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MasteringPhysics: Assignment Print View

Combining Truck PowerA loaded truck (truck 1) has a maximum engine power and is able to attain a maximum speed . Another truck (truck 2) has a maximum engine power and can attain a maximum speed of . The two trucks are then connected by a long cable, as shown. To solve this problem, assume that each truck, when not attached to another truck, has a speed that is limited only by wind resistance. Also assume (not very realistically) A) That the wind resistance is a constant force (a different constant for each truck though). i.e. It is independent of the speed at which the truck is going. B) That the wind resistance force on each truck is the same before and after the cable is connected, and, C) That the power that each truck's engine can generate is independent of the truck's speed.

Part AFind , the maximum speed of the two trucks when they are connected, assuming both engines are running at maximum power.

Hint A.1 Method for solvingHint not displayed

Hint A.2 Resistance force on truck 1Hint not displayed

Hint A.3 Net wind resistance on the two trucksHint not displayed

Hint A.4 Net power of the two trucksHint not displayed

http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=1465961 (1 of 74) [12/13/2010 7:03:11 PM]


Hint A.5 Solving for Hint not displayed

Express the maximum speed in terms of .

ANSWER: =

Correct

Note that truck 1 is going faster when in tow than when under its own power, and that truck 2 is going slower. This is consistent with having the cable connecting the trucks being subject to a tension.Anyone who has ever driven a truck, or closely watched one being driven, will know that this sort of arrangement is very unsafe and consequently is never used. However, train locomotives, which can be coupled together without cables, can combine their power in this way.

Delivering Rescue Supplies

You are a member of an alpine rescue team and must project a box of supplies, with mass , up an incline of constant slope angle so that it reaches a stranded skier who is a

vertical distance above the bottom of the incline. The incline is slippery, but there is some friction present, with kinetic friction coefficient .

Part AUse the work-energy theorem to calculate the minimum speed that you must give the box at the bottom of the incline so that it will reach the skier.

Hint A.1 How to approach the problemIn order to use the work-energy theorem,

,you need to find an expression for the total work done on the box and for the box's initial and final kinetic energies. At least one of these quantities will depend on the unknown initial speed of the box.

Hint A.2 Find the total work done on the box



What is , the total work done on the box between the moment it is projected and the moment it reaches the skier?

Hint A.2.1 Find the work done by gravityHint not displayed

Hint A.2.2 Find the work done by frictionHint not displayed

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

ANSWER: = Answer not displayed

Hint A.3 What is the initial kinetic energy?

Write , the initial kinetic energy of the box, in terms of the magnitude of its initial velocity and other given variables.

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


Hint A.4 What is the final kinetic energy?

If the box just reaches the skier, what is , the kinetic energy of the box when it arrives?

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


Express your answer in terms of some 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



Suppose that the right edge of the board is a distance from the boundary, as shown. When the board is at this position, what is the magnitude of the force of friction, , acting on the board (assuming that it's moving)?

Hint A.1.1 Fraction of board in region 2Hint not displayed

Hint A.1.2 Force of friction in region 1Hint not displayed

Express the force acting on the board in terms of , , , , , and .


Hint A.2 Work as integral of force

After you find the net force of friction that acts on the board, as a function of , to find the net work done by this force, you will need to perform the appropriate work integral,

The lower limit of this integral will be at . What will be the upper limit?

ANSWER: Upper limit at = Answer not displayed



Hint A.3 Direction of force of frictionDon't forget that the force of friction is directed opposite to the direction of the board's motion.

Hint A.4Formula for

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 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.)

Hint B.1 No accelerationHint not displayed

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

ANSWER: =

Correct

Power Dissipation Puts a Drag on Racing



The dominant form of drag experienced by vehicles (bikes, cars, planes, etc.) at operating speeds is called form drag. It increases quadratically with velocity (essentially because the amount of air you run into increases with and so does the amount of force you must exert on each small volume of air). Thus

,

where is the cross-sectional area of the vehicle and is called the coefficient of drag.

Part A

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

Hint A.1 How to approach the problemBecause the velocity of the car is constant, the drag force is also constant. Therefore, you can use the result that the power provided by a constant force to an object

moving with constant velocity is . Be careful to consider the relative

direction of the drag force and the velocity.

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 powerHint not displayed

Hint B.2 How is the algebra done?



Hint not displayed

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 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.

Work Done by a Spring

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 .

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.

Hint A.1 Spring force as a function of positionHint not displayed

Hint A.2 Integrand of the work integralHint not displayed

Hint A.3 Upper limit of the work integralHint not displayed

Express the work done by the spring in terms of and .

ANSWER: =

Correct

A Car with Constant Power

The engine in an imaginary sports car can provide constant power to the wheels over a range of speeds from 0 to 70 miles per hour (mph). At full power, the car can accelerate from zero to 31.0 in time 1.20 .

Part A



At full power, how long would it take for the car to accelerate from 0 to 62.0 ? Neglect friction and air resistance.

Hint A.1 Energy and powerIn the absence of friction, the constant power of the engine implies that the kinetic energy of the car increases linearly with time.

Hint A.2 Find the ratio of kinetic energies

Find the (numerical) ratio of the car's kinetic energy at time 62.0 to , the

kinetic energy at time 31.0 .


