C
Find a numerical value for
, the average density of the earth in kilograms per cubic meter. Use
the radius of the earth,
Calculate your answer to four significant digits.
, and a value of at the surface of
for
.
=5497
Correct
Newton's 3rd Law Discussed
Learning Goal: To understand Newton's 3rd law, which states that a physical interaction always generates a pair of
forces on the two interacting bodies.
In Principia, Newton wrote:
To every action there is always opposed an equal reaction: or, the mutual actions of two bodies upon each other are
always equal, and directed to contrary parts.
(translation by Cajori)
The phrase after the colon (often omitted from textbooks) makes it clear that this is a statement about the nature of
force. The central idea is that physical interactions (e.g., due to gravity, bodies touching, or electric forces) cause
forces to arise between pairs of bodies. Each pairwise interaction produces a pair of opposite forces, one acting on
each body. In summary, each physical interaction between two bodies generates a pair of forces. Whatever the
physical cause of the interaction, the force on body A from body B is equal in magnitude and opposite in direction to
the force on body B from body A.
Incidentally, Newton states that the word "action" denotes both (a) the force due to an interaction and (b) the
changes in momentum that it imparts to the two interacting bodies. If you haven't learned about momentum, don't
worry; for now this is just a statement about the origin of forces.
Mark each of the following statements as true or false. If a statement refers to "two bodies" interacting via some
force, you are not to assume that these two bodies have the same mass.
Harvaran Ghai
Part A
Every force has one and only one 3rd law pair force.
true
false
Correct
Part B
The two forces in each pair act in opposite directions.
true
false
Correct
Part C
The two forces in each pair can either both act on the same body or they can act on different bodies.
true
false
Correct
Part D
The two forces in each pair may have different physical origins (for instance, one of the forces could be due to
gravity, and its pair force could be due to friction or electric charge).
true
false
Correct
Part E
The two forces of a 3rd law pair always act on different bodies.
true
false
Correct
Part F
Given that two bodies interact via some force, the accelerations of these two bodies have the same magnitude but
opposite directions. (Assume no other forces act on either body.)
true
false
Correct
Newton's 3rd law can be summarixed as follows: A physical interaction (e.g., gravity) operates between two
interacting bodies and generates a pair of opposite forces, one on each body. It offers you a way to test for real
forces (i.e., those that belong on the force side of
)--there should be a 3rd law pair force operating on
some other body for each real force that acts on the body whose acceleration is under consideration.
Part G
According to Newton's 3rd law, the force on the (smaller) moon due to the (larger) earth is
greater in magnitude and antiparallel to the force on the earth due to the moon.
greater in magnitude and parallel to the force on the earth due to the moon.
equal in magnitude but antiparallel to the force on the earth due to the moon.
equal in magnitude and parallel to the force on the earth due to the moon.
smaller in magnitude and antiparallel to the force on the earth due to the moon.
smaller in magnitude and parallel to the force on the earth due to the moon.
Correct
A Book on a Table
A book weighing 5 N rests on top of a table.
Harvaran Ghai
Part A
A downward force of magnitude 5 N is exerted on the book by the force of
the table.
gravity
inertia
Correct
Part B
An upward force of magnitude _____ is exerted on the _____ by the table.
5 N / book
Correct
Part C
Do the downward force in Part A and the upward force in Part B constitute a 3rd law pair?
yes
no
Correct
Part D
The reaction to the force in Part A is a force of magnitude _____, exerted on the _____ by the _____. Its direction is
_____ .
5 N / earth / book / upward
Correct
Part E
The reaction to the force in Part B is a force of magnitude _____, exerted on the _____ by the _____. Its direction is
_____.
5 N / table / book / downward
Correct
Part F
Which of Newton's laws dictates that the forces in Parts A and B are equal and opposite?
Newton's 1st or 2nd law
Newton's 3rd law
Correct
Since the book is at rest, the net force on it must be zero (1st or 2nd law). This means that the force exerted on it by
the earth must be equal and opposite to the force exerted on it by the table.
Part G
Which of Newton's laws dictates that the forces in Parts B and E are equal and opposite?
Newton's 1st or 2nd law
Newton's 3rd law
Correct
Block on an Incline Adjacent to a Wall
A wedge with an inclination of angle rests next to a wall. A block of mass is sliding down the plane, as shown.
There is no friction between the wedge and the block or between the wedge and the horizontal surface.
Harvaran Ghai
Part A
Find the magnitude,
Express
, of the sum of all forces acting on the block.
in terms of and
, along with any necessary constants.
=
Correct
Part B
Find the magnitude,
Express
=
Correct
, of the force that the wall exerts on the wedge.
in terms of and
, along with any necessary constants.
Your answer to Part B could be expressed as either
as gets very small or as approaches 90 degrees (
or
. In either form, we see that
radians), the contact force between the wall and the wedge
goes to zero. This is what we should expect; in the first limit ( small), the block is accelerating very slowly, and all
horizontal forces are small. In the second limit ( about 90 degrees), the block simply falls vertically and exerts no
horizontal force on the wedge.
PSS 8.1: Dashing up the Slope
Learning Goal: To practice Problem-Solving Strategy 8.1 for problems involving the dynamics of an interacting
systems of objects.
A girl of mass
is walking up a slippery slope while pulling a sled of unknown mass; the slope makes an angle
with the horizontal. The coefficient of static friction between the girl's boots and the slope is ; the friction
between the sled and the slope is negligible. It turns out that the girl can pull the sled up the slope with acceleration
up to without slipping down the slope. Find the mass of the sled
. Assume that the rope connecting the girl and
the sled is kept parallel to the slope at all times.
Harvaran Ghai
MODEL:
Identify which objects are systems and which are part of the environment. Make simplifying assumptions.
VISUALIZE:


Pictorial representation: Show important points in the motion with a sketch. You may want to give each
system a separate coordinate system. Define symbols and identify what you are trying to find. Include
acceleration constraints as part of the pictorial model.
Physical representation: Identify all forces acting on each system and all action-reaction pairs. Draw a
separate free-body diagram for each system. Connect the force vectors of action-reaction pairs with dotted
lines. Use subscript labels to distinguish forces, such as
system.
SOLVE:




and
, that act independently on more than one
Use Newton's 2nd and 3rd laws:
Write the equations of Newton's 2nd law for each system using the force information from the free-body
diagrams.
Equate the magnitudes of action-reaction pairs.
Include the acceleration constraints, the friction model, and other quantitative information relevant to the
problem.
Solve for the acceleration, then use kinematics to find velocites and positions.
ASSESS:
Check that your result has the correct units, is reasonable, and answers the question.
Model
Start by making simplifying assumptions appropriate for the situation.
Part A
Which of the following objects qualify as systems in this problem?
A.
B.
C.
D.
E.
the slope
the girl
the earth
the sled
the air
List alphabetically all the letters corresponding to the systems. Do not use commas. For instance, if you think
the slope and sled qualify as systems, type AD.
BD
Correct
The slope, the earth, and the air all qualify as part of the environment. They each exert external forces on the two
systems (the sled and the girl). Unlike for the two systems, however, it is not important to keep track of all the forces
acting on elements of the environment.
Part B
Which of the following simplifying assumptions are reasonable?
A.
B.
C.
D.
E.
F.
The air resistance acting on the girl is negligible.
The air resistance acting on the girl equals the force of friction acting on her.
The air resistance acting on the sled is negligible.
The normal force acting on the sled is negligible.
The weight of the sled is a constant.
The weight of the sled increases as the sled accelerates.
List alphabetically all the letters corresponding to reasonable assumptions. Do not use commas. For instance,
if you think A and B are reasonable, type AB.
ACE
Correct
Part C
Which of the following simplifying assumptions are reasonable?
A.
B.
C.
D.
E.
F.
The rope connecting the sled and the girl is massless.
The rope connecting the sled and the girl is unstretchable.
The tension in the rope connecting the sled and the girl is zero.
The sled has the same acceleration as the girl.
The sled has greater acceleration than the girl.
The sled has smaller acceleration than the girl.
List alphabetically all the letters corresponding to reasonable assumptions. Do not use commas. For instance,
if you think A and B are reasonable, type AB.
ABD
Correct
Visualize
Now draw a sketch that includes the free-body diagrams for each system and the appropriate coordinate system. Use
your sketch to answer the following questions.
For all questions, assume that the slope angles downhill to the left:
.
