Mastering Physics Answers

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MP02: Motion DiagramsVelocity 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! arecombined 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 also called the position vector!

of objects as a function of time. "et and be velocity and acceleration, respectively.

Harvaran Ghai#onsider 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

$ou drop a power ball on the floor. The motion diagram of the ball is sketched in the figure . Indicate whether

the magnitude of the velocity of the ball is increasing, decreasing, or not changing.

increasing

decreasing

not changing

Correct

%hile the ball is in free fall, the magnitude of its velocity is increasing, so the ball is accelerating.

Part B

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http://session.masteringphysics.com/myct/problemPrint?backViewLink=1&assignmentProblemID=2505000



&ince the length of is directly proportional to the length of , the vector connecting each dot to the ne't 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 (ero.Use P, N, and Z for positive (upward), negative, and zero, respectively. Separate the letters for velocity and

acceleration with a coa.

),)

Correct

Part C )ow, 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

&ince the magnitude of the velocity of the ball is decreasing, the ball must be accelerating specifically, slowingdown!.

Part D

The ne't 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 (ero.Use P, N, and Z for positive (upward), negative, and zero, respectively. Separate the letters for velocity and


*,)Correct

Part EThe power ball has now reached its highest point above the ground and starts to descend again. The motion diagramrepresenting the velocity vectors is the same as that after the initial release, as shown in the figure of *art +. Indicate

whether the velocity and acceleration of the ball at its highest point are positive upward!, negative, or (ero.Use P, N, and Z for positive (upward), negative, and zero, respectively. Separate the letters for velocity and


,)Correct

These e'amples should show you that the velocity and acceleration can have opposite or similar signs or that one of

them can be (ero while the other has either sign. Try hard to think carefully about them as distinct physicalquantities when working with kinematics.

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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 AAt 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 and

at no time shown in the figure

Correct

Part B

At what time(s) do the rockets have the same x position?at time only

at time only

at times and



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Correct

Part CAt what time(s) do the two rockets have the same acceleration?

at time only

at time only

at times and



Correct

Part DThe motion of the rocket labeled A is an example of motion with uniform (i.e., constant) __________.

and nonzero

acceleration

velocity

displacement

time

Correct

Part EThe motion of the rocket labeled B is an example of motion with uniform (i.e., constant) __________.

and nonzero

acceleration

velocity

displacement

time

Correct

Part FAt what time(s) is rocket A ahead of rocket B?

before only

after only

before and after




between and

at no time(s) shown in the

figure

Correct

P !"!# $Al%ost& a Do'en Dia(ra%s

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

MODEL: Represent the moving object as a particle. Make simplifying assumptions when interpreting the problemstatement.

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.

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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 to indicate a point at which the acceleration is zero. Label the row of acceleration

vectors .

Model

It is appropriate to use the particle model for the car. You should also make some simplifying assumptions.

Part AWhich 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 BWhich 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 CWhich 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 DWhich of the diagrams best describes the position and the velocity of the car after the driver notices the policeman?

A

B

CCorrect

Part E

Which of these diagrams most accurately depicts the acceleration of the car during the events described in the

problem introduction?

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 ofthe 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 HWhich 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 )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 *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 DiagramsC+rved Motion Dia(ra%

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 AWhat 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


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Part BWhat 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 CIn which of the following other scenarios could the motion reasonably be represented by the motion diagram in the

introduction?

A. A weight placed on the rim of a bicycle wheel that is being held off the ground so it can rotate freely

B. An airplane pulling out of a dive

C. A race car rounding a turn

D. 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 (orwhatever 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 of the string needed for the

pendulum to have a period of 2 s.

Express 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.





Avera(e Velocity fro% a Position vs" Ti%e Gra,h

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 . We will use the notation to

indicate the average velocity over the time interval from to . Thus is the average velocity over the time

interval from 1 to 3 s.

Harvaran GhaiPart AFind the average velocity over the time interval from 0 to 1 second.

Answer to the nearest integer.

=0

Correct

Part BFind 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 .


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Give your answer to three significant figures.

=13.3

Correct

Note that is not equal to the simple arithmetic average of and , because they are averages

for time intervals of different length. You would have to double the weight given to because it is for an

interval twice as long.

Part DFind 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.

=6.7Correct

Part EObtaining 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 FFind the average velocity over the time interval from 2.5 to 6.0 seconds.

Express your answer to two significant figures.

=-8.6

Correct

R+nnin( and -alkin(

Tim and Rick both can run at speed and walk at speed , with . They set off together on a journey of

distance . Rick walks half of the distance and runs the other half. Tim walks half of the time and runs the other

half.

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 , , and .

=

Correct

Part CHow 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

Gra,h of v$t& for a ,orts Car

The graph shows the velocity of a sports car as a function of time . Use the graph to answer the following

questions.