Express your answer in seconds.

ANSWER: 4.80 Correct

Of course, neglecting friction, especially air friction, is completely unrealistic at such speeds.

Part BA more realistic car would cause the wheels to spin in a manner that would result in the ground pushing it forward with a constant force (in contrast to the constant power in Part A). If such a sports car went from zero to 31.0 in time 1.20 , how long would it take

to go from zero to 62.0 ?

Hint B.1 How to approach the problemHint not displayed

Express your answer numerically, in seconds.


This is probably the first and last time you will come across an imaginary car that goes slower than the real one!



The Work Done in Pulling a Supertanker

Two tugboats pull a disabled supertanker. Each tug exerts a constant force of 2.20×106 , one at an angle 16.0 west of north, and the other at an angle 16.0 east of north, as they pull the tanker a distance 0.680 toward the north.

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

Hint A.1 How to approach the problemThere are two ways to calculate the total work done on an object when several forces act on it. You can compute the quantities of work done on the object by each force and then add them together. Alternatively, you can compute the work done on the object by the net force acting on it. The hints that follow are meant to help you to calculate the total work using the first method.

Hint A.2 Find the work done by one tugboatHint not displayed

Express your answer in joules, to three significant figures.

ANSWER: 2.88×109 All attempts used; correct answer displayed

PSS 7.2 Problems Using Mechanical Energy II

Learning Goal: To practice Problem-Solving Strategy 7.2 Problems Using Mechanical Energy II.

The Great Sandini is a 60.0- circus performer who is shot from a cannon (actually a spring gun). You don’t find many men of his caliber, so you help him design a new gun. This new gun has a very large spring with a very small mass and a force constant of 1100

that he will compress with a force of 4400 . The inside of the gun barrel is coated with Teflon, so the average friction force will be only 40.0 during the 4.00 he moves in the barrel. At what speed will he emerge from the end of the barrel, 2.50 above his initial rest position?

Problem-Solving Strategy: Problems using mechanical energy II IDENTIFY the relevant concepts: The energy approach is useful in solving problems that involve elastic forces as well as gravitational forces, provided the additional elastic potential energy is included in the



potential energy . SET UP the problem using the following steps:

1. Decide what the initial and final states of the system are. Use the subscript 1 for the initial state and the subscript 2 for the final state. It helps to draw sketches.

2. Define your coordinate system, particularly the level at which . We suggest that you always choose the positive y direction to be upward because this is what assumes.

3. Identify all forces that do work, including those that can’t be described in terms of potential energy. A free-body diagram is always helpful.

4. List the unknown and known quantities, including the coordinates and velocities at each point. Decide which unknowns are your target variables. EXECUTE the solution as follows: Write expressions for the initial and final kinetic and potential energies—that is, , ,

, and . The potential energy now includes both the gravitational potential energy

and the elastic potential energy , where is the displacement of the

spring from its unstretched length. Then, relate the kinetic and potential energies and the work done by other forces, , using . If no other forces

do work, this expression becomes . It’s helpful to draw bar graphs showing the initial and final values of , , and . Then, solve to find whatever unknown quantity is required. EVALUATE your answer: Check whether your answer makes physical sense. Keep in mind that the work done by the gravitational and elastic forces is accounted for by their potential energies; the work of the other forces, , has to be included separately.

IDENTIFY the relevant concepts The problem involves a spring gun. Therefore, to account for the potential energy associated with its elastic force, the energy approach might be the best method to solve this problem.

SET UP the problem using the following steps

Part A



Below is a sketch of the initial state of the situation described in this problem. Draw the most suitable set of coordinate axes for this problem. Note that even though you can choose the level to be wherever you like, in most situations it is best to set the zero height to coincide with either the initial or final position, so that the calculations for the gravitational potential energy become easier. For this reason, in this particular problem place the origin of your coordinate axes on the black dot marking the performer's initial position. Draw only the positive portion of the coordinate axes.

Draw the vectors starting at the black dot. The location and orientation of the vectors will be graded. The length of the vectors will not be graded.

ANSWER:

View Correct

This is the coordinate system used in the rest of this problem. Note that since the origin of the axes coincides with the location of the performer's feet, all vertical distances are calculated relative to his feet, and not relative to his center of mass. Now, draw a sketch for the final state showing the performer at the exit of the gun barrel, and identify all the forces that do work on the performer as he travels from the initial to the final state.

Part BBelow is a list of variables representing some of the relevant quantities in this problem. Which ones are known quantities?

Check all that apply.

ANSWER: , magnitude of compressing force, final height, magnitude of friction, force constant of spring, mass of body in motion

, distance traveled between initial and final state, initial height, initial speed , final speed



All attempts used; correct answer displayed

Now, make sure that you list all the known quantities on your sketches for the initial and final states of the system. You have identified only one unknown, , the final speed of the performer. This is your target variable. However, as you work through the next part, you will find that there may be other unknown quantities that need to be found in order to solve the problem.

EXECUTE the solution as follows

Part CAt what speed will The Great Sandini emerge from the end of the gun barrel?