Part D
Which free-body diagram for the girl is correct? Note that the forces are not labeled; however, they should be
labeled on your diagram. You are looking for the correct number of forces in the correct directions. Don't worry
about relative magnitudes at this point.
a
b
c
d
e
Correct
Part E
Which free-body diagram for the sled is correct? Note that the forces are not labeled; however, they should be
labeled on your diagram. You are looking for the correct number of forces in the correct directions. Don't worry
about relative magnitudes at this point.
a
b
c
d
e
Correct
Part F
Which of these coordinate systems is most convenient for solving this problem? (The same coordinate system is
appropriate for both the sled and the girl.)
a
b
c
d
Correct
Part G
You should have identified the pairs of action-reaction forces on your free-body diagrams. Which of the following
pairs of forces form action-reaction pairs, according to Newton's 3rd law?
A.
B.
C.
D.
E.
F.
The weight of the girl and the normal force on the girl
The weight of the sled and the normal force on the sled
The weight of the girl and the weight of the sled
The force of friction on the girl and the tension of the string
The weight of the sled and the tension of the string
The weight of the sled and the gravitational force applied by the sled to the earth
List alphabetically all the letters corresponding to the action-reaction pairs in this problem. Do not use
commas. For instance, if you think A and B are both valid action-reaction pairs, type AB.
F
Correct
An action-reaction pair between objects A and B is always a pair of forces
and
. In our situation, if
we assume that the girl and the sled act directly on each other (a reasonable assumption since the mass of the string
is negligible), then the forces "girl on the sled" and "sled on the girl" would form an action-reaction pair.
Each of the other forces mentioned in this question does have a reaction force, of course. However, the objects on
which such reaction forces are acting are part of the environment: For instance, the reaction force to the weight of
the girl is the gravitational force applied by the girl to the earth, which is part of the environment.
There are many variations on how you might draw good pictorial and physical representations for this problem.
Here is one example.
Solve
Now use the information and the insights that you have accumulated to construct the necessary mathematical
expressions and to derive the solution.
Part H
Find the mass of the sled
.
Express the sled's mass in terms of the given quantities and , the magnitude of the acceleration due to
gravity.
=
Correct
Assess
When you work on a problem on your own, without the computer-provided feedback, only you can assess whether
your answer seems right. The following questions will help you practice the skills necessary for such an assessment.
Part I
Intuitively, what would happen if there were very little (or no) static friction between the girl and the slope?
Very little force would be required to pull a heavy sled up the slope.
The girl would slip down the slope and never be able to pull the sled up.
The girl would be able to pull the sled up the slope with a very large acceleration.
The girl would be able to pull the sled up only at constant velocity.
Correct
If is very close to or equal to zero, the formula you derived in the Solve section would give a negative value for
the mass of the sled. Since the mass of the sled must be positive, such an answer simply means that the formula is
not applicable: The girl would not be able to pull the sled up the slope, no matter how small the mass of the sled.
Part J
Intuitively, what would happen if the "slope" were horizontal and the mass of the sled were equal to the mass of the
girl?
The girl would be able to pull the sled with acceleration greater than .
The girl would slip along the surface and not be able to pull the sled.
The girl would be able to pull the sled with up to some maximum acceleration that depends on the friction
between her and the slope.
The girl would be able to pull the sled only at constant velocity.
Correct
If the slope is horizontal,
and
. Your formula then becomes
,
and, if
, it follows that the maximum acceleration is
.
Part K
Which of the following expressions have the dimensions of mass?
A.
B.
C.
D.
E.
List alphabetically all the letters corresponding to expressions with the correct dimensions. Do not use
commas. For instance, if you think A and B have the correct dimensions, type AB.
ACE
Correct
Note that trigonometric functions and the coefficient of static friction are dimensionless: They do not affect the
dimension or units of the final answer.
Pulling Three Blocks
Three identical blocks connected by ideal strings are being pulled along a horizontal frictionless surface by a
horizontal force
block has mass
.
.
The magnitude of the tension in the string between blocks B and C is
. Assume that each
Harvaran Ghai
Part A
What is the magnitude
of the force?
Express the magnitude of the force in terms of
.
=
Correct
Kinetic Friction in a Block-and-Pulley System
Consider the system shown in the figure
. Block A has weight
and block B has weight
. Once block B is
set into downward motion, it descends at a constant speed. Assume that the mass and friction of the pulley are
negligible.
Harvaran Ghai
Part A
Calculate the coefficient of kinetic friction
between block A and the table top.
=
Correct
Part B
A cat, also of weight
, falls asleep on top of block A. If block B is now set into downward motion, what is the
magnitude of its acceleration?
=
Correct
Two Masses, a Pulley, and an Inclined Plane
Block 1, of mass
, is connected over an ideal (massless and frictionless) pulley to block 2, of mass
, as
shown. Assume that the blocks accelerate as shown with an acceleration of magnitude and that the coefficient of
kinetic friction between block 2 and the plane is .
Harvaran Ghai
Part A
Find the ratio of the masses
.
Express your answer in terms of some or all of the variables ,
acceleration due to gravity .
, and , as well as the magnitude of the
=
Correct
The Impulse-Momentum Theorem
Learning Goal: To learn about the impulse-momentum theorem and its applications in some common cases.
Using the concept of momentum, Newton's second law can be rewritten as
, (1)
where
is the net force
acting on the object, and
is the rate at which the object's momentum is changing.
If the object is observed during an interval of time between times
(1) gives
and
, then integration of both sides of equation
. (2)
The right side of equation (2) is simply the change in the object's momentum
impulse of the net force and is denoted by
. The left side is called the
. Then equation (2) can be rewritten as
.
This equation is known as the impulse-momentum theorem. It states that the change in an object's momentum is
equal to the impulse of the net force acting on the object. In the case of a constant net force
direction of motion, the impulse-momentum theorem can be written as
. (3)
acting along the
Here , , and are the components of the corresponding vector quantities along the chosen coordinate axis. If
the motion in question is two-dimensional, it is often useful to apply equation (3) to the x and y components of
motion separately.
Harvaran Ghai
The following questions will help you learn to apply the impulse-momentum theorem to the cases of constant and
varying force acting along the direction of motion. First, let us consider a particle of mass moving along the x
axis. The net force
is acting on the particle along the x axis.
is a constant force.
Part A
The particle starts from rest at
that
. What is the magnitude of the momentum of the particle at time ? Assume
.
Express your answer in terms of any or all of
,
, and .
=
Correct
Part B
The particle starts from rest at
. What is the magnitude of the velocity of the particle at time ? Assume that
.
Express your answer in terms of any or all of
,
, and .
=
Correct
Part C
The particle has momentum of magnitude
at a certain instant. What is
seconds later?
Express your answer in terms of any or all of
=
,
,
, and
.
, the magnitude of its momentum
Correct
Part D
The particle has momentum of magnitude
at a certain instant. What is
, the magnitude of its velocity
seconds later?
Express your answer in terms of any or all of
,
,
, and
.
=
Correct
Let us now consider several two-dimensional situations.
A particle of mass
is moving in the positive x direction at speed . After a certain constant force is applied to the
particle, it moves in the positive y direction at speed
.
Part E
Find the magnitude of the impulse
Express your answer in terms of
delivered to the particle.
and . Use three significant figures in the numerical coefficient.
=
Correct
Part F
Which of the vectors below best represents the direction of the impulse vector
1
2
3
4
5
6
7
8
Correct
Part G
?
What is the angle between the positive y axis and the vector
as shown in the figure?
26.6 degrees
30 degrees
60 degrees
63.4 degrees
Correct
Part H
If the magnitude of the net force acting on the particle is
velocity,
, how long does it take the particle to acquire its final
in the positive y direction?
Express your answer in terms of
,
, and . If you use a numerical coefficient, use three significant figures.
=
Correct
So far, we have considered only the situation in which the magnitude of the net force acting on the particle was
either irrelevant to the solution or was considered constant. Let us now consider an example of a varying force
acting on a particle.
Part I
A particle of mass
kilograms is at rest at
seconds. A varying force
is acting on the particle between
seconds and
seconds. Find the
speed of the particle at
seconds.
Express your answer in meters per second to three significant figures.
=43
Correct
Problem 9.11
A 500 air-track glider collides with a spring at one end of the track. The figure
and the force exerted on the glider by the spring.
shows the glider's velocity
Harvaran Ghai
Part A
How long is the glider in contact with the spring?
0.167 s
Correct
Filling the Boat
A boat of mass 250
is coasting, with its engine in neutral, through the water at speed 3.00
rain. The rain is falling vertically, and it accumulates in the boat at the rate of 10.0
when it starts to
.
Harvaran Ghai
Part A
What is the speed of the boat after time 2.00
Express your answer in meters per second.
has passed? Assume that the water resistance is negligible.
