Assignment 3: Work and EnergyDue: 2:00am on Saturday, September 18, 2010

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Bungee Jumping

Kate, a bungee jumper, wants to jump off the edge of a bridge that spans a river below. Kate has amass , 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 apoint particle.

Use for the magnitude of the acceleration due to gravity.

Part A

How far below the bridge will Kate eventually be hanging, once she stops oscillating and comes finallyto rest? Assume that she doesn't touch the water.

Hint A.1 Decide how to approach the problem

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Hint A.2 Compute the force due to the bungee cord

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Express the distance in terms of quantities given in the problem introduction.

ANSWER:

=

Correct

Part B

If Kate just touches the surface of the river on her first downward trip (i.e., before the first bounce),what is the spring constant ? Ignore all dissipative forces.

Hint B.1 Decide how to approach the problem

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Hint B.2 Find the initial gravitational potential energy

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Hint B.3 Find the elastic potential energy in the bungee cord

Hint B.3 Find the elastic potential energy in the bungee cord

Hint not displayed

Express in terms of , , , and .

ANSWER:

=

Correct

Dancing Balls

Four balls, each of mass , are connected by four identical relaxed springs with spring constant . The

balls are simultaneously given equal initial speeds directed away from the center of symmetry of the

system.

Part A

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

ANSWER:a maximum

zero

neither a maximum nor zero

Correct

Part B

Use geometry to find , the distance each of the springs has stretched from its equilibrium position. (It

may help to draw the initial and the final states of the system.)

Express your answer in terms of , the maximum displacement of each ball from its initial

position.

ANSWER:

=

Correct

Correct

Part C

Find the maximum displacement of any one of the balls from its initial position.

Hint C.1 A useful equation

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

ANSWER:

=

Correct

If You Don't Want to Walk to the Kitchen...

As depicted in the applet, Albertine finds herself in a very odd contraption. She sits in a reclining chair, infront of a large, compressed spring. The spring is compressed 5.00 from its equilibrium position, and

a glass sits 19.8 from her outstretched foot.

Part A

For what value of the spring constant does Albertine just reach the glass without knocking it over?

Determine the answer "experimentally" by playing with the applet.

Express your answer in newtons per meter.

ANSWER: = 95.0

Correct

Part B

Assuming that Albertine's mass is 60.0 , what is , the coefficient of kinetic friction between the

chair and the waxed floor? Use = 9.80 for the magnitude of the acceleration due to gravity.

Assume that the value of found in Part A has three significant figures.

Note that if you did not assume that has three significant figures, it would be impossible to get three

significant figures for , since the length scale along the bottom of the applet does not allow you to

measure distances to that accuracy with different values of .

Hint B.1 How to approach the problem

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Hint B.2 Find Albertine's initial potential energy

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Hint B.3 Find the work done by nonconservative forces

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Express your answer to three significant figures.

ANSWER: = 0.102

Correct

Part C

The principle of conservation of energy states that energy is neither created nor destroyed. Which ofthe following describes the transformation of energy in this problem?

ANSWER:Conservation of energy does not apply to problems involvingnonconservative forces. Thus, the potential energy slowly disappearsduring Albertine's trip.

The potential energy was turned into Albertine's kinetic energy, whichwas then converted into internal (thermal) energy.

The potential energy was turned into Albertine's kinetic energy, which isnow stored in the floor as frictional potential energy.

The potential energy was turned into elastic frictional energy, creatingthe frictional force.

Correct

This applet shows how the energy transforms throughout Albertine's journey. Notice that herkinetic energy is never equal to her initial potential energy, because friction is acting even as thespring expands. Try changing the spring constant and observe how the transformation of energyis affected.

Loop the Loop

A roller coaster car may be approximated by ablock of mass . The car, which starts from

rest, is released at a height above the ground

and slides along a frictionless track. The carencounters a loop of radius , as shown.

Assume that the initial height is great enough

so that the car never loses contact with thetrack.