Hint C.1 Find expressions for the performer’s initial and final kinetic energies

Hint not displayed

Hint C.2 Find the performer’s initial and final gravitational potential energies

Hint not displayed

Hint C.3 Find the initial and final elastic potential energiesHint not displayed

Hint C.4 Find

Hint not displayed

Express your answer in meters per second to four significant figures.

ANSWER: =

15.46 All attempts used; correct answer displayed

EVALUATE your answer

Part D



To evaluate whether your result makes sense, it's useful to use bar graphs showing the initial and final values of kinetic and potential energies. These graphs will help you verify whether energy is conserved. The picture to the right is a bar graph showing the initial values of potential energy (gravitational potential energy + elastic potential energy), kinetic energy , and total energy

. Which of the following graphs shows the correct final values for , , and ?

ANSWER:

Correct

According to your calculations, the total energy decreases by 160 . You can verify that this equals the amount of energy lost to friction, which you previously computed as . So your results make sense. The initial elastic potential energy is for the most part transformed into gravitational and kinetic energy, with a small loss due to friction. In the absence of friction, energy would be conserved and The Great Sandini would emerge from the end of the barrel at an even higher speed.



A Mass-Spring System with Recoil and Friction

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

Find , the spring constant.

Hint 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.

Correct

Hint A.2 How does friction affect the system?



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


ANSWER: 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.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.Friction is a conservative force.Work done by friction is exactly equal to the negative of the energy dissipated by friction.

Correct

Hint A.3 Energy stored in a spring

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

.

Hint A.4 Compute the compression of the spring

By what distance does the object compress the spring?

Hint A.4.1 How to approach this questionUse 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 .

Hint A.4.2 The value of



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

Hint A.4.3 Find

What is the value of ?

Hint A.4.3.1 How to approach this partHint not displayed

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

ANSWER: =

Correct

Hint A.4.4 Find

What is the value of ?

Hint A.4.4.1 How to approach this partHint not displayed

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

ANSWER: =

Correct

Express in terms of , , and .

ANSWER:

=

Correct



Hint A.5 Putting it all togetherIn 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

The value of is again zero.

Hint A.7 Find for this part of the motion

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

Hint A.7.1 How to approach this partHint not displayed

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

ANSWER: =

Correct

Hint A.8 Find for this part of the motion



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

Hint A.8.1 How to approach this partHint not displayed

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

ANSWER: =

Correct

Express in terms of , , , and .

ANSWER:

=

Correct

Bungee Jumping

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.

Hint A.1 Decide how to approach the problemHere are three possible methods for solving this problem:a. 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 . b. 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 . c. 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 . Which of these options is the simplest, most accurate way to find given the information available?

ANSWER: abc

Correct

Hint 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?

Hint A.2.1 Find the extension of the bungee cordHint not displayed

Hint A.2.2 Formula for the force due to a stretched cordHint not displayed

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: =

Correct

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

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

ANSWER: =

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.

Hint B.1 Decide how to approach the problemHere are three possible methods for solving this problem:a. 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 . b. 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 . c. 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 . Which of these options is the simplest, most accurate way to find given the information available?

ANSWER: abc

Correct



Hint 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: =

Correct

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

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.3.1 Formula for elastic potential energyThe 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.

Hint B.3.2 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: =

Correct

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

ANSWER: =

Correct




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 in this vertical circle with negligible loss of total mechanical energy. At the top and bottom of the vertical 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.

Hint A.1 How to approach this problemHint not displayed

Hint A.2 Find the sum of forces at the bottom of the circleHint not displayed

Hint A.3 Find the acceleration at the bottom of the circleHint not displayed



Hint A.4 Find the tension at the bottom of the circleHint not displayed

Hint A.5 Find the sum of forces at the top of the circleHint not displayed

Hint A.6 Find the acceleration at the top of the circleHint not displayed

Hint A.7 Find the tension at the top of the circleHint not displayed

Hint A.8 Find the relationship between and Hint not displayed

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 done properly, 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 problemHint not displayed

Hint A.2 Find the change in potential energyHint not displayed

Hint A.3 Find the change in kinetic energyHint not displayed

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.2 Magnitude of the drag forceHint not displayed

Hint B.3 Relative direction of the drag force and velocityHint not displayed



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

ANSWER: =

Correct

Energy in a Spring Graphing Question

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 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.

ANSWER:

View All attempts used; correct answer displayed



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).

Hint B.1 Determine the sign of the initial elastic potential energyAt the instant the car is released, the spring is compressed. Therefore, is the spring's initial elastic potential energy positive, negative, or zero?

ANSWER: positivenegativezero

Correct

Hint B.2 Determine the sign of the initial kinetic energyIs the initial kinetic energy of the cart positive, negative, or zero?


Correct

Hint B.3 Determine the sign of the final elastic potential energyWhen the car reaches the equilibrium position of the spring, is the elastic potential energy positive, negative, or zero?


Correct

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 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).

ANSWER:

View Correct

Fun with a Spring Gun



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 AWhich of the following statements are true?

Hint A.1 Nonconservative forcesHint not displayed

Hint A.2 Forces acting on the ballHint not displayed


ANSWER: Mechanical energy is conserved because no dissipative forces perform work on the ball.The forces of gravity and the spring have potential energies associated with them.No conservative forces act in this problem after the ball is released from the spring gun.