2.78
Correct
Part B
Now assume that the boat is subject to a drag force
due to water resistance. Is the component of the total
momentum of the system parallel to the direction of motion still conserved?
yes
no
Correct
The boat is subject to an external force, the drag force due to water resistance, and therefore its momentum is not
conserved.
Part C
The drag is proportional to the square of the speed of the boat, in the form
. What is the acceleration of
the boat just after the rain starts? Take the positive axis along the direction of motion.
Express your answer in meters per second per second.
−1.80×10−2
Correct
PSS 9.1: Tools of the Trade
Learning Goal: To practice Problem-Solving Strategy 9.1 for problems involving conservation of momentum.
An astronaut performs maintenance work outside her spaceship when the tether connecting her to the spaceship
breaks. The astronaut finds herself at rest relative to the spaceship, at a distance from it. To get back to the ship,
she decides to sacrifice her favorite wrench and hurls it directly away from the spaceship at a speed relative to the
spaceship. What is the distance between the spaceship and the wrench by the time the astronaut reaches the
spaceship?
The mass of the astronaut is
; the mass of the wrench is
.
Harvaran Ghai
MODEL:
Clearly define the system.


If possible, choose a system that is isolated (
) or within which the interactions are sufficiently
short and intense that you can ignore external forces for the duration of the interaction (the impulse
approximation). Momentum is conserved.
If it is not possible to choose an isolated system, try to divide the problem into parts such that momentum is
conserved during one segment of the motion. Other segments of the motion can be analyzed using
Newton's laws or, as you'll learn in Chapters 10 and 11, conservation of energy.
Draw a before-and-after pictorial representation. Define symbols that will be used in the problem, list
known values, and identify what you are trying to find.
VISUALIZE:
The mathematical representation is based on the law of conservation of momentum:
form, this is
SOLVE:
,
ASSESS:
Check if your result has the correct units, is reasonable, and answers the question.
. In component
Model
We start by choosing the objects that would make up the system. In this case, it is possible to identify the system
that is isolated.
Part A
In addition to the astronaut, which of the following are components of the system that should be defined to solve the
problem?
A. the spaceship
B. the wrench
C. the earth
Enter the letter(s) of the correct answer(s) in alphabetical order. Do not use commas. For example, if you
think the system consists of all the objects listed, enter ABC.
B
Correct
Part B
Which of the following reasons best explains why the astronaut + wrench can be considered an isolated system?
The mass of the wrench is much smaller than that of the astronaut.
The force that the astronaut exerts on the wrench is very small.
The force that the astronaut exerts on the wrench is very large.
The force that the spaceship exerts on the wrench is very small.
The force that the spaceship exerts on the wrench is very large.
Correct
Visualize
Now draw a before-and-after pictorial representation including all the elements listed in the problem-solving
strategy. Be sure that your sketch is clear and includes all necessary symbols, both known and unknown. By the time
the astronaut reaches the spaceship, the wrench will have covered a certain distance; on your pictorial
representation, label this distance
.
Part C
After the wrench is thrown, the astronaut and the wrench move
in opposite directions.
in the same direction.
in perpendicular directions.
Correct
Part D
Which statement about
,
, and
is correct?
Correct
Here is an example of what a good before-and-after pictorial representation might look like for this problem.
Solve
Now use the information and the insights that you have accumulated to construct the necessary mathematical
expressions and to derive the solution.
Part E
Find the final distance between the spaceship and the wrench.
Express the distance in terms of the given variables. You may or may not use all of them.
=
Correct
Assess
When you work on a problem on your own, without the computer-provided feedback, only you can assess whether
your answer seems right. The following questions will help you practice the skills necessary for such an assessment.
Part F
Intuitively, which of the following statements are correct?
A. For realistic values of the quantities involved, it is possible that
.
B. If the astronaut threw a space pen instead of a wrench, the pen would travel further than the wrench would
in the time it takes the astronaut to reach the ship. (Assume the space pen weighs less than the wrench).
C. If the astronaut were more massive, the wrench would travel further in the time it takes the astronaut to
reach the ship.
Type the letters corresponding to the correct answers. Do not use commas. For instance, if you think that
only expressions C and D have the units of distance, type CD.
BC
Correct
could only be zero if
. As you can see from your answer, this would only happen if the mass of the
astronaut were zero, which is obviously unrealistic.
Part G
Which of the following mathematical expressions have the units of distance, where
and
are distances?
A.
B.
C.
D.
E.
F.
Type the letters corresponding to the correct answers. Do not use commas. For instance, if you think that
only expressions C and D have the units of distance, type CD.
ABE
Correct
Colliding Cars
In this problem we will consider the collision of two cars initially moving at right angles. We assume that after the
collision the cars stick together and travel off as a single unit. The collision is therefore completely inelastic.
Two cars of masses
and
collide at an intersection. Before the collision, car 1 was traveling eastward at a
speed of , and car 2 was traveling northward at a speed of .
After the collision, the two cars stick together
and travel off in the direction shown.
Harvaran Ghai
Part A
First, find the magnitude of
Express in terms of
,
, that is, the speed of the two-car unit after the collision.
, and the cars' initial speeds and .
=
Correct
Part B
Find the tangent of the angle .
Express your answer in terms of the momenta of the two cars,
and
.
=
Correct
Part C
Suppose that after the collision,
; in other words, is
The magnitudes of the momenta of the cars were equal.
The masses of the cars were equal.
The velocities of the cars were equal.
Correct
Collision at an Angle
. This means that before the collision:
Two cars, both of mass
, collide and stick together. Prior to the collision, one car had been traveling north at speed
, while the second was traveling at speed at an angle
collision, the two-car system travels at speed
south of east (as indicated in the figure). After the
at an angle east of north.
Harvaran Ghai
Part A
Find the speed
of the joined cars after the collision.
Express your answer in terms of and
.
=
Correct
Part B
What is the angle with respect to north made by the velocity vector of the two cars after the collision?
Express your answer in terms of
. Your answer should contain an inverse trigonometric function.
=
Correct
A Girl on a Trampoline
A girl of mass
second. At height
kilograms springs from a trampoline with an initial upward velocity of
meters above the trampoline, the girl grabs a box of mass
meters per
kilograms.
For this problem, use
meters per second per second for the magnitude of the acceleration due to gravity.
Harvaran Ghai
Part A
What is the speed
of the girl immediately before she grabs the box?
Express your answer numerically in meters per second.
=4.98
Correct
Part B
What is the speed
of the girl immediately after she grabs the box?
Express your answer numerically in meters per second.
=3.98
Correct
Part C
Is this "collision" elastic or inelastic?
elastic
inelastic
Correct
In inelastic collisions, some of the system's kinetic energy is lost. In this case the kinetic energy lost is converted to
heat energy in the girl's muscles as she grabs the box, and sound energy.
Part D
What is the maximum height
that the girl (with box) reaches? Measure
trampoline.
Express your answer numerically in meters.
with respect to the top of the
=2.81
Correct
Circling Ball
A ball of mass is attached to a string of length . It is being swung in a vertical circle with enough speed so that
the string remains taut throughout the ball's motion.
Assume that the ball travels freely in this vertical circle
with negligible loss of total mechanical energy. To avoid confusion, take the upward direction to be positive
throughout the problem. At the top and bottom of the vertical circle, label the ball's speeds and , and label the
corresponding tensions in the string
and
.
and
have magnitudes
and
.
Harvaran Ghai
Part A
Find
, the difference between the magnitude of the tension in the string at the bottom relative to that at the
top of the circle.
Express the difference in tension in terms of and . The quantities and should not appear in your final
answer.
=
Correct
The method outlined in the hints is really the only practical way to do this problem. If done properly, finding the
difference between the tensions,
, can be accomplished fairly simply and elegantly.
Bungee Jumping
Kate, a bungee jumper, wants to jump off the edge of a bridge that spans a river below. Kate has a mass
surface of the bridge is a height above the water. The bungee cord, which has length
straighten and then stretch as Kate falls.
, and the
when unstretched, will first
Assume the following:



The bungee cord behaves as an ideal spring once it begins to stretch, with spring constant .
Kate doesn't actually jump but simply steps off the edge of the bridge and falls straight downward.
Kate's height is negligible compared to the length of the bungee cord. Hence, she can be treated as a
point particle.
Use for the magnitude of the acceleration due to gravity.
Harvaran Ghai
Part A
How far below the bridge will Kate eventually be hanging, once she stops oscillating and comes finally to rest?
Assume that she doesn't touch the water.
Express the distance in terms of quantities given in the problem introduction.
=
Correct
Part B
If Kate just touches the surface of the river on her first downward trip (i.e., before the first bounce), what is the
spring constant ? Ignore all dissipative forces.