Part A

Find an expression for the kinetic energy of the car at the top of the loop.

Hint A.1 Find the potential energy at the top of the loop

Hint not displayed

Express the kinetic energy in terms of , , , and .

ANSWER: =

Correct

Part B

Find the minimum initial height at which the car can be released that still allows the car to stay in

contact with the track at the top of the loop.

Hint B.1 How to approach this part

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Hint B.2 Acceleration at the top of the loop

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Hint B.3 Normal force at the top of the loop

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Hint B.4 Solving for

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Express the minimum height in terms of .

ANSWER: =

Correct

For the car will still complete the loop, though it will require some normal reaction

even at the very top.For the car will just oscillate. Do you see this?

For , the cart will lose contact with the track at some earlier point. That is why

roller coasters must have a lot of safety features. If you like, you can check that the angle at

which the cart loses contact with the track is given by .

Work Raising an Elevator

Look at this applet. It shows an elevator with a small initial upward velocity being raised by a cable. Thetension in the cable is constant. The energy bar graphs are marked in intervals of 600 .

Part A

What is the mass of the elevator? Use for the magnitude of the acceleration of

gravity.

Hint A.1 Using the graphs

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Hint A.2 Needed formula

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Express your answer in kilograms to two significant figures.

ANSWER: = 60

Correct

Part B

Find the magnitude of the tension in the cable. Be certain that the method you are using will be

accurate to two significant figures.

Hint B.1 How to approach the problem

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Hint B.2 Find the change in mechanical energy

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Express your answer in newtons to two significant figures.

ANSWER: = 480

Correct

Power Dissipation Puts a Drag on Racing

The dominant form of drag experienced by vehicles (bikes, cars, planes, etc.) at operating speeds iscalled form drag. It increases quadratically with velocity (essentially because the amount of air you runinto 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.

Part A

Consider a vehicle moving with constant velocity . Find the power dissipated by form drag.

Hint A.1 How to approach the problem

Hint not displayed

Express your answer in terms of , , and speed .

ANSWER: =

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?

Hint B.1 Find the relationship between speed and power

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Hint B.2 How is the algebra done?

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Express the percent increase in top speed numerically to two significant figures.

ANSWER: = 3.2

Correct %

You'll note that your answer is very close to one-third of the percentage by which the power wasincreased. This dependence of small changes on each other, when the quantities are related byproportionalities of exponents, is common in physics and often makes a useful shortcut forestimations.

Pulling a Block on an Incline with Friction

A block of weight sits on an inclined plane as shown. A force of magnitude is applied to pull the

block up the incline at constant speed. Thecoefficient of kinetic friction between the planeand the block is .

Part A

What is the total work done on the block by the force of friction as the block moves a distance

up the incline?

Hint A.1 How to start

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Hint A.2 Find the magnitude of the friction force

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Express the work done by friction in terms of any or all of the variables , , , , , and .

ANSWER: =

Correct

Part B

What is the total work done on the block by the applied force as the block moves a distance

up the incline?

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

ANSWER: =

Correct

Now the applied force is changed so that instead of pulling the block up the incline, the force pulls theblock down the incline at a constant speed.

Part C

What is the total work done on the block by the force of friction as the block moves a distance

down the incline?

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

ANSWER: =

Correct

Part D

What is the total work done on the box by the appled force in this case?

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

ANSWER: =

Correct

Dragging a Board

A uniform board of length and mass lies near a boundary that separates two regions. In region 1,

the coefficient of kinetic friction between the board and the surface is , and in region 2, the coefficient

is . The positive direction is shown in the figure.

Part A

Find the net work done by friction in pulling the board directly from region 1 to region 2. Assume

that the board moves at constant velocity.

Hint A.1 The net force of friction

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Hint A.2 Work as integral of force

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Hint A.3 Direction of force of friction

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Hint A.4

Hint A.4 Formula for

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Express the net work in terms of , , , , and .