Correct

Part B



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

Hint B.1 Determine how to approach the problemHint not displayed

Hint B.2 Energy equationsHint not displayed

Hint B.3 Determine which two locations you should examineHint not displayed

Hint B.4 Find the initial energy of the systemHint not displayed

Hint B.5 Determine the final energyHint not displayed

Hint B.6 Creating an equationHint not displayed

ANSWER: = 4.78

Correct

Part C



Find the maximum height of the ball.

Hint C.1 Choose two locations to examineHint not displayed

Hint C.2 Find the initial energyHint not displayed

Hint C.3 Determine the final energyHint not displayed

Hint C.4 Creating an equationHint not displayed

Express your answer numerically, in meters.

ANSWER: = 1.17

Correct

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


ANSWER: reducing the spring constant increasing the spring constant decreasing the distance the spring is compressedincreasing the distance the spring is compresseddecreasing the mass of the ballincreasing the mass of the balltilting the spring gun so that it is at an angle degrees from the horizontal

Correct



Graphing Gravitational Potential Energy

A 1.00 ball is thrown directly upward with an initial speed of 16.0 .

A graph of the ball's gravitational potential energy vs. height, , for an arbitrary initial velocity is given in Part A. The zero point of gravitational potential energy is located at the height at which the ball leaves the thrower's hand.For this problem, take as the acceleration due to gravity.

Part A

Draw a line on the graph representing the total energy of the ball.

Hint A.1 How to approach the problemThe total energy is the sum of the kinetic energy and potential energy. You can compute the total energy at any point in the ball's trajectory, but the simplest method is to add the initial kinetic and potential energies just as the ball is thrown.

Hint A.2 Find the initial kinetic energy

When the ball first leaves the thrower’s hand, what is its kinetic energy ?

ANSWER: = 128

Correct

Hint A.3 Find the initial potential energy

What is the potential energy of the ball when it first leaves the thrower's hand?


Hint A.4 Shape of the total energy graph



As the ball ascends, does its total energy increase, decrease, or stay the same?

ANSWER: increasedecreasestay the same

Correct

The law of conservation of energy guarantees that the total energy of the ball remains constant throughout its motion. The increase in potential energy as the ball ascends is exactly balanced by the decrease in its kinetic energy.

ANSWER:

View Correct

Part BUsing the graph, determine the maximum height reached by the ball.

Hint B.1 Maximum heightThe 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. The ball does not have enough energy to rise above this point on the potential energy graph.

Express your answer to one decimal place.


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 CDraw a new gravitational potential energy vs. height graph to represent the gravitational potential energy if the ball had 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.

Hint C.1 SlopeThe gravitational potential energy is defined by

.In a graph of potential energy vs. height, is the slope.

Hint C.2 Determine the new gravitational potential energy

What is the gravitational potential energy for a 2.00- ball at a height of ?

Take as the acceleration due to gravity and express your answer to three decimal places.

ANSWER: = 100

Correct

The new graph of potential energy versus height must pass through the point .

ANSWER:

View Correct

For a ball with twice the mass, you should expect the plot of potential energy vs. height to have twice the slope.

Kinetic and Potential Energy of Baseball Graphing Question



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 ASketch a graph of the kinetic energy of the baseball.

Hint A.1 Determine the sign of the initial kinetic energyHint not displayed

Hint A.2 The shape of the kinetic energy graphHint not displayed

ANSWER:

View All attempts used; correct answer displayed

Part BBased 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 energyHint not displayed

Hint B.2 The shape of the gravitational potential energy graphHint not displayed

Hint B.3 Using conservation of energyHint not displayed



ANSWER:

View Correct

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

Hint C.1 Total energyHint not displayed

ANSWER:

View Correct

Loop the Loop



A roller coaster car may be approximated by a block of mass

. The car, which starts from rest, is released at a height above the ground and slides along a frictionless track. The car encounters a loop of radius , as shown. Assume that the initial height

is great enough so that the car never loses contact with the track.

Part AFind 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 loopWhat is the potential energy of the car when it is at the top of the loop? Define the gravitational potential energy to be zero at .

Express your answer in terms of and other given quantities.


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 partMeaning of "stay in contact" For the car to just stay in contact through the loop, without falling, the normal force that acts on the car when it's at the top of the loop must be zero (i.e., ). Find the velocity at the top such that the remaining force on the car i.e. its weight provides the necessary centripetal acceleration. If the velocity were any greater, you would additionally require some force from the track to provide the necessary centripetal acceleration. If the velocity were any less, the car would fall off the track.Use the above described condition to find the velocity and then the result from the above part to find the required height.

Hint B.2 Acceleration at the top of the loopHint not displayed

Hint B.3 Normal force at the top of the loopHint not displayed

Hint B.4 Solving for

Hint not displayed

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

.



Not Quite around the GlobeA large globe, with a radius of about 5 , was built in Italy between 1982 and 1987. Imagine that such a globe has a radius and a frictionless surface. A small block of mass

slides starts from rest at the very top of the globe and slides along the surface of the globe. The block leaves the surface of the globe when it reaches a height above the ground. The geometry of the situation is shown in the figure for an arbitrary height .

