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Course TUPH1061F07

Homework Ch. 7 Due 4 Oct.

Due at 12:00pm on Wednesday, October 10, 2007

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Bungee Jumping

Description: Determine the spring constant of a bungee cord using Newtons 2nd Law. Then compute length of cord needed

to avoid hitting the river below using conservation of energy.

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.

Part A

How far below the bridge will Kate eventually be hanging, once she stops oscillating and comes finally to rest? Assume that

she doesn't touch the water.

Part A.1 Decide how to approach the problem

Here are three possible methods for solving this problem:

No nonconservative forces are acting, so mechanical energy is conserved. Set Kate's gravitational potential energy at

the top of the bridge equal to the spring potential energy in the bungee cord (which depends on the cord's final length

) and solve for .

a.

Since nonconservative forces are acting, mechanical energy is not conserved. Set the spring potential energy in the

bungee cord (which depends on ) equal to Kate's gravitational potential energy plus the work done by dissipative

forces. Eliminate the unknown work, and solve for .

b.

When Kate comes to rest she has zero acceleration, so the net force acting on her must be zero. Set the spring force

due to the bungee cord (which depends on ) equal to the force of gravity and solve for .

c.

Which of these options is the simplest, most accurate way to find given the information available?

ANSWER:a b c

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

When Kate is at rest, what is the magnitude of the upward force the bungee cord exerts on her?

Part A.2.a Find the extension of the bungee cord

The upward force on Kate is due to the extension of the bungee cord. What is this extension?

Express your answer in terms of the cord's final (stretched) length and .

ANSWER: Extension =

Hint A.2.b Formula for the force due to a stretched cord

The formula for the force due to a stretched cord is

,

[ Print ]


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where is the spring constant of the cord and is the extension of the cord.

Express your answer in terms of the cord's final stretched length and quantities given in the problem introduction.

Your answer should not depend on Kate's mass .

ANSWER: =

Set this force equal to Kate's weight, and solve for .

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

ANSWER: =

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.

Part B.1 Decide how to approach the problem

Here are three possible methods for solving this problem:

Since nonconservative forces are ignored, mechanical energy is conserved. Set Kate's gravitational potential energy at

the top of the bridge equal to the spring potential energy in the bungee cord at the lowest point (which depends on )

and solve for .

a.

Nonconservative forces can be ignored, so mechanical energy is conserved. Set the spring potential energy in the

bungee cord (which depends on ) equal to Kate's gravitational potential energy at the top of the bridge plus the work

done by gravity as Kate falls. Compute the work done by gravity, then solve for .

b.

When Kate is being held just above the water she has zero acceleration, so the net force acting on her must be zero.

Set the spring force due to the bungee cord (which depends on ) equal to the force of gravity and solve for .

c.

Which of these options is the simplest, most accurate way to find given the information available?

ANSWER: a b c

Part B.2 Find the initial gravitational potential energy

What is Kate's gravitational potential energy at the moment she steps off the bridge? (Define the zero of gravitational

potential to be at the surface of the water.)

Express your answer in terms of quantities given in the problem introduction.

ANSWER: =

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


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What is the elastic potential energy stored in the bungee cord when Kate is at the lowest point of her first downward

trip?

Hint B.3.a Formula for elastic potential energy

The elastic potential energy of the bungee cord (which we are treating as an ideal spring) is

,

where is the amount by which the cord is stretched beyond its unstretched length.

Part B.3.b How much is the bungee cord stretched?

By how much is the bungee cord stretched when Kate is at a depth below the bridge?

Express your answer in terms of and .

ANSWER: =

Express your answer in terms of quantities given in the problem introduction.

ANSWER: =

Express in terms of , , , and .

ANSWER: =

Energy Required to Lift a Heavy Box

Description: Find force needed to lift a box with a single pulley. Then compare energy needed to lift box directly, versus

with a pulley.

As you are trying to move a heavy box of mass , you realize that it is too heavy for you to lift by yourself. There is no one

around to help, so you attach an ideal pulley to the box and a massless rope to the ceiling, which you wrap around the pulley.