Harvaran GhaiPart A

Find the maximum velocity of the car.

Express your answer in meters per second to the nearest integer.

=55

Correct

Part BDuring which time interval is the acceleration positive?

Indicate the most complete answer.

to

to

to

to

to

Correct



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Part C

Find the maximum acceleration of the car.

Express your answer in meters per second squared to the nearest integer.

=30Correct

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

Rearendin( Dra( 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 .

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Harvaran GhaiPart 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 BAssuming 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.

, =0.54,7.2 s, m

Correct





The blue curve shows how the car, initially at , continues at constant velocity (blue) and just barely touches the

accelerating drag car (red) at .

Motion of a hadow

A small source of light is located at a distance from a vertical wall. An opaque object with a height of

moves toward the wall with constant velocity 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 .

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=

Correct

Rocket Hei(ht

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 GhaiPart 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 numberequal to the magnitude of the acceleration due to gravity.

= a*t 1*((a*t 1)/g)-(.5g((a*t 1)/g)2)+(.5(t 1)2*a)

Correct

Part B

If the rocket's net acceleration is for , what is the maximum height the rocket will reach?

Express your answer numerically in meters, using .

=1470 mCorrect

A Flower Pot Fallin( Past a -indow

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 , , , , and .

=

Correct

Resolvin( Vector Co%,onents with Tri(ono%etry

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.

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Harvaran GhaiPart 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 with respect to the x axis as shown, named .

Write the components in the form x,y.

=

Correct

Notice that vectors and have the same form despite their placement with respect to the y axis on the drawing.

Part C

Find the components of the vector with length and angle as shown, named .

Express your answer in terms of and . Write the components in the form x,y.

=

Correct




Trackin( 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 . The

position vector has a magnitude of 360 and is located at exactly 40 above the horizon. The airplane is

tracked for another 123 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: . What are the

components of ?

Express in meters as an ordered pair, separating the x and z components with a comma, to two

significant figures.

=-1100,26

Correct


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Two Forces Actin( at a Point

Two forces, and , act at a point. has a magnitude of 9.80 and is directed at an angle of 60.0 above the

negative x axis in the second quadrant. has a magnitude of 6.40 and is directed at an angle of 53.2 below the

negative x axis in the third quadrant.

Harvaran Ghai

Part AWhat is the x component of the resultant force?

Express your answer in newtons.

-8.73

Correct

Part BWhat is the y component of the resultant force?


3.36

Correct

Part C

What is the magnitude of the resultant force?Express your answer in newtons.

9.36

Correct

A P+sh or a P+ll.

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 upon object B."

• "Force is applied to object B by object A."

• "Force due to object A is acting 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 oftension, 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 AWhich object exerts a downward force on the book?

the book itself

the earth

the surface of the

table

Correct



Part BThe downward force acting on the book is __________.

a contact force

a long-range

force

Correct

Part CWhat is the downward force acting on the book called?

tension

normal force

weight

friction

Correct

Part DWhich object exerts an upward force on the book?

the book itself

the earth

the surface of the

table

Correct

Part EThe upward force acting on the book is __________.

a contact force

a long-range

force

Correct

Part FWhat 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 GWhich 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 HThe 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 )

What is the force acting on the block and directed to the right called?

tension

normal force

weight

friction

Correct

Part *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 /The force acting on the block and directed to the left is __________.

a contact force

a long-range

force

Correct

Part 0What 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 MHow 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 1What 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).

Free2Body Dia(ra%s# )ntrod+ction

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.

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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. Draw a free-body

diagram for the puck. Which of the following forces are acting on the puck?

A. weight

B. friction

C. force of velocity

D. force of push

E. normal force



F. air drag

G. 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.

AECorrect

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. weight

B. friction

C. force of velocity

D. force of tension

E. normal force

F. air drag

G. acceleration





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. weight

B. kinetic friction

C. static friction

D. force of push

E. normal force



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




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




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


A. weight

B. kinetic friction

C. static friction

D. force of push

E. normal force

F. the force of velocity



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




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 )

Now consider a block being pushed up a smooth slope. The force pushing the block is parallel to the slope.


A. weight

B. kinetic friction

C. static friction

D. force of push

E. normal force


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 *

To solve for the acceleration of the blocks, you will have to draw the free-body diagrams for which objects?

A. the block of mass

B. the block of mass

C. the connecting string

D. the pulley

E. the table

F. the earth



AB

Correct

Part /

Draw the free-body diagram for the block of mass . How many forces are exerted on this block?

none

one

two



three

four

Correct

Your free-body diagram should look like that shown here.

Part 0

Draw the free-body diagram for the block of mass . How many forces are exerted on this block?

none

one

two

three

four

Correct



Your free-body diagram should look like that shown here.