Express in terms of
,
,
, and .
=
Correct
Dancing Balls
Four balls, each of mass , are connected by four identical relaxed springs with spring constant . The balls are
simultaneously given equal initial speeds directed away from the center of symmetry of the system.
Harvaran Ghai
Part A
As the balls reach their maximum displacement, their kinetic energy reaches __________.
a maximum
zero
neither a maximum nor zero
Correct
Part B
Use geometry to find , the distance each of the springs has stretched from its equilibrium position. (It may help to
draw the initial and the final states of the system.)
Express your answer in terms of
, the maximum displacement of each ball from its initial position.
=
Correct
Part C
Find the maximum displacement
Express
of any one of the balls from its initial position.
in terms of some or all of the given quantities , , and
=
Correct
Elastic Collision in One Dimension
.
Block 1, of mass
, moves across a frictionless surface with speed
, which is at rest (
).
speed . Assume that
object before the collision.
. It collides elastically with block 2, of mass
After the collision, block 1 moves with speed , while block 2 moves with
, so that after the collision, the two objects move off in the direction of the first
Harvaran Ghai
Part A
This collision is elastic. What quantities, if any, are conserved in this collision?
kinetic energy only
momentum only
kinetic energy and momentum
Correct
Part B
What is the final speed
Express in terms of
of block 1?
,
, and
.
of block 2?
,
, and
.
=
Correct
Part C
What is the final speed
Express in terms of
=
Correct
Sinking the 9-Ball
Jeanette is playing in a 9-ball pool tournament. She will win if she sinks the 9-ball from the final rack, so she needs
to line up her shot precisely. Both the cue ball and the 9-ball have mass , and the cue ball is hit at an initial speed
of . Jeanette carefully hits the cue ball into the 9-ball off center, so that when the balls collide, they move away
from each other at the same angle from the direction in which the cue ball was originally traveling (see figure).
Furthermore, after the collision, the cue ball moves away at speed , while the 9-ball moves at speed .
For the purposes of this problem, assume that the collision is perfectly elastic, neglect friction, and ignore the
spinning of the balls.
Harvaran Ghai
Part A
Find the angle that the 9-ball travels away from the horizontal, as shown in the figure.
Express your answer in degrees to three significant figures.
=45.0
Correct
Note that the angle between the final velocities of the two balls is
. It turns out that in any elastic
collision between two objects of equal mass, one of which is initially at rest, the angle between the final velocities of
the two objects will be ninety degrees.
Energy in a Spring Graphing Question
A toy car is held at rest against a compressed spring, as shown in the figure.
across the room. Let
When released, the car slides
be the initial position of the car. Assume that friction is negligible.
Harvaran Ghai
Part A
Sketch a graph of the total energy of the spring and car system. There is no scale given, so your graph should simply
reflect the qualitative shape of the energy vs. time plot. HORIZONTAL LINE
Correct
Part B
Sketch a plot of the elastic potential energy of the spring from the point at which the car is released to the
equilibrium position of the spring. Make your graph consistent with the given plot of total energy (the gray line
given in the graphing window). 4 POINTS GRADUALLY SLOPING DOWN
Correct
Part C
Sketch a graph of the car's kinetic energy from the moment it is released until it passes the equilibrium position of
the spring. Your graph should be consistent with the given plots of total energy (gray line in graphing window) and
potential energy (gray parabola in graphing window). 5 POINTS GRADUALLY SLOPING UP OPPOSITE OF
GIVEN
Correct
Graphing Gravitational Potential Energy
A 1.00
ball is thrown directly upward with an initial speed of 16.0
.
A graph of the ball's gravitational potential energy vs. height,
, for an arbitrary initial velocity is given in Part
A. The zero point of gravitational potential energy is located at the height at which the ball leaves the thrower's
hand.
For this problem, take
as the acceleration due to gravity.
Harvaran Ghai
Part A
Draw a line on the graph representing the total energy
of the ball. HORIZONTAL LINE AT 126
Correct
Part B
Using the graph, determine the maximum height reached by the ball.
Express your answer to one decimal place.
12.8
Correct
The ball reaches its maximum height when its velocity (and therefore kinetic energy) is zero, so all of its energy is
potential. This occurs at the height at which the total energy and potential energy graphs intersect.
Part C
Draw a new gravitational potential energy vs. height graph to represent the gravitational potential energy if the ball
had a mass of 2.00
reference.
Take
. The graph for a 1.00-
ball with an arbitrary initial velocity is provided again as a
as the acceleration due to gravity. (100,5) (150, 7.5)
Correct
For a ball with twice the mass, you should expect the plot of potential energy vs. height to have twice the slope.
Understanding Work and Kinetic Energy
Learning Goal: To learn about the Work-Energy Theorem and its basic applications.
In this problem, you will learn about the relationship between the work done on an object and the kinetic energy of
that object.
The kinetic energy of an object of mass moving at a speed is defined as
. It seems reasonable
to say that the speed of an object--and, therefore, its kinetic energy--can be changed by performing work on the
object. In this problem, we will explore the mathematical relationship between the work done on an object and the
change in the kinetic energy of that object.
Harvaran Ghai
First, let us consider a sled of mass being pulled by a constant, horizontal force of magnitude
horizontal surface. The sled is speeding up.
along a rough,
Part A
How many forces are acting on the sled?
one
two
three
four
Correct
Part B
The work done on the sled by the force of gravity is __________.
zero
negative
positive
Correct
Part C
The work done on the sled by the normal force is __________.
zero
negative
positive
Correct
Part D
The work done on the sled by the pulling force is __________.
zero
negative
positive
Correct
Part E
The work done on the sled by the force of friction is __________.
zero
negative
positive
Correct
Part F
The net work done on the sled is __________.
zero
negative
positive
Correct
Part G
In the situation described, the kinetic energy of the sled __________.
remains constant
decreases
increases
Correct
Let us now consider the situation quantitatively. Let the mass of the sled be
acting on the sled be
and the magnitude of the net force
. The sled starts from rest.
Consider an interval of time during which the sled covers a distance and the speed of the sled increases from to
. We will use this information to find the relationship between the work done by the net force (otherwise known
as the net work) and the change in the kinetic energy of the sled.
Part H
Find the net force
acting on the sled.
Express your answer in terms of some or all of the variables
, ,
, and
.
=
Correct
Part I
Find the net work
done on the sled.
Express your answer in terms of some or all of the variables
and .
=
Correct
Part J
Use
to find the net work
done on the sled.
Express your answer in terms of some or all of the variables
,
, and
.
=
Correct
Your answer can also be rewritten as
or
,
where
and
are, respectively, the initial and the final kinetic energies of the sled. Finally, one can write
.
This formula is known as the Work-Energy Theorem. The calculations done in this problem illustrate the
applicability of this theorem in a particlar case; however, they should not be interpreted as a proof of this theorem.
Nevertheless, it can be shown that the Work-Energy Theorem is applicable in all situations, including those
involving nonconstant forces or forces acting at an angle to the displacement of the object. This theorem is quite
useful in solving problems, as illustrated by the following example.
Here is a simple application of the Work-Energy Theorem.
Part K
A car of mass accelerates from speed to speed while going up a slope that makes an angle with the
horizontal. The coefficient of static friction is , and the acceleration due to gravity is . Find the total work
done on the car by the external forces.
Express your answer in terms of the given quantities. You may or may not use all of them.
=
Correct
Work Done by a Spring
Consider a spring, with spring constant , one end of which is attached to a wall.
unstretched, with the unconstrained end of the spring at position
.
The spring is initially
Harvaran Ghai
Part A
The spring is now compressed so that the unconstrained end moves from
to
. Using the work integral
,
find the work done by the spring as it is compressed.
Express the work done by the spring in terms of and
.
=
Correct
Work from a Constant Force
Learning Goal: To understand how to compute the work done by a constant force acting on a particle that moves in
a straight line.
In this problem, you will calculate the work done by a constant force. A force is considered constant if
independent of
. This is the most frequently encountered situation in elementary Newtonian mechanics.
is
Harvaran Ghai
Part A
Consider a particle moving in a straight line from initial point B to final point A, acted upon by a constant force
.
The force (think of it as a field, having a magnitude and direction at every position ) is indicated by a series
of identical vectors pointing to the left, parallel to the horizontal axis. The vectors are all identical only because the
force is constant along the path. The magnitude of the force is
A is
(of magnitude
, and the displacement vector from point B to point
, making and angle (radians) with the positive x axis). Find
, the work that the force
performs on the particle as it moves from point B to point A.
Express the work in terms of
in your answer.
