Chapter 07 Homework

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Chapter 7 Homewor k

Chapter 7 Homework Due: 10:00pm on Monday, March 17, 2014 You will receive no credit for for items you com plete after after the ass ignm ent is due . Grading Policy

Energy Required to Lift a Heavy Box As you y ou are are trying to move move a heav heavy y box of mass , you realize that it is too t oo heav heavy y for you to lift by yourself. There 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 grav gravity ity and neglect negle ct fr frict iction ion forces. forces.

Part A Once you hav have e pulled hard hard enough enough to start the t he box moving moving upward, upward, what is the magnitude

of the upward upward force

you must apply to the rope to start raising the box with constant velocity? Express the magnitude magni tude of the force force in terms of

, the the mas ma ss of the box.

Hint 1. What force must be applied to the box to keep it moving at a constant speed? Once you have pulled hard enough to start the box moving upward, 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:  =

Hint 2. What force does the pulley exert on the box? If you tak take e the tensi tension on in the rope to be

, what what is

, the magnitude of the net upwar upward d force that the pulley

exerts on the box? Express your answer in terms of

.

ANSWER: http://sessi on.master ing physi cs.com/myct/assi g nmentPr i ntView?assi g nmentID= 2808817

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 =

Hint 3. Find the tension in the rope Find the tens tension ion in the rope in terms of

, the force force wit with h which which you are are pulling upward. upward.

ANSWER:  =

Hint 4. Putting it all together On your own or using the previous hints, you should have found equations for he following: 1. the force force needed needed to lift the box at constant velocity, velocity, in terms of its mass, 2. the relationshi relationship p between the force on the box due to the pulley pulley and the tension in the rope, and 3. the relationship relationshi p between the force applied applied to the rope and the tension in the rope. rope. 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.

ANSWER:  =

Correct

Part B Consider Con sider lifting a box of mass

to a heigh heightt

using two diff differe erent nt methods: lifting lifting the box directly or lifting the box

using a pulley (as in the previous part). What is

, the ratio of the wor work k don done e lifting the box directly to the wor work k done lifting the box with a pulley?

Express the ratio numerically.

Hint 1. Definition of work In each cas case, e, the amount of work

you do is equal to the force

you apply tim times es the dist distance ance

over ov er

 which you apply the force: force: .

Hint 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. http://sessi on.master ing physi cs.com/myct/assi g nmentPr i ntView?assi g nmentID= 2808817

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ANSWER:  = 2

Hint 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 force is applie applied d when lifting lifting the box with the pulley? Express the ratio of distances numerically.

when using the pulley Hint 1. Find the distance when Find

, the dist distance ance ov over er which you must apply force when lifti lifting ng the box using the pulley.

Express your answe answe r in term terms s of

, the total total he ight that that the box is lifted.

Pulling ng the rope a short distance Hint 1. Pulli Using the pully, imagine that you pull the end of the rope a short dist dist ance  will actually rise a distance dist ance

upward.. The upward The box

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

ANSWER:  =

when lifting directly Hint 2. Find the distance when When lifti lifting ng the box directl directly, y, the dist distance ance ov over er which force is applied, applied, distance

, is equal equal to the vertic ertical al

that the box is raised.

ANSWER:  = 0.500

ANSWER:  = 1

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Correct No matter which method you use to lift the box, its grav gravitational itational potential energy energy will increase by neglecting neglecti ng friction, fricti on, you will always need to do an amount of work equal to

. So,

to lift it.

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 encoun ters a loop of rad radius ius , as shown shown.. Assume that that the initial init ial height is great great enough so that the car nev never er loses contact with the track.

Part A Find an expression for the kinetic energy of the car at the top of the loop. Express the kinetic ene rgy in terms of

,

,

, and

.

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

ANSWER:  =

Correct

ANSWER:

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 =

Correct

Part B Find the minimum initial height height

at which which the car can be be released released that still allows the car to stay in contact with

the track at the top of the loop. Express the mini mum height in terms of

.

Hint 1. How to approach this part Meaning of "stay in contact" For the car to just  to just  stay  stay in contact through the loop, without falling, the normal force that acts on the car  when it's at the  when t he top of the loop must be zero z ero (i.e., (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.

Acceleration eration at the top of the loop Hint 2. Accel Assuming Ass uming that the speed of the car at the top of the loop is

, and and that the car stays on the track, find

the acceleration of the car. Take the positive y  positive  y  direction  direction to be upward. Express your answer in terms of

and any other quantitie s give given n in the proble m introduction.