You pull up on the rope to lift the box.

Use for the magnitude of the acceleration due to gravity and neglect

friction forces.


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Part A

What is the magnitude of the upward force you must apply to the rope to start raising the box with constant velocity?

Part A.1 What force must be applied to the box to keep it moving at a constant speed?

What is the magnitude of the force that the pulley must exert on the box so that it moves at a constant speed?

Express your answer in terms of the mass of the box.

ANSWER: =

Part A.2 What force does the pulley exert on the box?

If you take the tension in the rope to be , what is , the magnitude of the net upward force that the pulley exerts on the

box?

Express your answer in terms of .

ANSWER: =

Part A.3 Find the tension in the rope

Find the tension in the rope in terms of , the force with which you are pulling upward.

ANSWER: =

Hint A.4 Putting it all together

On your own or using the previous hints, you should have found equations for he following:

the force needed to lift the box in terms of its mass,I.

the relationship between the force on the box due to the pulley and the tension in the rope, andII.

the relationship between the force applied to the rope and the tension in the rope.III.

Use two of these equations to eliminate the force applied by the pulley and the tension in the rope. You should then be able

to express the force applied on the rope in terms of the mass of the box.

Express the magnitude of the force in terms of , the mass of the box.

ANSWER:

=

Part B

Consider lifting a box of mass to a height using two different methods: lifting the box directly or lifting the box using a

pulley (as in the previous part).

What is , the ratio of the work done lifting the box directly to the work done lifting the box with a pulley?

Hint B.1 Definition of work


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In each case, the amount of work you do is equal to the force you apply times the distance over which you apply the

force:

.

Part B.2 Ratio of the forces

What is the ratio of the force needed to lift the box directly to the force needed to lift the box using the pulley?

Express your answer numerically.

ANSWER: =

2

Part B.3 Ratio of the distances

What is the ratio of the distance over which force is applied when lifting the box directly to the distance over which force is

applied when lifting the box with the pulley?

Part B.3.a Find the distance when using the pulley

Find , the distance over which you must apply force when lifting the box using the pulley.

Hint B.3.a.i Pulling the rope a short distance

Using the pully, imagine that you pull the end of the rope a short distance upward. The box will actually rise a

distance . (Draw a picture if you have trouble visualizing this.)

Express your answer in terms of , the total height that the box is lifted.

ANSWER: =

Hint B.3.b Find the distance when lifting directly

When lifting the box directly, the distance over which force is applied, , is equal to the vertical distance that the box

is raised.

Express the ratio of distances numerically.

ANSWER: =

0.500

Express the ratio numerically.

ANSWER: =

1

No matter which method you use to lift the box, its gravitational potential energy will increase by . So, neglecting

friction, you will always need to do an amount of work equal to to lift it.


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Fun with a Spring Gun

Description: A ball is launched vertically from a spring gun. Use conservation of energy to compute the velocity of the ball

as a function of height, and to compute the maximum height reached by the ball.

A spring-loaded toy gun is used to shoot a ball of mass straight up in the air, as shown in the figure. The spring

has spring constant . If the spring is compressed a distance of

25.0 centimeters from its equilibrium position and then released, the

ball reaches a maximum height (measured from the equilibrium

position of the spring). There is no air resistance, and the ball never touches

the inside of the gun. Assume that all movement occurs in a straight line up

and down along the y axis.

Part A

Which of the following statements are true?

Mechanical energy is conserved because no dissipative forces perform work on the ball.A.

The forces of gravity and the spring have potential energies associated with them.B.

No conservative forces act in this problem after the ball is released from the spring gun.C.

Hint A.1 Nonconservative forces

Dissipative, or nonconservative, forces are those that always oppose the motion of the object on which they act. Forces such

as friction and drag are dissipative forces.

Hint A.2 Forces acting on the ball

The ball is acted on by the spring force only when the two are in contact. The force of tension in the spring is a conservative

force. Also, the ball is always acted on by gravity, which is also a conservative, or nondissipative, force.

Enter the letter(s) of the correct statements in alphabetical order. For example, if A and C are correct, enter AC.