3nderstandin( 1ewton4s 0awsHarvaran Ghai

Part AAn 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 BIf 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: (directed to the left) and (directed to the right). What canyou say about the block's motion?

It must be moving to the left.

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. Inthis 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 DA 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

accelerationCorrect

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

Conce,t+al 5+estions on 1ewton4s !st and 6nd 0aws

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 AWhat 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.

http://void%280%29/

http://void%280%29/






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 BWhat 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 CWhat 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 iszero.

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 DWhich 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?



35

Correct

Part FWhich of these diagrams may possibly correspond to the situation at point C on the motion diagram?



24

Correct

Part G

Which of these diagrams correspond to a situation where the moving object (not necessarily the one shown in themotion diagram) is changing its velocity?



12345

Correct

Consider the following situations:

A. A car is moving along a straight road at a constant speed.

B. A car is moving along a straight road while slowing down.

C. A car is moving along a straight road while speeding up.

D. A hockey puck slides along a smooth (i.e., frictionless) icy surface.

E. A hockey puck slides along a rough concrete surface.

F. A cockroach is speeding up from rest.

G. A rock is thrown horizontally; air resistance is negligible.

H. A rock is thrown horizontally; air resistance is substantial.

I. A rock is dropped vertically; air resistance is negligible.

J. A rock is dropped vertically; air resistance is substantial.

Part HWhich 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 )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 *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 -orld2Class ,rinterWorld-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 exert on the starting blocks to produce this

acceleration?


=810

Correct

Part BWhich 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 )ncline

A block lies on a plane raised an angle from the horizontal. Three forces act upon the block: , the force of

gravity; , the normal force; and , the force of friction. The coefficient of friction is large enough to prevent the

block from sliding .

Harvaran GhaiPart 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 and

Correct

Part BWhich forces lie along the axes of the coordinate system b, in which the y axis is vertical?

http://void%280%29/





only

only

only

and

and

and

and and

Correct

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 appearing in both the x

and y equations, each multiplied by a trigonometric function.

Part CBecause 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 DBecause 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.


Correct

Part E

To find the magnitude of the normal force, you must express in terms of since is an unknown. Using the

equations you found in the two previous parts, find an expression for involving 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 , which leads immediately to the result obtained here

for .

CONCLUSION: A thoughtful examination of which coordinate system to choose can save a lot of algebra.

Han(in( 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 and makes an angle of with the ceiling.

Cable 2 has tension and makes an angle of 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 , , and , as well as the magnitude of the

acceleration due to gravity .

=

Correct


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Pro7le% 8"!!Harvaran Ghai

Part A

An astronaut's weight on earth is 805 . What is his weight on Mars, where

309 N

Correct

Pro7le% 8"!6

A woman has a mass of .

Harvaran GhaiPart AWhat is her weight on earth?

539 N

Correct

Part B

What is her mass on the moon, where

55.0 kg

Correct

Part CWhat is her weight on the moon?

89.1 N

Correct

Pro7le% 8"!9

The figure shows the velocity graph of a passenger in an elevator.

http://void%280%29/









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

P+shin( a Chair alon( the Floor

A chair of weight 120 lies atop a horizontal floor; the floor is not frictionless. You push on the chair with a force

of = 35.0 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.


=143

Correct






Board P+lled :+t fro% +nder a Bo;

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 for the

magnitude of the friction force between the board and the box.

Harvaran GhaiPart 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 , , , , and . Do not include in your

answer.

=

Correct

Friction Force on a Dancer on a Draw7rid(eA 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




http://void%280%29/





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 GhaiPart ABefore 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 .

=0Correct

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 of the frictional force now?

Express your answer in terms of some or all of the variables , , and/or . The angle should not appear

in your answer.

=

Correct



http://void%280%29/

http://void%280%29/

http://void%280%29/



Part CThen, 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 , , and/or . The angle should not appearin your answer.

=

Correct

Part DThe 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 now?

Express your answer in terms of some or all of the variables , , , , and/or .

=

Correct

kydivin(

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 AAt 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 directeddownward.

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 BAt 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?


=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 is about 0.250 .

Using this value for , 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 DWhen 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 --3000 .) Immediately

after deploying the parachute, does the skydiver have a nonzero acceleration?

No; the sky diver keeps falling at constant

speed.

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 acting on the sky diver immediately after she has opened the parachute?


=1.88×105

Correct

Part F

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 :7<ect Acceleratin( on a Ra%,

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 and is another vector that can lie in any direction. If it is longer but

in the same direction, will be parallel to . On the other hand, if has the same magnitude as







but is in a slightly different direction, then will be perpendicular to . In general, can differ from

in both magnitude and direction, hence can have any direction relative to .

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 questionsconcern 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 GhaiPart A

Which direction best approximates the direction of when the object is at position 1?

straight up

downward to the left

downward to the

right

straight down

Correct

Part B


straight up

upward to the right


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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


upward to the right

to the right

straight down

downward to the

right

Correct

Pro7le% =">

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 = 4.50 ?