ANSWER:  =

Hint 3. Normal force at the top of the loop Suppose the car st stays ays on the track and has speed an express expression ion for for

at the top of the loop. Use Newton's 2nd law to find

, the magnitude of of the normal force that the loop exerts on the car when when the car is at at the

top of the loop. Express your answer in terms of

,

,

, and

.

Hint 1. Find the sum of forces at the top of the loop Find the sum of the forces acting on the car at the top of the loop. Remember that the positive y  positive y  direction is upward . http://sessi on.master ing physi cs.com/myct/assi g nmentPr i ntView?assi g nmentID= 2808817

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Express your answer in terms of

,

, and

.

ANSWER:  =

ANSWER:

 =

Solving ng for Hint 4. Solvi The req requireme uirement nt to stay in contact results in an expression for

in terms of

and . Substit Substitute ute this into

your express expression ion for kinet kinetic ic energy, found in Part A, to determine a relation between

and

.

ANSWER:  =

Correct For

the car will still st ill complet e the loop, though it will require some normal reaction even even at the very

top. For

the car will just oscillate. oscillat e. Do you see this?

For

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

must have a lot  of  of safety features. If you like, you can check that the angle at which the cart loses contact with the track is given given by

. Where

is the angle measured counterclock countercl ockwise wise from

the horizontal positive x  positive  x -axis, -axis, where the origin of the x  the  x -axis -axis is at the center of the loop.

Work and Potential Energy on a Sliding Block with Friction A block of weight sit s on a plane incli inclined ned at an angle and the block is .

as shown. The coefficient of kinet kinetic ic frict friction ion between the plane

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A force force

Chapter 7 Homewor k

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

Part A What is the wor work k

done on the block by the for force ce of friction as the block mov moves es a distance

Express your answe answerr in terms of some or al alll of the follow ing:

,

,

,

up the incline?

.

Hint 1. A formula for work The work done by a constant force is given by the dot product of the force vector with the vector representing the displacement over which the force is applied.

Hint 2. Find the magnitude of the frictional force What is the magnitude

of the frictional for force? ce?

Express your answer in terms of

,

, and

.

Compute pute the normal force Hint 1. Com Find the magnitude

of the normal force on the block block..

Express your answer in terms of

and

.

ANSWER:  =

ANSWER:  = http://sessi on.master ing physi cs.com/myct/assi g nmentPr i ntView?assi g nmentID= 2808817

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Correct

ANSWER:  =

Correct

Part B What is the work

done by the applied force of magnitude

?

Express your answe answerr in terms of some or al alll of the follow ing:

,

,

,

.

ANSWER:  =

Correct

Part C What is the change in the potential ener energy gy of the block,

, af after ter it has bee been n pushed a distance

Express your answe answerr in terms of some or al alll of the follow ing:

,

,

,

up the incline?

.

ANSWER:  =

Correct

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

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pulls the block block down the

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Part D What is the change in potential ener energy gy of the block,

, as it mov moves a distance

Express your answe answerr in terms of some or al alll of the follow ing:

,

,

,

.

,

,

,

.

,

,

,

.

down the down  the incline?

ANSWER:  =

Correct

Part E What is the work

done by the applied force of magnitude

?

Express your answe answerr in terms of some or al alll of the follow ing: ANSWER:  =

Correct

Part F What is the wor work k

done on the block by the frictional for force? ce?

Express your answe answerr in terms of some or al alll of the follow ing: ANSWER:

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 =

Correct

Exercise 7.5 A baseb baseball all is is thr throw own n fro from m the roo rooff of of 20.7 20.7 angle of 57.1 abo abov ve the hor horizontal. izontal.

-tall -ta ll build building ing wi with th an an initia initiall velo velocity city of mag magnitu nitude de 13 13.1 .1

and an d dire directed cted at at an

Part A What is the speed of the ball just before it strikes the ground? Use energy methods and ignore air resistance. ANSWER: = 24.0

Correct

Part B What is t he answer answer for for part part (A) if the initial velocity velocity is at an angle angle of 57.1 below  the  t he horizontal? horizontal? ANSWER: = 24.0

Correct

Part C If the effects of air resistance are included, will part (A) or (B) give the higher speed? ANSWER: The part (A) will give the higher speed. The part (B) will give the higher speed.