Part AConsider what happens at the moment when the block leaves the surface of the globe. Which of the following statements are correct? a. The net acceleration of the block is directed straight down. b. The component of the force of gravity toward the center of the globe is equal to the magnitude of the normal force. c. The force of gravity is the only force acting on the block.

Hint A.1 How is the normal force changing?Hint not displayed



ANSWER: a onlyb onlyc onlya and ba and cb and ca and b and c

Correct

Part BWhich of the following statements is also true at the moment when the block leaves the surface of the globe?

ANSWER: The centripetal acceleration is zero. The normal force is zero. The net acceleration of the block is parallel to its velocity. The kinetic energy of the block equals its potential energy.

Correct

Part CUsing Newton's 2nd law, find , the speed of the block at the critical moment when the block leaves the surface of the globe.Assume that the height at which the block leaves the surface of the globe is .

Hint C.1 How to approach this problemSince the normal force goes to zero at the critical moment when the block leaves the surface of the globe, it is the radial component of the gravitational force that generates the entire centripetal acceleration at this point. Use this fact and Newton's 2nd law to relate the acceleration due to gravity and the centripetal acceleration.

Hint C.2 Find the centripetal acceleration



What is , the magnitude of the centripetal acceleration of the block when its speed is ? Assume that the block has not lost contact with the globe.

Hint C.2.1 Formula for centripetal accelerationHint not displayed


ANSWER:

=

Correct

Hint C.3 Find the radial component of the gravitational force

What is , the magnitude of the radial component of the gravitational force on the block when the block is at the position indicated in the figure?

Express your answer in terms of , , and .

ANSWER: =

Correct

Hint C.4 What is ?

Having found , you now need to find in terms of (the height of

the block) and . You need to find a right triangle where is the included angle and

is the hypoteneuse. Using this triangle, what is ?

Give your answer in terms of and .

ANSWER: =

Correct

Express the speed in terms of , , and , the magnitude of the accleration due to gravity. Do not use in your answer.



ANSWER: =

Correct

Part DUse the law of conservation of energy to find . This will give you a difference expression for than you found in the previous part.

Hint D.1 How to apply conservation of energyThe law of conservation of energy states that

.You may assume that the initial velocity of the block is negligible, so that the block's initial kinetic energy is zero. The final kinetic energy of the block can be easily expressed in terms of and . The initial and final potential energies of the block simply depend on the height of the block above the ground (or any other reference point).


ANSWER: =

Correct

Part E

Find , the height from the ground at which the block leaves the surface of the globe.

Hint E.1 How to approach this questionHint not displayed

Express in terms of .

ANSWER: =

Correct

Projectile Motion and Conservation of Energy Ranking Task



Part ASix baseball throws are shown below. In each case the baseball is thrown at the same initial speed and from the same height above the ground. Assume that the effects of air resistance are negligible. Rank these throws according to the speed of the baseball the instant before it hits the ground.


Rank from largest to smallest. To rank items as equivalent, overlap them.

ANSWER:

View Correct

This answer is best understood in terms of conservation of energy. The initial energy of the ball is independent of the direction in which it is thrown. The initial and final potential energies of the ball are the same regardless of the trajectory. Therefore, the final kinetic energy, and therefore the final speed, of the ball must be the same no matter in what direction it is thrown.

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 is negligibly small), and the center of the box is a distance 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 a horizontal distance .

Part ABy 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 the marble 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 marbleUse 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



Use kinematics to find , the time it takes the second marble to hit the ground after it is shot off the table.


ANSWER:

=

Correct

Hint A.4 Combining equations and solving for

The kinematic equation for the motion along the x axis is . Using the expressions for and from the previous hints, solve for in terms of the quantities

, , , , and .

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

ANSWER:

=

Correct

Part BNow imagine that the second child does not know the mass of the marble, the height of the table above the floor, or the spring constant. Find an expression for that depends only on and distance measurements.

Hint B.1 Compute

Use your answer to Part A to write in terms of , , , , , and .





ANSWER:

=

Correct

Shooting a Block up an Incline

A block of mass is placed in a smooth-bored spring gun at the bottom of the incline so that it compresses the spring by an amount . The spring has spring constant . The incline makes an angle with the horizontal and the coefficient of kinetic friction between the block and the incline is . The block is released, exits the muzzle of the gun, and slides up an incline a total distance .

Part A

Find , the distance traveled along the incline by the block after it exits the gun. Ignore friction when the block is inside the gun. Also, assume that the uncompressed spring is just at the top of the gun (i.e., the block moves a distance while inside of the gun). Use

for the magnitude of acceleration due to gravity.

Hint A.1 How to approach the problem



This is an example of a problem that would be very difficult using only Newton's laws and calculus. Instead, use the Work-Energy Theorem: , where

is the final energy, is the initial energy, and is the work done on the system by external forces. Let the gravitational potential energy be zero before the spring is released. Then, is the potential energy due to the spring, is the

potential energy due to gravity, and is the work done by friction. Once you've set up this equation completely, solve for .

Hint A.2 Find the initial energy of the block

Find the initial energy of the block. Take the gravitational potential energy to be zero before the spring is released.

Hint A.2.1 Potential energy of a compressed springHint not displayed

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

ANSWER: =

Correct

Hint A.3 Find the work done by friction

Find , the work done by friction on the block.