12.6 m/s

Correct

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 = 4.50 ?

11.4 counterclockwise from the +x axis

Correct

Pro<ectile Motion T+torial

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 it takes the projectile to reach its maximum height.

Express in terms of , , and (the magnitude of the acceleration due to gravity).

=

Correct


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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 (often called the range) traveled in the x direction; in other words, find where the projectile

lands.

Express the range in terms of , , and .

=

Correct

The actual formula for is less important than how it is obtained:

1. Consider the x and y motion separately.

2. Find the time of flight from the y-motion

3. 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.

Hori'ontal Cannon on a Cliff

A cannonball is fired horizontally from the top of a cliff. The cannon is at height above ground level, and the

ball is fired with initial horizontal speed .








Harvaran Ghai

Part A

Assume that the cannon is fired at and that the cannonball hits the ground at time . What is the y

position of the cannonball at the time ?

Express the y position of the cannonball in terms of . The quantities and should not appear in your

answer.

=

Correct

Part B

Given that the projectile lands a distance from the cliff, as shown, find the initial speed of the projectile, .

Express the initial speed in terms of , , and .

=

Correct

Part C

What is the y position of the cannonball when it is a distance from the hill?

Express the position of the cannonball in terms of only.

=

Correct




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Pro7le% ="!>

A boat takes 3.70 to travel 15.0 down a river, then 5.30 to return.

Harvaran Ghai

Part AHow fast is the river flowing?

0.612 km/h

Correct

Crossin( a River

A swimmer wants to cross a river, from point A to point B. The distance (from A to C) is 200 , the

distance (from C to B) is 150 , and the speed of the current in the river is 5 . Suppose that the

swimmer makes an angle of (0.785 ) with respect to the line from A to C, as indicated in

the figure.

Harvaran GhaiPart 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

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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.

,eed of a B+llet

A bullet is shot through two cardboard disks attached a distance apart to a shaft turning with a rotational period

, as shown.

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 thatthe bullet must travel through the set of disks within a single revolution.

=

Correct


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Direction of Acceleration of Pend+l+%

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


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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 C+rve? and Flat C+rve 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.

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Harvaran Ghai

Part AWhat 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 , , , 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 , , , as well as the magnitude of the


=

Correct

At the Test TrackYou 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 Pend+l+% )

A bob of mass is suspended from a fixed point with a massless string of length (i.e., it is a pendulum). You

are to investigate the motion in which the string moves in a cone with half-angle .

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 , , and , as well as the acceleration due to

gravity .

=

Correct


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Part BHow 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 , , and , as well as the acceleration due to

gravity .

Correct

Pro7le% @"!

Harvaran Ghai

Part AWhat is the acceleration due to gravity of the sun at the distance of the earth's orbit?

6.00×10−3

Correct

Pro7le% @"!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 AWhat is the car's speed at the bottom of the dip?

14.0

Correct

Pro7le% @"6>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 BWhat is the tangential component of the acceleration at that point?

0.296

Correct

A atellite in :r7it

A satellite used in a cellular telephone network has a mass of 1850 and is in a circular orbit at a height of700 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 , the mass of the earth to be = 5.97×1024 ,

and the radius of the Earth to be = 6.38×

10

6

.


=1.47×104

Correct

Part BWhat 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 from the center of the planet, where . (The object is

located inside of the planet.)

Harvaran GhaiPart A

Find an expression for the magnitude of the acceleration due to gravity, , inside the planet.

Express the acceleration due to gravity in terms of , , , and , the universal gravitational constant.

=

Correct

Part B

Rewrite your result for in terms of , the gravitational acceleration at the surface of the planet, times a

function of R.


=

Correct

Notice that increases linearly with , rather than being proportional to . This assures that it is zero at the

center of the planet, as required by symmetry.

Part C

Find a numerical value for , the average density of the earth in kilograms per cubic meter. Use for

the radius of the earth, , and a value of at the surface of .

Calculate your answer to four significant digits.

=5497







Correct

1ewton4s >rd 0aw Disc+ssed

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) causeforces 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 GhaiPart AEvery 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 DThe 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 EThe two forces of a 3rd law pair always act on different bodies.

true

false

Correct

Part FGiven 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 isgreater 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 Ta7le

A book weighing 5 N rests on top of a table.

Harvaran Ghai

Part AA downward force of magnitude 5 N is exerted on the book by the force of

the table

gravity

inertia

.

Correct

Part BAn upward force of magnitude _____ is exerted on the _____ by the table.

5 N / book

Correct

Part CDo the downward force in Part A and the upward force in Part B constitute a 3rd law pair?

yes

no

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Correct

Part DThe 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 EThe 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 FWhich 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 GWhich 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 )ncline Ad<acent to a -all

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.