,
, and . Remember to use radians, not degrees, for any angles that appear
=
Correct
This result is worth remembering! The work done by a constant force of magnitude
with respect to the direction of motion along a straight path of length
, which acts at an angle of
, is
. This equation
correctly gives the sign in this problem. Since is the angle with respect to the positive x axis (in radians),
; hence
.
Part B
Now consider the same force
vector
acting on a particle that travels from point A to point B.
now points in the opposite direction as it did in Part A. Find the work
Express your answer in terms of
=
Correct
,
, and .
done by
The displacement
in this case.
The Work Done in Pulling a Supertanker
Two tugboats pull a disabled supertanker. Each tug exerts a constant force of 1.60×10 6
, one at an angle 14.0
west of north, and the other at an angle 14.0 east of north, as they pull the tanker a distance 0.890
north.
toward the
Harvaran Ghai
Part A
What is the total work done by the two tugboats on the supertanker?
Express your answer in joules.
2.76×109
Correct
Work on a Sliding Block
A block of weight
shown.
sits on a frictionless inclined plane, which makes an angle with respect to the horizontal, as
A force of magnitude
, applied parallel to the incline, pulls the block up the plane at constant speed.
Harvaran Ghai
Part A
The block moves a distance
up the incline. The block does not stop after moving this distance but continues to
move with constant speed. What is the total work
done on the block by all forces? (Include only the work done
after the block has started moving, not the work needed to start the block moving from rest.)
Express your answer in terms of given quantities.
=0
Correct
Part B
What is
, the work done on the block by the force of gravity as the block moves a distance up the incline?
Express the work done by gravity in terms of the weight and any other quantities given in the problem
introduction.
=
Correct
Part C
What is
, the work done on the block by the applied force
Express your answer in terms of
as the block moves a distance
up the incline?
and other given quantities.
=
Correct
Part D
What is
, the work done on the block by the normal force as the block moves a distance
plane?
Express your answer in terms of given quantities.
up the inclined
=0
Correct
Potential Energy Graphs and Motion
Learning Goal: To be able to interpret potential energy diagrams and predict the corresponding motion of a
particle.
Potential energy diagrams for a particle are useful in predicting the motion of that particle. These diagrams allow
one to determine the direction of the force acting on the particle at any point, the points of stable and unstable
equilibrium, the particle's kinetic energy, etc.
Consider the potential energy diagram shown.
The curve represents the value of potential energy as a
function of the particle's coordinate . The horizontal line above the curve represents the constant value of the total
energy of the particle
. The total energy
is the sum of kinetic (
) and potential (
The key idea in interpreting the graph can be expressed in the equation
) energies of the particle.
where
is the x component of the net force as function of the particle's coordinate . Note the negative sign: It
means that the x component of the net force is negative when the derivative is positive and vice versa. For instance,
if the particle is moving to the right, and its potential energy is increasing, the net force would be pulling the particle
to the left.
If you are still having trouble visualizing this, consider the following: If a massive particle is increasing its
gravitational potential energy (that is, moving upward), the force of gravity is pulling in the opposite direction (that
is, downward).
If the x component of the net force is zero, the particle is said to be in equilibrium. There are two kinds of
equilibrium:


Stable equilibrium means that small deviations from the equilibrium point create a net force that
accelerates the particle back toward the equilibrium point (think of a ball rolling between two hills).
Unstable equilibrium means that small deviations from the equilibrium point create a net force that
accelerates the particle further away from the equilibrium point (think of a ball on top of a hill).
In answering the following questions, we will assume that there is a single varying force acting on the particle
along the x axis. Therefore, we will use the term force instead of the cumbersome x component of the net force.
Harvaran Ghai
Part A
The force acting on the particle at point A is __________.
directed to the right
directed to the left
equal to zero
Correct
Consider the graph in the region of point A. If the particle is moving to the right, it would be "climbing the hill," and
the force would "pull it down," that is, pull the particle back to the left. Another, more abstract way of thinking
about this is to say that the slope of the graph at point A is positive; therefore, the direction of
Part B
The force acting on the particle at point C is __________.
directed to the right
directed to the left
equal to zero
Correct
Part C
The force acting on the particle at point B is __________.
directed to the right
directed to the left
equal to zero
Correct
The slope of the graph is zero; therefore, the derivative
, and
.
Part D
The acceleration of the particle at point B is __________.
directed to the right
directed to the left
equal to zero
Correct
If the net force is zero, so is the acceleration. The particle is said to be in a state of equilibrium.
Part E
If the particle is located slightly to the left of point B, its acceleration is __________.
directed to the right
directed to the left
equal to zero
Correct
is negative.
Part F
If the particle is located slightly to the right of point B, its acceleration is __________.
directed to the right
directed to the left
equal to zero
Correct
As you can see, small deviations from equilibrium at point B cause a force that accelerates the particle further away;
hence the particle is in unstable equilibrium.
Part G
Name all labeled points on the graph corresponding to unstable equilibrium.
List your choices alphabetically, with no commas or spaces; for instance, if you choose points B, D, and E,
type your answer as BDE.
BF
Correct
Part H
Name all labeled points on the graph corresponding to stable equilibrium.
List your choices alphabetically, with no commas or spaces; for instance, if you choose points B, D, and E,
type your answer as BDE.
DH
Correct
Part I
Name all labeled points on the graph where the acceleration of the particle is zero.
List your choices alphabetically, with no commas or spaces; for instance, if you choose points B, D, and E,
type your answer as BDE.
BDFH
Correct
Your answer, of course, includes the locations of both stable and unstable equilibrium.
Part J
Name all labeled points such that when a particle is released from rest there, it would accelerate to the left.
List your choices alphabetically, with no commas or spaces; for instance, if you choose points B, D, and E,
type your answer as BDE.
AE
Correct
Part K
Consider points A, E, and G. Of these three points, which one corresponds to the greatest magnitude of acceleration
of the particle?
A
E
G
Correct
Kinetic energy
If the total energy of the particle is known, one can also use the graph of
kinetic energy of the particle since
to draw conclusions about the
.
As a reminder, on this graph, the total energy
is shown by the horizontal line.
Part L
What point on the graph corresponds to the maximum kinetic energy of the moving particle?
D
Correct
It makes sense that the kinetic energy of the particle is maximum at one of the (force) equilibrium points. For
example, think of a pendulum (which has only one force equilibrium point--at the very bottom).
Part M
At what point on the graph does the particle have the lowest speed?
B
Correct
As you can see, many different conclusions can be made about the particle's motion merely by looking at the graph.
It is helpful to understand the character of motion qualitatively before you attempt quantitative problems. This
problem should prove useful in improving such an understanding.
Potential Energy Calculations
Learning Goal: To understand the relationship between the force and the potential energy changes associated with
that force and to be able to calculate the changes in potential energy as definite integrals.
Imagine that a conservative force field is defined in a certain region of space. Does this sound too abstract? Well,
think of a gravitational field (the one that makes apples fall down and keeps the planets orbiting) or an electrostatic
field existing around any electrically charged object.
If a particle is moving in such a field, its change in potential energy does not depend on the particle's path and is
determined only by the particle's initial and final positions. Recall that, in general, the component of the net force
acting on a particle equals the negative derivative of the potential energy function along the corresponding axis:
.
Therefore, the change in potential energy can be found as the integral
,
where
is the change in potential energy for a particle moving from point 1 to point 2,
the particle at a given point of its path, and
is the net force acting on
is a small displacement of the particle along its path from 1 to 2.
Evaluating such an integral in a general case can be a tedious and lengthy task. However, two circumstances make it
easier:
1.
2.
Because the result is path-independent, it is always possible to consider the most straightforward way to
reach point 2 from point 1.
The most common real-world fields are rather simply defined.
In this problem, you will practice calculating the change in potential energy for a particle moving in three common
force fields.
Note that, in the equations for the forces,
is the unit vector in the x direction, is the unit vector in the y direction,
and is the unit vector in the radial direction in case of a spherically symmetrical force field.
Harvaran Ghai
Part A
Consider a uniform gravitational field (a fair approximation near the surface of a planet). Find
,
where
and
Express your answer in terms of
=
Correct
, ,
, and
.
.
Part B
Consider the force exerted by a spring that obeys Hooke's law. Find
,
where
,
and the spring constant is positive.
Express your answer in terms of ,
, and
.
=
Correct
Part C
Finally, consider the gravitational force generated by a spherically symmetrical massive object. The magnitude and
direction of such a force are given by Newton's law of gravity:
,
where
;
,
, and
are constants; and
. Find
.
Express your answer in terms of
,
,
,
, and
.