ANSWER:

=

Correct

This answer makes sense because it is as if the board spent half its time in region 1, and half inregion 2, which on average, it in fact did.

Part B

What is the total work done by the external force in pulling the board from region 1 to region 2?(Again, assume that the board moves at constant velocity.)

Hint B.1 No acceleration

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

ANSWER:

=

Correct

Circling Ball

A ball of mass is attached to a string of length . It is being swung in a vertical circle with enough

speed so that the string remains taut throughout the ball's motion. Assume that the ball travels freely inthis vertical circle with negligible loss of totalmechanical energy. At the top and bottom of thevertical circle, the ball's speeds are and ,

and the corresponding tensions in the string are and . and have magnitudes and

.

Part A

Find , the difference between the magnitude of the tension in the string at the bottom relative

to that at the top of the circle.

to that at the top of the circle.

Hint A.1 How to approach this problem

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Hint A.2 Find the sum of forces at the bottom of the circle

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Hint A.3 Find the acceleration at the bottom of the circle

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Hint A.4 Find the tension at the bottom of the circle

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Hint A.5 Find the sum of forces at the top of the circle

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Hint A.6 Find the acceleration at the top of the circle

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Hint A.7 Find the tension at the top of the circle

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Hint A.8 Find the relationship between and

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Express the difference in tension in terms of and . The quantities and should not

appear in your final answer.

ANSWER: =

Correct

The method outlined in the hints is really the only practical way to do this problem. If doneproperly, finding the difference between the tensions, , can be accomplished fairly simply

and elegantly.

Drag on a Skydiver

A skydiver of mass jumps from a hot air balloon and falls a distance before reaching a terminal

velocity of magnitude . Assume that the magnitude of the acceleration due to gravity is .

Part A

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

Hint A.1 How to approach the problem

Hint A.1 How to approach the problem

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Hint A.2 Find the change in potential energy

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Hint A.3 Find the change in kinetic energy

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

ANSWER:

=

Correct

Part B

Find the power supplied by the drag force after the skydiver has reached terminal velocity .

Hint B.1 How to approach the problem

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Hint B.2 Magnitude of the drag force

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Hint B.3 Relative direction of the drag force and velocity

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Express the power in terms of quantities given in the problem introduction.

ANSWER: =

Correct

Shooting a ball into a box

Two children are trying to shoot a marble of mass into a small box using a spring-loaded gun that is

fixed on a table and shoots horizontally from the edge of the table. The edge of the table is a height

above the top of the box (the height of which isnegligibly small), and the center of the box is adistance from the edge of the table. The

spring has a spring constant . The first child

compresses the spring a distance and finds

that the marble falls short of its target by ahorizontal distance .

Part A

By what distance, , should the second child compress the spring so that the marble lands in the

middle of the box? (Assume that height of the box is negligible, so that there is no chance that themarble will hit the side of the box before it lands in the bottom.)

Hint A.1 General method for finding

For this part of the problem, you don't need to consider the first child's toss. (The quantities and

should not appear in your answer.) Consider the energy conservation and kinematic relations for

the marble, and solve for its range, , in terms of , , , and .

Hint A.2 Initial speed of the marble

Use conservation of energy to find the initial speed, , of the second marble.

Express your answer in terms of , , and .

ANSWER:

=

Correct

Hint A.3 Time for the marble to hit the ground

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Hint A.4 Combining equations and solving for

Hint not displayed

Express the distance in terms of , , , , and .

ANSWER:

=

Correct

Part B

Now imagine that the second child does not know the mass of the marble, the height of the tableabove the floor, or the spring constant. Find an expression for that depends only on and

distance measurements.

Hint B.1 Compute

Hint not displayed

Hint not displayed

Express in terms of , , and .

ANSWER:

=

Correct

Score Summary:

Your score on this assignment is 96.9%.You received 38.74 out of a possible total of 40 points.