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Harvaran GhaiPart A

Find the magnitude, , of the sum of all forces acting on the block.

Express in terms of and , along with any necessary constants.

=

Correct

Part B

Find the magnitude, , of the force that the wall exerts on the wedge.Express in terms of and , along with any necessary constants.

=

Correct

Your answer to Part B could be expressed as either or . In either form, we see that

as gets very small or as approaches 90 degrees ( 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.

P "!# Dashin( +, the lo,e







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 accelerationup 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 and , that act independently on more than one

system.

SOLVE: 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 AWhich of the following objects qualify as systems in this problem?

A. the slope

B. the girl

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C. the earth

D. the sled

E. the air

List alphabetically all the letters corresponding to the systems. Do not use commas. For instance, if you thinkthe 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 BWhich of the following simplifying assumptions are reasonable?

A. The air resistance acting on the girl is negligible.B. The air resistance acting on the girl equals the force of friction acting on her.

C. The air resistance acting on the sled is negligible.

D. The normal force acting on the sled is negligible.

E. The weight of the sled is a constant.

F. 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.

ACECorrect

Part CWhich of the following simplifying assumptions are reasonable?

A. The rope connecting the sled and the girl is massless.

B. The rope connecting the sled and the girl is unstretchable.

C. The tension in the rope connecting the sled and the girl is zero.

D. The sled has the same acceleration as the girl.

E. The sled has greater acceleration than the girl.

F. 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 DWhich 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.

ab

c

d

e

Correct

Part EWhich 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 worryabout relative magnitudes at this point.



a

b

c

d

e

Correct

Part FWhich 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 GYou 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. The weight of the girl and the normal force on the girl

B. The weight of the sled and the normal force on the sled

C. The weight of the girl and the weight of the sled

D. The force of friction on the girl and the tension of the string

E. The weight of the sled and the tension of the string

F. 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 usecommas. 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 )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 *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 /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.

ACECorrect

Note that trigonometric functions and the coefficient of static friction are dimensionless: They do not affect the

dimension or units of the final answer.

P+llin( Three BlocksThree identical blocks connected by ideal strings are being pulled along a horizontal frictionless surface by a

horizontal force . The magnitude of the tension in the string between blocks B and C is . Assume that each

block has mass .

Harvaran GhaiPart A

What is the magnitude of the force?

Express the magnitude of the force in terms of .

=

Correct


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/inetic Friction in a Block2and2P+lley yste%

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 arenegligible.

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 P+lley? and an )nclined 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 .

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Harvaran GhaiPart A

Find the ratio of the masses .

Express your answer in terms of some or all of the variables , , and , as well as the magnitude of the


=

Correct

The )%,+lse2Mo%ent+% Theore%

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 and , then integration of both sides of equation

(1) gives

. (2)






The right side of equation (2) is simply the change in the object's momentum . The left side is called the

impulse of the net force and is denoted by . 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 acting along the

direction of motion, the impulse-momentum theorem can be written as

. (3)

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 . What is the magnitude of the momentum of the particle at time ? Assume

that .

Express your answer in terms of any or all of , , and .

=

Correct

Part B

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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 , the magnitude of its momentum

seconds later?

Express your answer in terms of any or all of , , , and .

=

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 delivered to the particle.

Express your answer in terms of 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 , how long does it take the particle to acquire its final

velocity, 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 waseither irrelevant to the solution or was considered constant. Let us now consider an example of a varying force

acting on a particle.

Part )

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

Pro7le% "!!

A 500 air-track glider collides with a spring at one end of the track. The figure shows the glider's velocity

and the force exerted on the glider by the spring.

Harvaran Ghai

Part AHow long is the glider in contact with the spring?

0.167 s

Correct

Fillin( the Boat

A boat of mass 250 is coasting, with its engine in neutral, through the water at speed 3.00 when it starts to

rain. The rain is falling vertically, and it accumulates in the boat at the rate of 10.0 .

Harvaran GhaiPart A

What is the speed of the boat after time 2.00 has passed? Assume that the water resistance is negligible.

Express your answer in meters per second.

2.78

Correct




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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?

yesno

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

P "!# 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.

VISUALIZE: Draw a before-and-after pictorial representation. Define symbols that will be used in the problem, listknown values, and identify what you are trying to find.

SOLVE: The mathematical representation is based on the law of conservation of momentum: . In component

form, this is

,

ASSESS: Check if your result has the correct units, is reasonable, and answers the question.

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 AIn 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 BWhich 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 CAfter 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

AssessWhen 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 FIntuitively, 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

Collidin( 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 , that is, the speed of the two-car unit after the collision.

Express in terms of , , and the cars' initial speeds and .

=

Correct

Part B

Find the tangent of the angle .