=
Correct
As you can see, the change in potential energy of the particle can be found by integrating the force along the
particle's path. However, this method, as we mentioned before, does have an important restriction: It can only be
applied to a conservative force field. For conservative forces such as gravity or tension the work done on the particle
does not depend on the particle's path, and the potential energy is the function of the particle's position.
In case of a nonconservative force--such as a frictional or magnetic force--the potential energy can no longer be
defined as a function of the particle's position, and the method that you used in this problem would not be
applicable.
A Mass-Spring System with Recoil and Friction
An object of mass is traveling on a horizontal surface. There is a coefficient of kinetic friction
object and the surface. The object has speed when it reaches
between the
and encounters a spring. The object compresses
the spring, stops, and then recoils and travels in the opposite direction. When the object reaches
trip, it stops.
on its return
Harvaran Ghai
Part A
Find , the spring constant.
Express in terms of
,
, , and .
=
Correct
Dragging a Board
A uniform board of length and mass
lies near a boundary that separates two regions. In region 1, the coefficient
of kinetic friction between the board and the surface is , and in region 2, the coefficient is . The positive
direction is shown in the figure.
Harvaran Ghai
Part A
Find the net work
done by friction in pulling the board directly from region 1 to region 2. Assume that the board
moves at constant velocity.
Express the net work in terms of
, ,
,
, and
.
=
Correct
This answer makes sense because it is as if the board spent half its time in region 1, and half in region 2, which on
average, it in fact did.
Part B
What is the total work done by the external force in pulling the board from region 1 to region 2? (Again, assume that
the board moves at constant velocity.)
Express your answer in terms of
, ,
,
, and
.
=
Correct
Drag on a Skydiver
A skydiver of mass jumps from a hot air balloon and falls a distance before reaching a terminal velocity of
magnitude . Assume that the magnitude of the acceleration due to gravity is .
Harvaran Ghai
Part A
What is the work
done on the skydiver, over the distance
Express the work in terms of
, ,
, by the drag force of the air?
, and the magnitude of the acceleration due to gravity .
=
Correct
Part B
Find the power
supplied by the drag force after the skydiver has reached terminal velocity .
Express the power in terms of quantities given in the problem introduction.
=
Correct
Power Dissipation Puts a Drag on Racing
The dominant form of drag experienced by vehicles (bikes, cars, planes, etc.) at operating speeds is called form
drag. It increases quadratically with velocity (essentially because the amount of air you run into increases with and
so does the amount of force you must exert on each small volume of air). Thus
,
where
is the cross-sectional area of the vehicle and
is called the coefficient of drag.
Harvaran Ghai
Part A
Consider a vehicle moving with constant velocity
Express your answer in terms of
,
. Find the power dissipated by form drag.
, and speed .
=
Correct
Part B
A certain car has an engine that provides a maximum power
. Suppose that the maximum speed of the car,
limited by a drag force proportional to the square of the speed (as in the previous part). The car engine is now
modified, so that the new power
is 10 percent greater than the original power (
.
Assume the following:


The top speed is limited by air drag.
The magnitude of the force of air drag at these speeds is proportional to the square of the speed.
By what percentage,
, is the top speed of the car increased?
Express the percent increase in top speed numerically to two significant figures.
=3.2 %
Correct
You'll note that your answer is very close to one-third of the percentage by which the power was increased. This
dependence of small changes on each other, when the quantities are related by proportionalities of exponents, is
common in physics and often makes a useful shortcut for estimations.
, is
Energy of a Spacecraft
Very far from earth (at
), a spacecraft has run out of fuel and its kinetic energy is zero. If only the
gravitational force of the earth were to act on it (i.e., neglect the forces from the sun and other solar system objects),
the spacecraft would eventually crash into the earth. The mass of the earth is
and its radius is
. Neglect air
resistance throughout this problem, since the spacecraft is primarily moving through the near vacuum of space.
Harvaran Ghai
Part A
Find the speed
of the spacecraft when it crashes into the earth.
Express the speed in terms of
,
, and the universal gravitational constant
.
=
Correct
Part B
Now find the spacecraft's speed when its distance from the center of the earth is
Express the speed in terms of and .
, where
.
=
Correct
Orbiting Satellite
A satellite of mass
density . (
constant.
is in a circular orbit of radius
around a spherical planet of radius
is measured from the center of the planet, not its surface.) Use
made of a material with
for the universal gravitational
Harvaran Ghai
Part A
Find the kinetic energy of this satellite,
.
Express the satellite's kinetic energy in terms of
,
,
,
,
, and .
=
Correct
Part B
Find , the gravitational potential energy of the satellite. Take the gravitational potential energy to be zero for an
object infinitely far away from the planet.
Express the satellite's gravitational potential energy in terms of
,
,
=
Correct
Part C
What is the ratio of the kinetic energy of this satellite to its potential energy?
Express
in terms of parameters given in the introduction.
=-0.500
Correct
,
,
, and .
The result of this problem may be expressed as
where
is the exponent of the force law (i.e.
). This is a specical case of a general and powerful theroem of advanced classical
mechanics known as the Virial Theorem. The theorem applies to the average of the kinetic and potential energies of
of any one or multiple objects moving over any closed (or almost closed) path that returns very close to itself
provided that all objects interact via potentials with the same power law dependence on their separation. Thus it
applies to stars in a galaxy, or masses tied together with springs (where
since the force law is
).
Problem 12.28
The space shuttle is in a 250
-high circular orbit. It needs to reach a 700
Hubble Space Telescope for repairs. The shuttle's mass is 8.00×104
-high circular orbit to catch the
.
Harvaran Ghai
Part A
How much energy is required to boost it to the new orbit?
1.53×1011 J
Correct
An Exhausted Bicyclist
An exhausted bicyclist pedals somewhat erratically, so that the angular velocity of his tires follows the equation
,
where represents time (measured in seconds).
Harvaran Ghai
Part A
There is a spot of paint on the front tire of the bicycle. Take the position of the spot at time
to be at angle
radians with respect to an axis parallel to the ground (and perpendicular to the axis of rotation of the tire) and
measure positive angles in the direction of the tire's rotation. What angular displacement has the spot of paint
undergone between time 0 and 2 seconds?
Express your answer in radians.
=0.793
Correct
Part B
Express the angular displacement undergone by the spot of paint at
seconds in degrees.
=45.5
Correct
Part C
What distance has the spot of paint moved in 2 seconds if the radius of the tire is 50 centimeters?
Express your answer in centimeters.
=39.7
Correct
Part D
Which one of the following statements describes the motion of the spot of paint at
seconds?
The angular acceleration of the spot of paint is constant and the magnitude of the angular speed is decreasing.
The angular acceleration of the spot of paint is constant and the magnitude of the angular speed is increasing.
The angular acceleration of the spot of paint is positive and the magnitude of the angular speed is decreasing.
The angular acceleration of the spot of paint is positive and the magnitude of the angular speed is increasing.
The angular acceleration of the spot of paint is negative and the magnitude of the angular speed is decreasing.
The angular acceleration of the spot of paint is negative and the magnitude of the angular speed is increasing.
Correct
A Spinning Grinding Wheel
At time
32.0
433
a grinding wheel has an angular velocity of 23.0
until a circuit breaker trips at time
. It has a constant angular acceleration of
= 1.90 . From then on, the wheel turns through an angle of
as it coasts to a stop at constant angular deceleration.
Harvaran Ghai
Part A
Through what total angle did the wheel turn between
Express your answer in radians.
534
Correct
and the time it stopped?
Part B
At what time does the wheel stop?
Express your answer in seconds.
12.2
Correct
Part C
What was the wheel's angular acceleration as it slowed down?
Express your answer in radians per second per second.
-8.11
Correct
Finding Torque
A force
of magnitude
, making an angle with the x axis, is applied to a particle located at point A, at Cartesian
coordinates (0, 0) in the figure. The vector and the four reference points (i.e., A, B, C, and D) all lie in the xy
plane. Rotation axes A - D lie parallel to the z axis and pass through each respective reference point.
The torque
of a force
acting on a particle having a position vector
with respect to a reference point (thus
points from the reference point to the point at which the force acts) is equal to the cross product of
. The magnitude of the torque is
is perpendicular to both
and
. For this problem
to clockwise rotation. You must express
in terms of ,
, where
is the angle between
and
and
,
; the direction of
; negative torque about a reference point corresponds
, and/or
when entering your answers.
Harvaran Ghai
Part A
What is the torque due to force about the point A?
Express the torque about point A at Cartesian coordinates (0, 0).
=0
Correct
Part B
What is the torque
due to force
about the point B? (B is the point at Cartesian coordinates (0, ), located a
distance from the origin along the y axis.)