ANSWER: AB

Part B

Find the muzzle velocity of the ball (i.e., the velocity of the ball at the spring's equilibrium position ).

Part B.1 Determine how to approach the problem

What physical relationship can you use to solve this problem? Choose the best answer.

ANSWER: kinematics equations Newton's second law law of conservation of energy conservation of

momentum

Note that the law of conservation of energy applies to closed systems. In this case, such a closed system consists of the

ball and the spring (and, technically, the Earth, but we will follow the traditional, somewhat imprecise, language and

will assume that it is the ball that has gravitaitonal potential energy, not the system "ball-Earth.")


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Hint B.2 Energy equations

Recall that kinetic energy is given by the equation

,

where is the speed of the object and is the object's mass.

Gravitational potential energy is given by

,

where is the object's height measured from .

The elastic potential energy of a spring is given by

,

where is the spring constant and is the spring's displacement from equilibrium.

Part B.3 Determine which two locations you should examine

Pick the two points along the ball's path that would be most useful to compare in order to find the solution to this problem.

Choose from among the following three points:

, the location of the ball when the spring is compressed.A.

, the equilibrium position of the spring.B.

, the maximum height that the ball reaches above the point .C.

Enter the letters that correspond to the two points in alphabetical order.

ANSWER: AB

Because you do not know enough information about the ball at , you need to compare the energy at

to the energy at to find .

Part B.4 Find the initial energy of the system

A useful statement of mechanical energy conservation relating the initial and final kinetic ( ) and potential ( ) energies is

.

In this situation, the initial position is and the final position is , which is the equilibrium position of the

spring. What kind(s) of energy does the system "spring-ball" have at the initial position?

ANSWER: kinetic only elastic potential only gravitational potential only kinetic and gravitational potential

kinetic and elastic potential elastic and gravitational potentials

Keep in mind that at the equilibrium position of the spring. The inital position defined at will have

negative gravitational potential energy.


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Part B.5 Determine the final energy

A useful statement of mechanical energy conservation relating the initial and final kinetic ( ) and potential ( ) energies is

.

In this situation, the initial position is and the final position is , which is the equilibrium position of the

spring. What kind(s) of energy does the system "spring-ball" have at the final position?



Hint B.6 Creating an equation

From the hints you now know what kinds of energy are present at the initial and final positions chosen for the ball in this

part of the problem. You also know that

.

It has been determined that is zero and consists of two terms: gravitational potential energy and elastic

potential energy. In addition, is zero.

ANSWER: =

Part C

Find the maximum height of the ball.

Part C.1 Choose two locations to examine

Pick the two points along the ball's movement that would be most useful to compare in order to find a solution to this

problem. Choose from among the following three points:

, the location of the ball when the spring is compressed.A.

, the equilibrium position of the spring.B.

, the maximum height that the ball reaches measured from .C.

Enter the letters that correspond to the two points in alphabetical order.

ANSWER: AC

BC

You could compare to either or . It is probably most convenient to use for

comparison because using requires that you know the energy at the equilibrium position of the spring. Of course,

you do know it, as long as you got that part of the problem correct. For the remainder of the problem, we will use

and .

Part C.2 Find the initial energy


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A useful statement of mechanical energy conservation is

.

Recall that in the problem statement, is set to correspond to the equilibrium position of the spring. Therefore, in this

situation, the initial location is at and the final position should be taken as .

What kind(s) of energy does the ball have at the initial location?



Part C.3 Determine the final energy

A useful statement of mechanical energy conservation is

.

In this situation, the initial location is at , and the final position should be taken as . What kind(s) of

energy does the ball have at ?

Part C.3.a Find the speed of the ball at the top of its trajectory

What is the speed of the ball at the top of its trajectory?

Hint C.3.a.i Motion in the vertical direction

Recall from kinematics that a ball travels upward until its speed decreases to zero, at which point it starts falling back to

Earth.

Express your answer numerically, in meters per second.

ANSWER: =

Recall that . Because the ball has zero veolicty at the peak of its trajectory, it has no kinetic energy.