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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 . This means that before the collision:

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 An(le

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 south of east (as indicated in the figure). After the

collision, the two-car system travels at speed 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 .

=

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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 Tra%,oline

A girl of mass kilograms springs from a trampoline with an initial upward velocity of meters per

second. At height meters above the trampoline, the girl grabs a box of mass kilograms.

For this problem, use meters per second per second for the magnitude of the acceleration due to gravity.

Harvaran GhaiPart 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


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Part B

What is the speed of the girl immediately after she grabs the box?

Express your answer numerically in meters per second.

=3.98Correct

Part CIs 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 with respect to the top of the

trampoline.

Express your answer numerically in meters.

=2.81

Correct

Circlin( 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 .

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Harvaran GhaiPart 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.

B+n(ee *+%,in(

Kate, a bungee jumper, wants to jump off the edge of a bridge that spans a river below. Kate has a mass , and the

surface of the bridge is a height above the water. The bungee cord, which has length when unstretched, will first

straighten and then stretch as Kate falls.

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 GhaiPart 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 BIf 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

Dancin( 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

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Part AAs the balls reach their maximum displacement, their kinetic energy reaches __________.

a maximum

zero

neither a maximum nor

zeroCorrect

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 of any one of the balls from its initial position.

Express in terms of some or all of the given quantities , , and .

=

Correct

Elastic Collision in :ne Di%ension

Block 1, of mass , moves across a frictionless surface with speed . It collides elastically with block 2, of mass

, which is at rest ( ). After the collision, block 1 moves with speed , while block 2 moves with

speed . Assume that , so that after the collision, the two objects move off in the direction of the first

object before the collision.

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Harvaran GhaiPart AThis 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 of block 1?

Express in terms of , , and .

=

Correct

Part C

What is the final speed of block 2?


=

Correct

inkin( the 2Ball








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.

Ener(y in a ,rin( Gra,hin( 5+estion

A toy car is held at rest against a compressed spring, as shown in the figure. When released, the car slides

across the room. Let be the initial position of the car. Assume that friction is negligible.

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Harvaran GhaiPart ASketch 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 BSketch 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 CSketch 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

Gra,hin( Gravitational Potential Ener(y

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 ballhad a mass of 2.00 . The graph for a 1.00- ball with an arbitrary initial velocity is provided again as a

reference.

Take 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.

3nderstandin( -ork and /inetic Ener(y

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 reasonableto 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 along a rough,

horizontal surface. The sled is speeding up.

Part AHow many forces are acting on the sled?

one

two

three

four

Correct

Part BThe work done on the sled by the force of gravity is __________.

zero

negative

positive

Correct

Part CThe work done on the sled by the normal force is __________.

zero

negative

positive

Correct

Part DThe work done on the sled by the pulling force is __________.

zero

negative

positive

Correct

Part EThe work done on the sled by the force of friction is __________.

zero

negative

positive



Correct

Part FThe net work done on the sled is __________.

zero

negative

positive

Correct

Part GIn 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 and the magnitude of the net force

acting on the sled be . 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.


=

Correct

Part )

Find the net work done on the sled.

Express your answer in terms of some or all of the variables and .

=

Correct






Part *

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 /

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





-ork Done 7y a ,rin(

Consider a spring, with spring constant , one end of which is attached to a wall. The spring is initially

unstretched, with the unconstrained end of the spring at position .

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

-ork fro% 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 is

independent of . This is the most frequently encountered situation in elementary Newtonian mechanics.

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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 , and the displacement vector from point B to point

A is (of magnitude , 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 , , and . Remember to use radians, not degrees, for any angles that appear

in your answer.

=

Correct

This result is worth remembering! The work done by a constant force of magnitude , which acts at an angle of

with respect to the direction of motion along a straight path of length , 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 acting on a particle that travels from point A to point B. The displacement

vector now points in the opposite direction as it did in Part A. Find the work done by in this case.





=

Correct

The -ork Done in P+llin( a +,ertanker

Two tugboats pull a disabled supertanker. Each tug exerts a constant force of 1.60×106 , 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 toward the

north.

Harvaran Ghai

Part AWhat is the total work done by the two tugboats on the supertanker?

Express your answer in joules.

2.76×109

Correct

-ork on a lidin( Block

A block of weight sits on a frictionless inclined plane, which makes an angle with respect to the horizontal, as

shown. A force of magnitude , applied parallel to the incline, pulls the block up the plane at constant speed .

Harvaran GhaiPart A

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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.

=0Correct

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 as the block moves a distance up the incline?

Express your answer in terms of 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 up the inclined

plane?

Express your answer in terms of given quantities.

=0

Correct

Potential Ener(y Gra,hs 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 ( ) energies of the particle.

The key idea in interpreting the graph can be expressed in the equation

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.

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Harvaran GhaiPart AThe 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 is negative.