Express the torque about point B in terms of
, ,
,
, and/or other given coordinate data.
=
Correct
Part C
What is the torque
along the x axis?
about the point C, located at a position given by Cartesian coordinates ( , 0), a distance
Express the torque about point C in terms of
, ,
,
, and/or other given coordinate data.
=
Correct
Part D
What is the torque
axis?
about the point D, located at a distance
from the origin and making an angle
with the x
Express the torque about point D in terms of
, ,
,
, and/or other given coordinate data.
=
Correct
Note that the cross product
which simplifies to
can also be expressed as a third-order determinant
when
and
lie in the xy plane.
Pivoted Rod with Unequal Masses
A thin rod of mass
and length
is allowed to pivot freely about its center, as shown in the diagram. A
small sphere of mass
is attached to the left end of the rod, and a small sphere of mass
is attached to the right
end. The spheres are small enough that they can be considered point particles. The gravitational force acts
downward, with the magnitude of the gravitational acceleration equal to .
Harvaran Ghai
Part A
What is the moment of inertia of this assembly about the axis through which it is pivoted?
Express the moment of inertia in terms of
=
Correct
,
,
, and
. Remember, the length of the rod is
, not
.
Part B
Suppose the rod is held at rest horizontally and then released. (Throughout the remainder of this problem, your
answer may include the symbol , the moment of inertia of the assembly, whether or not you have answered the
first part correctly.)
What is the angular acceleration of the rod immediately after it is released?
Take the counterclockwise direction to be positive. Express in terms of some or all of the variables
,
,
,
, , and .
=
Correct
Pulling a String to Accelerate a Wheel
A bicycle wheel is mounted on a fixed, frictionless axle, as shown
wheel's rim, and a constant horizontal force
starting at time
of magnitude
. A massless string is wound around the
starts pulling the string from the top of the wheel
when the wheel is not rotating. Suppose that at some later time the string has been pulled
through a distance . The wheel has moment of inertia
, where is a dimensionless number less than 1,
is the wheel's mass, and is its radius. Assume that the string does not slip on the wheel.
Harvaran Ghai
Part A
Find , the angular acceleration of the wheel, which results from pulling the string to the left. Use the standard
convention that counterclockwise angular accelerations are positive.
Express the angular acceleration,
, in terms of
, ,
, and (but not
).
=
Correct
Part B
The force pulling the string is constant; therefore the magnitude of the angular acceleration
constant for this configuration.
Find the magnitude of the angular velocity
of the wheel when the string has been pulled a distance
Note that there are two ways to find an expression for
Express the angular velocity
of the wheel is
.
; these expressions look very different but are equivalent.
of the wheel in terms of the displacement
, the magnitude
of the applied
force, and the moment of inertia of the wheel , if you've found such a solution. Otherwise, following the
hints for this part should lead you to express the angular velocity of the wheel in terms of the displacement
, the wheel's radius , and
.
=
Correct
This solution can be obtained from the equations of rotational motion and the equations of motion with constant
acceleration. An alternate approach is to calculate the work done over the displacement
by the force
and equate
this work to the increase in rotational kinetic energy of rotation of the wheel
Part C
Find , the speed of the string after it has been pulled by
Express the speed of the string in terms of
,
over a distance
, , and
.
; do not include
,
, or
in your answer.
=
Correct
Note that this is the speed that an object of mass
force with constant magnitude
(which is less than
.
A Bar Suspended by Two Vertical Strings
) would attain if pulled a distance
by a
A rigid, uniform, horizontal bar of mass
and length
is supported by two identical massless strings.
Both
strings are vertical. String A is attached at a distance
from the left end of the bar and is connected to the
ceiling; string B is attached to the left end of the bar and is connected to the floor. A small block of mass
is
supported against gravity by the bar at a distance from the left end of the bar, as shown in the figure.
Throughout this problem positive torque is that which spins an object counterclockwise. Use for the magnitude of
the acceleration due to gravity.
Harvaran Ghai
Part A
Find
, the tension in string A.
Express the tension in string A in terms of ,
,
,
,
, and .
=
Correct
Part B
Find
, the magnitude of the tension in string B.
Express the magnitude of the tension in string B in terms of
,
,
, and .
=
Correct
Part C
If the bar and block are too heavy the strings may break. Which of the two identical strings will break first?
string A
string B
Correct
Part D
If the mass of the block is too large and the block is too close to the left end of the bar (near string B) then the
horizontal bar may become unstable (i.e., the bar may no longer remain horizontal).
What is the smallest possible value of
Express your answer for
such that the bar remains stable (call it
in terms of
,
,
, and
)?
.
=
Correct
Part E
Note that since
, as computed in the previous part, is not necessarily positive. If
stable no matter where the block of mass
is placed on it.
Assuming that
,
, and
are held fixed, what is the maximum block mass
stable? In other words, what is the maximum block mass such that
Answer in terms of
,
, and
, the bar will be
for which the bar will always be
?
.
=
Correct
A Person Standing on a Leaning Ladder
A uniform ladder with mass
and length
rests against a smooth wall.
A do-it-yourself enthusiast of mass
stands on the ladder a distance from the bottom (measured along the ladder). The ladder makes an angle with
the ground. There is no friction between the wall and the ladder, but there is a frictional force of magnitude
between the floor and the ladder.
is the magnitude of the normal force exerted by the wall on the ladder, and
is the magnitude of the normal force exerted by the ground on the ladder. Throughout the problem, consider
counterclockwise torques to be positive. None of your answers should involve (i.e., simplify your trig functions).
Harvaran Ghai
Part A
What is the minimum coeffecient of static friction
does not slip?
Express
=
Correct
in terms of
,
,
,
required between the ladder and the ground so that the ladder
, and .
Part B
Suppose that the actual coefficent of friction is one and a half times as large as the value of
. Under these circumstances, what is the magnitude of the force of friction
. That is,
that the floor applies to
the ladder?
Express your answer in terms of
and .
,
,
,
, , and . Remember to pay attention to the relation of force
=
Correct
A Rolling Hollow Sphere
A hollow spherical shell with mass 1.80
the horizontal.
rolls without slipping down a slope that makes an angle of 39.0 with
Harvaran Ghai
Part A
Find the magnitude of the acceleration
of the center of mass of the spherical shell.
Take the free-fall acceleration to be
= 9.80
.
=3.70
Correct
Part B
Find the magnitude of the frictional force acting on the spherical shell.
Take the free-fall acceleration to be
= 9.80
.
=4.44
Correct
The frictional force keeps the spherical shell stuck to the surface of the slope, so that there is no slipping as it rolls
down. If there were no friction, the shell would simply slide down the slope, as a rectangular box might do on an
inclined (frictionless) surface.
Part C
Find the minimum coefficient of friction
slope.
=0.324
Correct
needed to prevent the spherical shell from slipping as it rolls down the
Weight and Wheel
Consider a bicycle wheel that initially is not rotating. A block of mass
fall a distance
is attached to the wheel and is allowed to
. Assume that the wheel has a moment of inertia about its rotation axis.
Harvaran Ghai
Part A
Consider the case that the string tied to the block is attached to the outside of the wheel, at a radius
. Find
, the angular speed of the wheel after the block has fallen a distance
Express
in terms of
, ,
,
, for this case.
, and .
=
Correct
Part B
Now consider the case that the string tied to the block is wrapped around a smaller inside axle of the wheel of radius
. Find
Express
, the angular speed of the wheel after the block has fallen a distance
in terms of
, ,
,
, for this case.
, and .
=
Correct
Part C
Which of the following describes the relationship between
and
?
Correct
This is related to why gears are found on the inside rather than the outside of a wheel.
Record and Turntable
Learning Goal: To understand how to use conservation of angular momentum to solve problems involving
collisions of rotating bodies.
Consider a turntable to be a circular disk of moment of inertia rotating at a constant angular velocity
around an axis through the center and perpendicular to the plane of the disk (the disk's "primary axis of
symmetry"). The axis of the disk is vertical and the disk is supported by frictionless bearings. The motor of the
turntable is off, so there is no external torque being applied to the axis.
Another disk (a record) is dropped onto the first such that it lands coaxially (the axes coincide). The moment of
inertia of the record is
. The initial angular velocity of the second disk is zero.
There is friction between the two disks.
After this "rotational collision," the disks will eventually rotate with the same angular velocity.
Harvaran Ghai
Part A
What is the final angular velocity,
Express
in terms of
,
, of the two disks?
, and
.
=
Correct
Part B
Because of friction, rotational kinetic energy is not conserved while the disks' surfaces slip over each other. What is
the final rotational kinetic energy,
, of the two spinning disks?