Hint C.4 Creating an equation

From the above hints, you now know what kind of energy is present at the inital and final positions chosen for the ball in

this part of the problem. You know that

.

It was determined that is zero and that consists of two terms: gravitational potential energy and elastic

potential energy. In addition, is zero.

Express youramswer numerically, in meters.


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ANSWER: =

In this problem you practiced applying the law of conservation of mechanical energy to a physical situation to find the

muzzle velocity and the maximum height reached by the ball.

Part D

Which of the following actions, if done independently, would increase the maximum height reached by the ball?

reducing the spring constant A.

increasing the spring constant B.

decreasing the distance the spring is compressedC.

increasing the distance the spring is compressedD.

decreasing the mass of the ballE.

increasing the mass of the ballF.

tilting the spring gun so that it is at an angle degrees from the horizontalG.

Enter the letters that correspond to the correct answer(s) in alphabetical order.

ANSWER: BDE

Spring and Projectile

Description: A spring gun is used to launch a ball off of a table with a ramp at the edge. Problem contains a number of

multiple choice questions about changes in kinetic and potential energy (including an energy diagram), then asks for analytic

expressions for the speed of the ball when it leaves the ramp and when it hits the floor.

A child's toy consists of a block that attaches to a table with a suction cup, a spring connected to that block, a ball, and a

launching ramp. The spring has a spring constant , the ball has a mass , and the ramp rises a height above the table, the

surface of which is a height above the floor.

Initially, the spring rests at its equilibrium length. The spring then is

compressed a distance , where the ball is held at rest. The ball is then

released, launching it up the ramp. When the ball leaves the launching ramp

its velocity vector makes an angle with respect to the horizontal.

Throughout this problem, ignore friction and air resistance.

Part A

Relative to the initial configuration (with the spring relaxed), when the spring has been compressed, the ball-spring system

has

ANSWER: gained kinetic energy

gained potential energy

lost kinetic energy


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Part B

As the spring expands (after the ball is released) the ball-spring system

ANSWER: gains kinetic energy and loses potential energy

gains kinetic energy and gains potential energy

loses kinetic energy and gains potential energy

loses kinetic energy and loses potential energy

Part C

As the ball goes up the ramp, it





Part D

As the ball falls to the floor (after having reached its maximum height), it





Part E

Which of the graphs shown best represents the potential energy of the ball-spring system as a function of the ball's horizontal

displacement? Take the "zero" on the distance axis to represent the point at which the spring is fully compressed. Keep in

mind that the ball is not attached to the spring, and neglect any recoil of

the spring after the ball loses contact with it.

ANSWER: CC

Part F

Calculate , the speed of the ball when it leaves the launching ramp.


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Hint F.1 General approach

Find an expression for the mechanical energy (kinetic plus potential) of the spring and ball when the spring is compressed.

Then find an expression for the mechanical energy of the ball when it leaves the launching ramp. ( will be an unknown in

this expression.) Since energy is conserved, you can set these two expressions equal to each other, and solve for .

Part F.2 Find the initial mechanical energy

Find the total mechanical energy of the ball-spring system when the spring is fully compressed. Take the gravitational

potential energy to be zero at the floor.

Hint F.2.a What contributes to the mechanical energy?

The total initial mechanical energy is the sum of the potential energy of the spring, the gravitational potential energy, plus

any initial kinetic energy of the ball.

ANSWER: =

Part F.3 Find the mechanical energy at the end of the ramp

Find the total mechanical energy of the ball when it leaves the launching ramp. (At this point, assume that the spring is

relaxed and has no stored potential energy.) Again, take the gravitational potential energy to be zero at the floor.

Express your answer in terms of and other given quantities.

ANSWER: =

Hint F.4 Is energy conserved?

Because no nonconservative forces act on the system, energy is conserved:

Express the speed of the ball in terms of , , , , , and/or .

ANSWER:

=

Part G

With what speed will the ball hit the floor?

Hint G.1 General approach

Find an expression for the initial mechanical energy (kinetic plus potential) of the spring and ball. Then find an expression

for the mechanical energy of the ball when it hits the floor. ( will be an unknown in this expression.) Since energy is

conserved, you can set these two expressions equal to each other, and solve for .