Part BThe force acting on the particle at point C is __________.

directed to the

right


equal to zero

Correct

Part CThe force acting on the particle at point B is __________.

directed to theright


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


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 EIf the particle is located slightly to the left of point B, its acceleration is __________.

directed to theright


equal to zero

Correct

Part FIf the particle is located slightly to the right of point B, its acceleration is __________.

directed to the

right


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 GName 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 HName all labeled points on the graph corresponding to stable equilibrium.



DH

Correct

Part )Name all labeled points on the graph where the acceleration of the particle is zero.



BDFH

Correct







Your answer, of course, includes the locations of both stable and unstable equilibrium.

Part *

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,


AE

Correct

Part /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 to draw conclusions about the

kinetic energy of the particle since

.

As a reminder, on this graph, the total energy is shown by the horizontal line.

Part 0What 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 MAt 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 Ener(y Calc+lations

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, is the net force acting

on the particle at a given point of its path, and 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. Because the result is path-independent , it is always possible to consider the most straightforward way to

reach point 2 from point 1.

2. 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 AConsider a uniform gravitational field (a fair approximation near the surface of a planet). Find

,

where

and .

Express your answer in terms of , , , and .

=

Correct

Part BConsider the force exerted by a spring that obeys Hooke's law. Find

,

where

,

and the spring constant is positive.


=

Correct

Part CFinally, 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 Mass2,rin( yste% with Recoil and Friction

An object of mass is traveling on a horizontal surface. There is a coefficient of kinetic friction between the

object and the surface. The object has speed when it reaches and encounters a spring. The object compresses

the spring, stops, and then recoils and travels in the opposite direction. When the object reaches on its return

trip, it stops.

Harvaran Ghai

Part A

Find , the spring constant.

Express in terms of , , , and .

=

Correct

Dra((in( a BoardA 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.

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Harvaran GhaiPart 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 BWhat 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

Dra( on a kydiver

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 GhaiPart A







What is the work done on the skydiver, over the distance , by the drag force of the air?

Express the work in terms of , , , 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 Dissi,ation P+ts a Dra( on Racin(

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 . Find the power dissipated by form drag.

Express your answer in terms of , , and speed .

=

Correct

Part B

A certain car has an engine that provides a maximum power . Suppose that the maximum speed of the car, , is

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.

Ener(y of a ,acecraft

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 , where .

Express the speed in terms of and .

=

Correct

:r7itin( atellite










A satellite of mass is in a circular orbit of radius around a spherical planet of radius made of a material with

density . ( is measured from the center of the planet, not its surface.) Use for the universal gravitational

constant.

Harvaran GhaiPart 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 , , , , , and .

=

Correct

Part CWhat is the ratio of the kinetic energy of this satellite to its potential energy?

Express in terms of parameters given in the introduction.



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=-0.500

Correct

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

).

Pro7le% !6"6

The space shuttle is in a 250 -high circular orbit. It needs to reach a 700 -high circular orbit to catch the

Hubble Space Telescope for repairs. The shuttle's mass is 8.00×104 .

Harvaran GhaiPart A

How much energy is required to boost it to the new orbit?1.53×1011 J

Correct

An E;ha+sted 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 ,innin( Grindin( -heel









At time a grinding wheel has an angular velocity of 23.0 . It has a constant angular acceleration of

32.0 until a circuit breaker trips at time = 1.90 . From then on, the wheel turns through an angle of

433 as it coasts to a stop at constant angular deceleration.

Harvaran Ghai

Part A

Through what total angle did the wheel turn between and the time it stopped?

Express your answer in radians.

534

Correct

Part B

At what time does the wheel stop?Express your answer in seconds.

12.2

Correct

Part CWhat was the wheel's angular acceleration as it slowed down?

Express your answer in radians per second per second.

-8.11

Correct

Findin( Tor+e

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 and ,

. The magnitude of the torque is , where is the angle between and ; the direction of

is perpendicular to both and . For this problem ; negative torque about a reference point corresponds








to clockwise rotation. You must express in terms of , , and/or when entering your answers.

Harvaran GhaiPart 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 about the point C, located at a position given by Cartesian coordinates ( , 0), a distance

along the x axis?

Express the torque about point C in terms of , , , , and/or other given coordinate data.

=

Correct



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Part D

What is the torque about the point D, located at a distance from the origin and making an angle with the x

axis?

Express the torque about point D in terms of , , , , and/or other given coordinate data.

=

Correct

Note that the cross product can also be expressed as a third-order determinant

which simplifies to when and lie in the xy plane.

Pivoted Rod with 3ne+al 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 GhaiPart A

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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 , , , and . Remember, the length of the rod is , not .

=

Correct

Part BSuppose 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

P+llin( a trin( to Accelerate a -heel

A bicycle wheel is mounted on a fixed, frictionless axle, as shown . A massless string is wound around the

wheel's rim, and a constant horizontal force of magnitude starts pulling the string from the top of the wheel

starting at time 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.