Correct

Bungee Jumping Jumping http://sessi on.master ing physi cs.com/myct/assi g nmentPr i ntView?assi g nmentID= 2808817

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Kat e, a bungee Kate, bungee jumper, wants to jump off the edge of a bridge bridge that spans a riv river er below. Kate has a mass , and the surface of the bridge is a height abov above e the water. The bungee cord, which which has has length when unstret unstretched, ched, will first straighten and then stretch as Kate falls. Assume Ass ume the follo following: wing: The Th e bungee bungee cord behav behaves as an ideal spring once once it begins to stretch, stret ch, with wit h spring const ant . 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 acc acceleration eleration due due to grav gravity ity..

Part A How far below the bridge will Kate eventually be hanging, once she stops oscillating and comes finally to rest? Assume that she doesn't touch the water. Express the distance in terms of quantities given in the problem introduction.

Hint 1. Decide how to approach the problem Here are three possible methods for solving this problem: 1. No nonconserv nonconservative ative forces are act acting, ing, so mechanic al energy energy is conserv c onserved. ed. Set Kate's K ate's gravitational potential energy at the top of the bridge equal to the spring potential energy in the bungee cord (which (which depends depends on the cord's final length ) and solve for . 2. Since Sinc e nonconserv nonconservative ative forces are act acting, ing, mechanical mechanic al energy energy is not conserv conserved. ed. Set the spring potential energy in the bungee cord (which (which depends on on ) equal equal to Kate's gravit gravitational ational potential potenti al energy plus plus the work done done by by dissipati diss ipativ ve forces. forces. Elim E liminate inate the unknown work, and solve for . 3. When Kate comes to rest s he has has zero acceleration, so the net force force acting on her her must be zero. Set t he spring force force due to the bungee cord (which (which depends on ) equal equal to the force force of gravit grav ity y and sol solv ve for .

Whic h of these options is the sim simplest plest,, most accurate way way to find

given giv en the information av available? ailable?

ANSWER: a b c

Correct

Compute pute the force due to the bungee cord cor d Hint 2. Com When Kat Kate e is at rest, what is the magnitude

of the upwar upward d force the bungee cord exerts on her?

Express your answer in terms of the cord's final stretched stretched length le ngth

and quantitie s give given n in the

problem proble m introduction. introduction. Your answe answerr should not depe nd on Kate's ma mas ss http://sessi on.master ing physi cs.com/myct/assi g nmentPr i ntView?assi g nmentID= 2808817

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Hint 1. 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) (stretched) le length ngth

and

.

ANSWER: Extension =

Hint 2. Formula for the force due to a stretched cord The formula for the force due to a stretched cord is ,  where  whe re

is the spring constant of the cord c ord and and

is t he extension extension of  of the cord.

ANSWER:  =

Incorrect; Try Again; 5 attempts remaining

ANSWER:  =

Correct

Part B If Kate just touches the surface of the river on her first downward trip (i.e., before the first bounce), what is the spring const ant ? Ignor Ignore e all diss dissipative ipative forces. Express

in terms of

,

,

, and

.

Hint 1. Decide how to approach the problem Here are three possible methods for solving this problem: 1. Since Sinc e nonconserva nonconservative tive forces are ignored, mechanical energy is conserv cons erved. ed. Set Kate's K ate'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 (which depends depends on ) and solve for . 2. Nonconservative Nonconservative forces can be ignored, so mechanical energy is conserv c onserved. ed. Set the t he spring potential energy in the bungee cord (which (which depends on on ) equal equal to Kate's gravit gravitational ational potential potenti al energy at the top of the bridge plus the work done by gravity as Kate falls. Compute the work done by grav gravity ity,, then solve for . http://sessi on.master ing physi cs.com/myct/assi g nmentPr i ntView?assi g nmentID= 2808817

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3. When Kate is being held just above above the water water she has zero acceleration, so s o the net force force acting act ing on her must be zero. Set the t he spring force due to the bungee cord (which (which depends on ) equal to the force force of grav gravit ity y and solve solve for for .

Whic h of these options is the sim simplest plest,, most accurate way way to find

given giv en the information av available? ailable?