Express the final kinetic energy in terms of , , and the initial kinetic energy
angular velocities should appear in your answer.
of the two-disk system. No
=
Correct
Some of the energy was converted into heat and sound as the frictional force, torque acted, stopping relative motion.
Part C
Assume that the turntable deccelerated during time
before reaching the final angular velocity (
is the time
interval between the moment when the top disk is dropped and the time that the disks begin to spin at the same
angular velocity). What was the average torque,
, acting on the bottom disk due to friction with the record?
Express the torque in terms of
.
=
,
,
, and
Correct
Problem 13.87
During most of its lifetime, a star maintains an equilibrium size in which the inward force of gravity on each atom is
balanced by an outward pressure force due to the heat of the nuclear reactions in the core. But after all the hydrogen
"fuel" is consumed by nuclear fusion, the pressure force drops and the star undergoes a gravitational collapse until it
becomes a neutron star. In a neutron star, the electrons and protons of the atoms are squeezed together by gravity
until they fuse into neutrons. Neutron stars spin very rapidly and emit intense pulses of radio and light waves, one
pulse per rotation. These "pulsing stars" were discovered in the 1960s and are called pulsars.
Harvaran Ghai
Part A
A star with the mass
and size
of our sun rotates once every 34.0 days.
After undergoing gravitational collapse, the star forms a pulsar that is observed by astronomers to emit radio pulses
every 0.100 . By treating the neutron star as a solid sphere, deduce its radius.
6.46×104 m
Correct
Part B
What is the speed of a point on the equator of the neutron star? Your answer will be somewhat too large because a
star cannot be accurately modeled as a solid sphere.
4.06×106 m/s
Correct
Analyzing Simple Harmonic Motion
This Error! Hyperlink reference not valid. shows two masses on springs, each accompanied by a graph of its
position versus time.
Harvaran Ghai
Part A
What is an expression for
, the position of mass I as a function of time? Assume that position is measured in
meters and time is measured in seconds.
Express your answer as a function of . Express numerical constants to three significant figures.
=
Correct
Part B
What is
, the position of mass II as a function of time? Assume that position is measured in meters and time is
measured in seconds.
Express your answer as a function of . Express numerical constants to three significant figures.
=
Correct
Harmonic Oscillator Acceleration
Learning Goal: To understand the application of the general harmonic equation to finding the acceleration of a
spring oscillator as a function of time.
One end of a spring with spring constant is attached to the wall. The other end is attached to a block of mass .
The block rests on a frictionless horizontal surface. The equilibrium position of the left side of the block is defined
to be
. The length of the relaxed spring is
.
The block is slowly pulled from its equilibrium position to some position
the block is released with zero initial velocity.
The goal of this problem is to determine the acceleration of the block
and
.
along the x axis. At time
as a function of time in terms of ,
It is known that a general solution for the position of a harmonic oscillator is
,
where
,
, and
are constants.
Your task, therefore, is to determine the values of
connection between
and
,
, and
to find the acceleration.
in terms of ,
,and
and then use the
,
,
Harvaran Ghai
Part A
Combine Newton's 2nd law and Hooke's law for a spring to find the acceleration of the block
time.
Express your answer in terms of ,
, and the coordinate of the block
as a function of
.
=
Correct
The negative sign in the answer is important: It indicates that the restoring force (the tension of the spring) is always
directed opposite to the block's displacement. When the block is pulled to the right from the equilibrium position,
the restoring force is pulling back, that is, to the left--and vice versa.
Part B
Using the fact that acceleration is the second derivative of position, find the acceleration of the block
function of time.
Express your answer in terms of
, , and
=
Correct
Part C
Find the angular frequency
.
Express your answer in terms of and
=
Correct
.
.
as a
Note that the angular frequency
and, therefore, the period of oscillations
characteristics of the system ( and
amplitude of the motion.
depend only on the intrinsic physical
). Frequency and period do not depend on the initial conditions or the
Energy of Harmonic Oscillators
Learning Goal: To learn to apply the law of conservation of energy to the analysis of harmonic oscillators.
Systems in simple harmonic motion, or harmonic oscillators, obey the law of conservation of energy just like all
other systems do. Using energy considerations, one can analyze many aspects of motion of the oscillator. Such an
analysis can be simplified if one assumes that mechanical energy is not dissipated. In other words,
,
where
is the total mechanical energy of the system,
is the kinetic energy, and
is the potential energy.
Harvaran Ghai
As you know, a common example of a harmonic oscillator is a mass attached to a spring. In this problem, we will
consider a horizontally moving block attached to a spring. Note that, since the gravitational potential energy is not
changing in this case, it can be excluded from the calculations.
For such a system, the potential energy is stored in the spring and is given by
,
where is the force constant of the spring and
is the distance from the equilibrium position.
The kinetic energy of the system is, as always,
,
where
is the mass of the block and is the speed of the block.
We will also assume that there are no resistive forces; that is,
.
Consider a harmonic oscillator at four different moments, labeled A, B, C, and D, as shown in the figure
Assume that the force constant , the mass of the block,
the following questions.
, and the amplitude of vibrations,
.
, are given. Answer
Part A
Which moment corresponds to the maximum potential energy of the system?
A
B
C
D
Correct
Part B
Which moment corresponds to the minimum kinetic energy of the system?
A
B
C
D
Correct
When the block is displaced a distance
from equilibrium, the spring is stretched (or compressed) the most, and the
block is momentarily at rest. Therefore, the maximum potential energy is
course,
. Recall that
. At that moment, of
. Therefore,
.
In general, the mechanical energy of a harmonic oscillator equals its potential energy at the maximum or minimum
displacement.
Part C
Consider the block in the process of oscillating.
at the equilibrium position.
at the amplitude displacement.
moving to the right.
If the kinetic energy of the block is increasing, the block must be
moving to the left.
moving away from equilibrium.
moving toward equilibrium.
Correct
Part D
Which moment corresponds to the maximum kinetic energy of the system?
A
B
C
D
Correct
Part E
Which moment corresponds to the minimum potential energy of the system?
A
B
C
D
Correct
When the block is at the equilibrium position, the spring is not stretched (or compressed) at all. At that moment, of
course,
. Meanwhile, the block is at its maximum speed (
then be written as
. Recall that
and that
.
Recalling what we found out before,
,
). The maximum kinetic energy can
at the equilibrium position. Therefore,
we can now conclude that
,
or
.
Part F
At which moment is
A
?
B
C
D
Correct
Part G
Find the kinetic energy
of the block at the moment labeled B.
Express your answer in terms of and
.
=
Correct
Energy of a Spring
An object of mass
attached to a spring of force constant oscillates with simple harmonic motion. The maximum
displacement from equilibrium is
and the total mechanical energy of the system is
.
Harvaran Ghai
Part A
What is the system's potential energy when its kinetic energy is equal to
?
Correct
Part B
What is the object's velocity when its potential energy is
?
Correct
Gravity on Another Planet
After landing on an unfamiliar planet, a space explorer constructs a simple pendulum of length 55.0
explorer finds that the pendulum completes 108 full swing cycles in a time of 136 .
. The
Harvaran Ghai
Part A
What is the value of the acceleration of gravity on this planet?
Express your answer in meters per second per second.
=13.7
136/108…2pi/ans…ans*0.55
Correct
The Fish Scale
The scale of a spring balance reading from 0 to 205
has a length of 13.5
the spring oscillates vertically at a frequency of 2.95
. A fish hanging from the bottom of
.
Harvaran Ghai
Part A
Ignoring the mass of the spring, what is the mass
of the fish?
Express your answer in kilograms.
=4.42
Correct
Vertical Mass-and-Spring Oscillator
A block of mass is attached to the end of an ideal spring. Due to the weight of the block, the block remains at rest
when the spring is stretched a distance
from its equilibrium length.
The spring has an unknown spring
constant .
Harvaran Ghai
Part A
What is the spring constant ?
Express the spring constant in terms of given quantities and , the magnitude of the acceleration due to
gravity.
=
Correct
Part B
Suppose that the block gets bumped and undergoes a small vertical displacement. Find the resulting angular
frequency of the block's oscillation about its equilibrium position.
Express the frequency in terms of given quantities and , the magnitude of the acceleration due to gravity.
=
Correct
It may seem that this result for the frequency does not depend on either the mass of the block or the spring constant,
which might make little sense. However, these parameters are what would determine the extension
when the block is hanging:
of the spring
.
One way of thinking about this problem is to consider both and as unknowns. By measuring and (both fairly
simple measurements), and knowing the mass, you can determine the value of the spring constant and the
acceleration due to gravity experimentally.