Hint A.3.1 How to compute workHint not displayed

Express in terms of , , , , and .

ANSWER: =

Correct

Hint A.4 Find the final energy of the block



Find an expression for the final energy of the block (the energy when it has traveled a distance up the incline). Assume that the gravitational potential energy of the block is zero before the spring is released and that the block moves a distance inside of the gun.

Hint A.4.1 What form does the energy take?Hint not displayed

Your answer should contain and .

ANSWER: =

Correct

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

ANSWER:

=

Correct

Sliding In Socks

Suppose that the coefficient of friction between your feet and the floor, while wearing socks, is 0.250. Knowing this, you decide to get a running start and then slide across the floor.

Part A

If your speed is 3.00 when you start to slide, what distance will you slide before stopping?

Express your answer in meters.


Part B



Now, suppose that your young cousin sees you sliding and takes off her shoes so that she can slide as well (assume her socks have the same coefficient of friction as yours). Instead of getting a running start, she asks you to give her a push. So, you push her with a force of 125 over a distance of 1.00 . If her mass is 20.0 , what distance does she slide (i.e., how far does she move after the push ends)? Remember that the friction force is acting anytime that she is moving.


Express your answer in meters.

ANSWER: = 1.55 Correct

Spring and Projectile

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 lost potential energy

Correct

Part BAs 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

Correct

Part CAs the ball goes up the ramp, it


Correct

Part D



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


Correct

Part EWhich 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:

Correct

Part F



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

Hint F.1 General approachHint not displayed

Hint F.2 Find the initial mechanical energyHint not displayed

Hint F.3 Find the mechanical energy at the end of the rampHint not displayed

Hint F.4 Is energy conserved?Hint not displayed

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

ANSWER:

=

Correct

Part GWith what speed will the ball hit the floor?

Hint G.1 General approachHint not displayed

Hint G.2 Initial mechanical energyHint not displayed

Hint G.3 Find the final mechanical energyHint not displayed

Hint G.4 Is energy conserved?Hint not displayed



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

ANSWER:

=

Correct

Spring Gun

A spring-loaded toy gun is used to shoot a ball straight up in the air. The ball reaches a maximum height , measured from the equilibrium position of the spring.

Part AThe same ball is shot straight up a second time from the same gun, but this time the spring is compressed only half as far before firing. How far up does the ball go this time? Neglect friction. Assume that the spring is ideal and that the distance by which the spring is compressed is negligible compared to .

Hint A.1 Potential energy of the springHint not displayed

Hint A.2 Potential energy of the ballHint not displayed



ANSWER: height =

Correct

Springs in Two Dimensions

The ends of two identical springs are connected. Their unstretched lengths are negligibly small and each has spring constant . After being connected, both springs are stretched an amount and their free ends are anchored at and as shown . The point where the springs are connected to each other is now pulled to the position ( , ). Assume that ( , ) lies in the first quadrant.

Part AWhat is the potential energy of the two-spring system after the point of connection has been moved to position ( , )? Keep in mind that the unstretched length of each spring is much less than and can be ignored (i.e., ).

Hint A.1 An important property of the potential energyHint not displayed

Hint A.2 Potential energy of the left-hand springHint not displayed

Express the potential in terms of , , , and .



ANSWER: =

Correct

Part B

Find the force on the junction point, the point where the two springs are attached to

each other.


Hint B.2 Components of the force vectorHint not displayed

Express as a vector in terms of the unit vectors and .

ANSWER: =

Correct

Notice how much more difficult it would have been to obtain the force via vector addition (computing the two components of the force from each spring, then adding them). This is the power of scalar potential functions: They allow you to simply add up the contributions, without having to worry about vectors or coordinate axes. By taking the gradient of the potential, you automatically obtain the desired vector quantities.

Work and Potential Energy on a Sliding Block with Friction



A block of weight sits on a plane inclined at an angle as shown. The coefficient of kinetic friction between the plane and the block is .

A force is applied to push the block up the incline at constant speed.

Part A

What is the work done on the block by the force of friction as the block moves a distance up the incline?

Hint A.1 A formula for workHint not displayed

Hint A.2 Find the magnitude of the frictional forceHint not displayed

Express your answer in terms of some or all of the following: , , , .

ANSWER: =

Correct

Part B



What is the work done by the applied force of magnitude ?


ANSWER: =

Correct

Part C

What is the change in the potential energy of the block, , after it has been pushed a distance up the incline?


ANSWER: =

All attempts used; correct answer displayed

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

Part D



What is the change in potential energy of the block, , as it moves a distance down the incline?


ANSWER: =

Answer Requested

Part E

What is the work done by the applied force of magnitude ?


ANSWER: =

Correct

Part F

What is the work done on the block by the frictional force?


ANSWER: =

Answer Requested

Work on a Sliding Box



A box of mass is sliding along a horizontal surface.

Part A

The box leaves position with speed . The box is slowed by a constant frictional force until it comes to rest at position .Find , the magnitude of the average frictional force that acts on the box. (Since you don't know the coefficient of friction, don't include it in your answer.)