Hint G.2 Initial mechanical energy

For the initial mechanical energy, you can use either the expression you found for the mechanical energy of the ball at the

top of the ramp or that for the total mechanical energy of the ball plus spring just before the ball was launched. These two

expressions are equal.


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Part G.3 Find the final mechanical energy

Find the total mechanincal energy of the ball when it hits the floor.

Express your answer in terms of and other given quantities.

ANSWER: =

Hint G.4 Is energy conserved?

Only conservative forces (gravity, spring) are acting on the ball, so energy is conserved: .

Express the speed in terms of , , , , , and/or .

ANSWER:

=

Energy in a Spring Graphing Question

Description: Draw kinetic, elastic potential, and total energy plots of a mass and spring system. (graphing applet)

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.

Part A

Sketch a graph of the total energy of the spring and car system. There is no scale given, so your graph should simply reflect

the qualitative shape of the energy vs. time plot.

ANSWER:

View

Part B


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Sketch a plot of the elastic potential energy of the spring from the point at which the car is released to the equilibrium

position of the spring. Make your graph consistent with the given plot of total energy (the gray line given in the graphing

window).

Part B.1 Determine the sign of the initial elastic potential energy

At the instant the car is released, the spring is compressed. Therefore, is the spring's initial elastic potential energy positive,

negative, or zero?

ANSWER: positive negative zero

Part B.2 Determine the sign of the initial kinetic energy

Is the initial kinetic energy of the cart positive, negative, or zero?


Part B.3 Determine the sign of the final elastic potential energy

When the car reaches the equilibrium position of the spring, is the elastic potential energy positive, negative, or zero?


Hint B.4 The shape of the elastic potential energy graph

The elastic potential energy of a spring with spring constant that is stretched or compressed to position is given by

,

where is the equilibrium position of the spring.

ANSWER:

View

Part C

Sketch a graph of the car's kinetic energy from the moment it is released until it passes the equilibrium position of the spring.

Your graph should be consistent with the given plots of total energy (gray line in graphing window) and potential energy

(gray parabola in graphing window).

ANSWER:


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View

Kinetic and Potential Energy of Baseball Graphing Question

Description: Graph kinetic, potential, and total energy over time of a vertically thrown baseball. (graphing applet)

A baseball is thrown directly upward at time and is caught again at time . Assume that air resistance is so small

that it can be ignored and that the zero point of gravitational potential energy is located at the position at which the ball leaves

the thrower's hand.

Part A

Sketch a graph of the kinetic energy of the baseball.

Part A.1 Determine the sign of the initial kinetic energy

At the instant the ball leaves the thrower's hand, is its kinetic energy positive, negative, or zero?


Hint A.2 The shape of the kinetic energy graph

The ball's speed decreases linearly from its initial value, which we can denote by , because of the constant acceleration

due to gravity. The velocity of the ball can be described by the equation

.

Since kinetic energy depends on the square of velocity, how does the kinetic energy vary with time?

Also, note that the ball reaches its maximum height halfway between the time that it leaves the thrower's hand and the

moment it is caught. What is the speed of the ball when it reaches the maximum height?

ANSWER:

View

Part B

Based on the graph of kinetic energy given (gray curve in the graphing window), sketch a graph of the baseball's gravitational

potential energy.

Hint B.1 Initial gravitational potential energy


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The point at which the ball leaves the thrower's hand is defined to be the origin of the y axis, and the gravitational potential

energy of the ball depends on the ball's height above the origin.

Hint B.2 The shape of the gravitational potential energy graph

The potential energy of the ball is proportional to its height, and the height of the ball can be described by the equation

.

Hint B.3 Using conservation of energy

Since there are no nonconservative forces acting on the ball, the total energy must remain the same throughout the motion.

Therefore, your graph of potential energy should be shaped such that potential energy plus kinetic energy does not change

during the motion.

ANSWER:

View

Part C

Based on the kinetic and potential energy graphs given, sketch a graph of the baseball's total energy.