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Harvaran GhaiPart 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 of the wheel is

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 ; these expressions look very different but are equivalent.

Express the angular velocity 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 over a distance .

Express the speed of the string in terms of , , , and ; do not include , , or in your answer.

=

Correct






Note that this is the speed that an object of mass (which is less than ) would attain if pulled a distance by a

force with constant magnitude .

A Bar +s,ended 7y Two Vertical trin(s

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.


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Express the magnitude of the tension in string B in terms of , , , and .

=

Correct

Part CIf 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 DIf 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 such that the bar remains stable (call it )?

Express your answer for in terms of , , , and .

=

Correct

Part E

Note that since , as computed in the previous part, is not necessarily positive. If , the bar will be

stable no matter where the block of mass is placed on it.

Assuming that , , and are held fixed, what is the maximum block mass for which the bar will always be

stable? In other words, what is the maximum block mass such that ?

Answer in terms of , , and .

=

Correct

A Person tandin( on a 0eanin( 0adder








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, considercounterclockwise 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 required between the ladder and the ground so that the

ladder does not slip?

Express in terms of , , , , and .

=

Correct

Part B

Suppose that the actual coefficent of friction is one and a half times as large as the value of . That is,

. Under these circumstances, what is the magnitude of the force of friction that the floor applies to

the ladder?

Express your answer in terms of , , , , , and . Remember to pay attention to the relation of force

and .

=

Correct



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A Rollin( Hollow ,here

A hollow spherical shell with mass 1.80 rolls without slipping down a slope that makes an angle of 39.0 with

the horizontal.

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 CFind the minimum coefficient of friction needed to prevent the spherical shell from slipping as it rolls down the

slope.

=0.324

Correct

-ei(ht and -heel

Consider a bicycle wheel that initially is not rotating. A block of mass is attached to the wheel and is allowed to

fall a distance . Assume that the wheel has a moment of inertia about its rotation axis.

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Harvaran Ghai

Part AConsider 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 , for this case.


=

Correct

Part BNow consider the case that the string tied to the block is wrapped around a smaller inside axle of the wheel of radius

. Find , the angular speed of the wheel after the block has fallen a distance , for this case.


=

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 T+rnta7le

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 GhaiPart A

What is the final angular velocity, , of the two disks?


=

Correct

Part B


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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 of the two-disk system. No

angular velocities should appear in your answer.

=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

Pro7le% !>"@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 BWhat 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







Analy'in( i%,le Har%onic Motion

This Error! Hyperlink reference not valid. shows two masses on springs, each accompanied by agraph 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

Har%onic :scillator 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 along the x axis. At time ,

the block is released with zero initial velocity.

The goal of this problem is to determine the acceleration of the block as a function of time in terms of , ,

and .









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 , , and in terms of , ,and and then use the

connection between and to find the acceleration.

Harvaran Ghai

Part A

Combine Newton's 2nd law and Hooke's law for a spring to find the acceleration of the block as a function of

time.

Express your answer in terms of , , and the coordinate of the block .

=

Correct

The negative sign in the answer is important: It indicates that the restoring force (the tension of the spring) is alwaysdirected 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 as a

function of time.



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Harvaran GhaiAs 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, , and the amplitude of vibrations, , are given. Answer

the following questions.

Part AWhich moment corresponds to the maximum potential energy of the system?

A

B



C

D

Correct

Part BWhich 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 . At that moment, of

course, . Recall that . Therefore,

.

In general, the mechanical energy of a harmonic oscillator equals its potential energy at the maximum or minimum

displacement.

Part CConsider the block in the process of oscillating.

If the kinetic energy of the block is increasing, the block must

be

at the equilibrium position.

at the amplitude displacement.

moving to the right.

moving to the left.

moving away from

equilibrium.

moving toward equilibrium.

Correct

Part DWhich moment corresponds to the maximum kinetic energy of the system?

A

B





C

D

Correct

Part EWhich 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 ( ). The maximum kinetic energy can

then be written as . Recall that and that at the equilibrium position. Therefore,

.

Recalling what we found out before,

,

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

Ener(y of a ,rin(

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 . The

explorer finds that the pendulum completes 108 full swing cycles in a time of 136 .

Harvaran Ghai

Part AWhat 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 cale

The scale of a spring balance reading from 0 to 205 has a length of 13.5 . A fish hanging from the bottom of

the spring oscillates vertically at a frequency of 2.95 .

Harvaran GhaiPart A

Ignoring the mass of the spring, what is the mass of the fish?

Express your answer in kilograms.

=4.42

Correct

Vertical Mass2and2,rin( :scillator

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 BSuppose 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 of the spring

when the block is hanging: .

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.



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