Hint A.2 Find the initial kinetic energyHint not displayed

Hint A.3 Find the final kinetic energyHint not displayed

Hint A.4 Find the work done by frictionHint not displayed

Express the frictional force in terms of , , and .

ANSWER:

=

Correct

Part B



After the box comes to rest at position , a person starts pushing the box, giving it a speed .When the box reaches position (where ), how much work has the person done on the box?Assume that the box reaches after the person has accelerated it from rest to speed .


Hint B.2 Find the work done by frictionHint not displayed

Hint B.3 Find the change in kinetic energyHint not displayed

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

ANSWER:

=

Correct

PSS 9.1 Rotational Energy

Learning Goal: To practice Problem-Solving Strategy 9.1 Rotational Energy.

A frictionless pulley has the shape of a uniform solid disk of mass 4.00 and radius 25.0

. A 1.90 stone is attached to a very light wire that is wrapped around the rim of the pulley, and the stone is released from rest. As it falls down, the wire unwinds without



stretching or slipping, causing the pulley to rotate. How far must the stone fall so that the pulley has 4.40 of kinetic energy?

Problem-Solving Strategy: Rotational energy IDENTIFY the relevant concepts: You can use work–energy relationships and conservation of energy to find relationships involving position and motion of a rigid body rotating around a fixed axis. The energy method is usually not helpful for problems that involve elapsed time.SET UP the problem using the following steps:

1. When using the energy approach, sketch the initial and final states of the system. Include the positions and velocities on your sketch.

2. Define your coordinate system, particularly the level at which . You will use it to compute gravitational potential energies. Choose the positive direction to be upward because this is what the equation assumes.

3. Identify all forces that do work that can’t be described in terms of potential energy. A free-body diagram is always helpful.

4. List the unknown and known quantities, including the coordinates and velocities at each point. Decide which unknowns are your target variables.

5. Many problems involve a rope or cable wrapped around a rotating rigid body, which functions as a pulley. In these situations, a point on the pulley that contacts the rope has the same linear speed as the rope, provided the rope doesn’t slip on the pulley. You can then take advantage of the following equations that relate the linear speed and tangential acceleration of a point on a rigid body to the angular velocity and angular acceleration of the body:

EXECUTE the solution as follows: Write the expressions for the initial and final kinetic and potential energies ( , , ,

and ) and the nonconservative work (if any). Rotational kinetic energy

is expressed in terms of the body's moment of inertia and its angular speed

. Substitute these expressions into (if nonconservative



work is done) or (if only conservative work is done) and solve for the target variable(s).EVALUATE your answer: As always, check whether your answer makes physical sense.

IDENTIFY the relevant concepts This problem describes a system where a rigid body, the pulley, rotates around a fixed axis while a second body, the stone, moves vertically. Since no information about elapsed time is provided, energy considerations must be used to find relationships involving the position and the motion of the system. To simplify the problem, assume that the wire is massless and it unwinds without stretching or slipping, so that there is no relative motion between the wire and the pulley. Note that the pulley is assumed to be rotating around its fixed axis without friction.

SET UP the problem using the following steps

Part AWhat forces do work on this system?

Hint A.1 How to approach this problemHint not displayed


ANSWER: frictional forcenormal forcegravitational forcetension

Correct

There is friction between the cable and the pulley, producing the no-slip conditions. However, frictional forces do no work because no mechanical energy is lost by the cable moving relative to the pulley. Since no other forces besides gravity do work on this system, .

Part B



Compare the magnitude of the stone's velocity and the magnitude of the pulley's linear velocity at the point of contact between the wire and pulley.

ANSWER: The magnitude of the velocity of the stone is less than that of the point of contact.The magnitude of the velocity of the stone is the same as that of the point of contact.The magnitude of the velocity of the stone is greater than that of the point of contact.

Correct

Here is a sketch of the initial and final states of the system.

In the initial state, the stone is at rest, initially located at . In the final state, the

stone is located at and the stone and pulley are moving. The target variable is .

EXECUTE the solution as follows

Part C



How far must the stone fall so that the pulley has 4.40 of kinetic energy?

Hint C.1 How to approach the problemHint not displayed

Hint C.2 Find the initial gravitational potential energyHint not displayed

Hint C.3 Find the total initial kinetic energyHint not displayed

Hint C.4 Find the final gravitational potential energyHint not displayed

Hint C.5 Find the total final kinetic energyHint not displayed

Hint C.6 Setting the initial energy equal to the final energyHint not displayed

Express your answer numerically in meters to three significant figures.

ANSWER: = 0.460

Correct

The potential energy that the stone loses is converted into the stone's and the pulley's kinetic energy.

EVALUATE your answer

Part D



To see if your results are reasonable, you can compare the final velocity of the stone as it falls down unwinding the wire from the pulley, to the velocity the stone would have if falling the same distance while unconnected to the pulley. What is the velocity of an untethered stone after falling 0.460 from rest?

Hint D.1 Using the proper kinematic equationsTo find the speed of the stone during free fall, set the change in potential energy of the stone

to the kinetic energy after falling the given distance

.

Express your answer numerically in meters per second.