Hint C.1 Total energy

The total energy of the baseball is the sum of its kinetic energy and gravitational potential energy.

ANSWER:

View

A Mass-Spring System with Recoil and Friction

Description: A object moving on a surface with friction compresses a spring, then recoils back and stops at exactly the

spring's equilibrium position. Use conservation of energy, explicitly including energy dissipated by friction, to find the spring

constant.

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.

Part A


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Find , the spring constant.

Part A.1 Why does the object stop?

Why does the object come to rest when it returns to ?

Although more than one answer may be true of the system, you must choose the answer that explains why the object

ultimately comes to a stop.

ANSWER: When the object reaches the second time all of its initial energy has gone into the compression

and extension of the spring.

When the object reaches the second time all of its initial energy has been dissipated by friction.

is an equilibrium position and at this point the spring exerts no force on the object.

At the force of friction exactly balances the force exerted by the spring on the object.

Part A.2 How does friction affect the system?

Indicate whether the following statements regarding friction are true or false.

Work done by friction is equal to , where is the mass of an object, is the magnitude of the acceleration

due to gravity, is the coefficient of kinetic friction, and is the distance the object has traveled.

A.

Energy dissipated by friction is equal to , where is the coefficient of friction, is the acceleration due to

gravity, is the mass of the object, and is the amount of time (since encountering the spring) the object has been

moving.

B.

Friction is a conservative force.C.

Work done by friction is exactly equal to the negative of the energy dissipated by friction.D.

Enter the letters of the correct statements in alphabetical order. For example, if statements A and C are correct,

enter AC.

ANSWER: AD

Hint A.3 Energy stored in a spring

The potential energy stored in a spring having constant that is compressed a distance is

.

Part A.4 Compute the compression of the spring

By what distance does the object compress the spring?

Hint A.4.a How to approach this question

Use the fact that

to solve for the distance the spring was compressed.

Look at the initial condition when the object originally hits the spring and the final condition when the object returns to

.


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Hint A.4.b The value of

In its final position, the object is not moving. Also the spring is not compressed. Therefore .

Part A.4.c Find

What is the value of ?

Hint A.4.c.i How to approach this part

Initially the spring is uncompressed, so the only contribution to the system's energy comes from the kinetic energy of the

object.

Express your answer in terms of some or all of the variables , , , and and , the acceleration due to gravity.

ANSWER: =

Part A.4.d Find

What is the value of ?

Hint A.4.d.i How to approach this part

The only nonconservative force in the system is the frictional force between the object and the surface it's on. If the

object moves through a distance , the work done by friction is

Express your answer in terms of some or all of the variables , , , and and , the acceleration due to gravity.

ANSWER: =

Express in terms of , , and .

ANSWER: =

Hint A.5 Putting it all together

In the previous part, at the two ends of the motion considered, the spring had no energy, so was not part of the equation.

However, you were able to find a relation for in terms of the known quantities. To obtain an equation involving , use

conservation of energy again,

,

but this time, take the initial condition to be the moment when the spring is at its maximum compression and the final

condition to be the moment when the spring returns to . So now can be written in terms of and other

variables.

Hint A.6 The value of


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The value of is again zero.

Part A.7 Find for this part of the motion

What is the value of for this part of the motion?

Hint A.7.a How to approach this part

Since the spring is at its maximum compression, the object must be momentarily at rest. So the only contribution to the

energy is from the potential energy of the spring.

Express your answer in terms of and , the spring constant, so that you end up with an equation containing .

ANSWER: =

Part A.8 Find for this part of the motion

What is the value of for this part of the motion?

Hint A.8.a How to approach this part

The only nonconservative force in the system is the frictional force between the object and the surface it's on. If the object

moves through a distance , the work done by friction is

.

Express your answer in terms of , , , and , the acceleration due to gravity.

ANSWER: =

Express in terms of , , , and .

ANSWER:

=

Summary 2 of 7 items complete (28.57% avg. score)10 of 35 points

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math.temple.educmartoff/teaching/ph87f07/solns...Created Date 10/10/2007 4:24:47 AM