ANSWER: =

3.01 Answer Requested

While solving Part C. you may have calculated the value of the final speed of the stone to be 2.10 . Now you found that, if untethered, the stone would move faster. This is reasonable because when the stone is connected to the pulley through the wire, the change in potential energy of the stone must equal the sum of the change in kinetic energy of the stone and the pulley. When the stone is free to fall on its own, instead, the same change in potential energy of the stone must equal only the change of its kinetic energy. Thus, the change in kinetic energy of the stone is larger when the stone is in free fall, causing the stone to reach a higher speed than if it were connected to the pulley.

Kinetic Energy of a Dumbbell

This problem illustrates the two contributions to the kinetic energy of an extended object: rotational kinetic energy and translational kinetic energy. You are to find the total kinetic energy of a dumbbell of mass when it is rotating with angular speed and its center of mass is moving translationally with speed . Denote the dumbbell's moment of



inertia about its center of mass by . Note that if you approximate

the spheres as point masses of mass each located a distance

from the center and ignore the moment of inertia of the connecting rod, then the moment of inertia of the dumbbell is given by , but this fact will not be necessary for this problem.

Part A

Find the total kinetic energy of the dumbbell.


Hint A.2 Find the rotational kinetic energyHint not displayed

Hint A.3 Find the translational kinetic energyHint not displayed

Express your answer in terms of , , , and .

ANSWER: =

Correct

Part B



The rotational kinetic energy term is often called the kinetic energy in the center of mass, while the translational kinetic energy term is called the kinetic energy of the center of mass.You found that the total kinetic energy is the sum of the kinetic energy in the center of mass plus the kinetic energy of the center of mass. A similar decomposition exists for angular and linear momentum. There are also related decompositions that work for systems of masses, not just rigid bodies like a dumbbell.It is important to understand the applicability of the formula . Which of the following conditions are necessary for the formula to be valid?


ANSWER: The velocity vector must be perpendicular to the axis of rotation.The velocity vector must be perpendicular or parallel to the axis of rotation.The moment of inertia must be taken about an axis through the center of mass.

Correct

Kinetic Energy of a Rotating Wheel

A simple wheel has the form of a solid cylinder of radius with a mass uniformly distributed throughout its volume. The wheel is pivoted on a stationary axle through the axis of the cylinder and rotates about the axle at a constant angular speed. The wheel rotates full revolutions in a time interval .

Part A

What is the kinetic energy of the rotating wheel?

Hint A.1 What is the formula for rotational kinetic energy?Hint not displayed

Hint A.2 Moment of inertia of the wheelHint not displayed

Hint A.3 Find the angular velocityHint not displayed

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



ANSWER:

=

Correct

Rotational Kinetic Energy and Conservation of Energy Ranking

TaskThe five objects of various masses, each denoted , all have the same radius. They are all rolling at the same speed as they approach a curved incline.

Part ARank the objects based on the maximum height they reach along the curved incline.

Hint A.1 Using energy conservationHint not displayed

Hint A.2 Moment of inertiaHint not displayed

Rank from largest to smallest. To rank items as equivalent, overlap them.



ANSWER:

View Correct

A Rolling Hollow Sphere

A hollow spherical shell with mass 1.75 rolls without slipping down a slope that makes an angle of 38.0 with the horizontal.

Part AFind the magnitude of the acceleration of the center of mass of the spherical shell.


Hint A.2 Translational motion in the x directionHint not displayed

Hint A.3 Torque on the spherical shellHint not displayed

Hint A.4 Moment of inertiaHint not displayed

Hint A.5 Relation between the translational and angular accelerationsHint not displayed

Take the free-fall acceleration to be = 9.80 .



ANSWER: = 3.62

Correct

Part BFind the magnitude of the frictional force acting on the spherical shell.


Take the free-fall acceleration to be = 9.80 .

ANSWER: = 4.22

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.

Hint C.1 How to approach the problemHint not displayed

ANSWER: = 0.313

Correct

Unwinding Cylinder



A cylinder with moment of inertia about its center of mass, mass , and radius has a string wrapped around it which is tied to the ceiling . The cylinder's vertical position as a function of time is .

At time the cylinder is released from rest at a height above the ground.

Part AThe string constrains the rotational and translational motion of the cylinder. What is the relationship between the angular rotation rate and , the velocity of the center of mass of the cylinder?Remember that upward motion corresponds to positive linear velocity, and counterclockwise rotation corresponds to positive angular velocity.

Hint A.1 Key to the constrained motionHint not displayed

Hint A.2 Velocity of contact pointHint not displayed

Express in terms of and other given quantities.

ANSWER: =

Correct

Part B



In similar problems involving rotating bodies, you will often also need the relationship between angular acceleration, , and linear acceleration, . Find in terms of and .

ANSWER: =

Correct

Part CSuppose that at a certain instant the velocity of the cylinder is . What is its total kinetic energy, , at that instant?

Hint C.1 Rotational kinetic energyHint not displayed

Hint C.2 Rotational kinetic energy in terms of Hint not displayed

Hint C.3 Translational kinetic energyHint not displayed


ANSWER:

=

Correct

Part D



Find , the cylinder's vertical velocity when it hits the ground.

Hint D.1 Initial energyHint not displayed

Hint D.2 Energy conservationHint not displayed

Express , in terms of , , , , and .

ANSWER:

=

Correct


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