ANSWER: a b c

gravitational potential energy Hint 2. Find the initial gravitational What is Kate's grav gravitat itational ional potenti potential al energy

at the moment she st steps eps off the bridge? (Def (Define ine 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:  =

Hint 3. Find the elastic potential energy in the bungee cord What is the elastic potential ener energy gy

stored in the bung bungee ee cord whe when n Kate is at the lowe lowest st point of her

first downward trip? Express your answer in terms of quantities given in the problem introduction.

Hint 1. Formula for elastic potential energy The elastic potential energy of the bungee cord (which we are treating as an ideal spring) is ,  where  whe re

is t he amount amount by which the cord is s tretched beyond its unstretched unst retched length. length.

Hint 2. How much is the bungee cord stretched? By how much is the bungee cord st stretched retched when Kat Kate e is at a depth Express your answer in terms of

and

below the bridge?

.

ANSWER:  =

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ANSWER:  =

ANSWER:

 =

Correct

Dancing Balls Four balls, each of mass , are are connect connected ed by four identic al relaxed springs wit with h spring const ant . The The balls are are simultaneously given given equal equal initial speeds speeds directed awa away y from from the center of of symmetry of the system. syst em.

Part A As the balls rea reach ch their t heir maximum displacement, their kinetic ener energy gy reaches _________ __________. _. ANSWER: a maximum zero neither a maximum nor zero

Correct

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Part B Use geometry geometry to find , the distance each of the springs springs has stretched from from its equilibrium position. position. (It may help to draw the initial and the final states of the system.) Express your answe answerr in terms of , the max imum dis di spla placeme ceme nt of ea ch bal balll from its initia l position. position. ANSWER:  =

Correct

Part C Find the maximum displacement Express

of any one one of the balls balls from from its initial position. position.

in terms of some or al alll of the given quantitie s ,

, and

.

Hint 1. A useful equation The Th e equation equati on

could be useful. If you are familiar with this equation, you most likely have seen the expression applied to a single mass on a single spring. For the situation with four balls and four masses, you will need to consider carefully which quantities to use in this expression.

ANSWER:  =

Correct

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

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, measured

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Part A The 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 by which which the spring is compressed compressed is negligible negligible compare compared d to .

Hint 1. Potential energy of the spring The potential energy of a spring is proportional to the square of the distance the spring is compressed. The spring was compressed half the distance, so the mass, when launched, has one quarter of the energy as in the first trial.

Hint 2. Potential energy of the ball At the highest point in the ball's trajectory, all of the spring's potential energy has been converted into gravitational potential energy of the ball.

ANSWER: height =

Correct

Stretching a Spring As illustrated in the figu figure, re, a spring with spring constant equilibrium position of the spring.

is stretc stretched hed fr from om

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to

, whe where re

is the

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Part A During which interval is the largest amount of energy required to stretch the spring?

Hint 1. How to approach the problem The force exerted on a spring to stretch or compress it from equilibrium is given by Hooke's law: ,  where  whe re

is the displacement of the spring from from equilibrium. equilibrium. Notice Notic e that t his force varies varies in magnitude: as

increases so does the magnitude of the force. On a graph of force as a function of position, the total work done by the force is represented by the area under the curve between the initial and final positions. Plot a graph of force versus displac displacement ement and compare the areas under the curv curve e from to , to , and

to

.

ANSWER: From

to

From

to

From

to

The energy required is the same in all three intervals.

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Correct A graph of the force exerted on the spring versus the displacement of the spring is shown in the figure.

Recall that on a graph of force as a function of position, the work done by the force is represented by the area under the curve. The work done by the hand in the first segment to pull the spring from to is represented by a single triangle. The The area under the second segment from

to

third segment from

is three times tim es larger than the first segment, and the area under the

to

is five five times ti mes larger than in the first segment. segment . So more energy is required

to pull the spring through the third segment.

Part B A spring is stretched fr from om compressed fr from om

to

to

, whe where re

is the equilibr equilibrium ium position of the spring. It is then

. What can be said abou aboutt the ener energy gy requ required ired to stretc stretch h or compress the

spring?

Hint 1. How to approach the problem Recall that on a graph of force as a function of position, the work done by the force is represented by the area "under" the curve, or more accurately, the area between the curve and the horizontal axis. Plot a graph of force versus displ displacement acement and compare the areas "under" the curv curve e from to and to .

ANSWER: More energy is required to stretch the spring than to compress it. The same amount of energy is required to either stretch or compress the spring. Less energy is required to stretch the spring than to compress it.

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Correct The work done to stretch or compress a spring from equilibrium is given by ,  where  where

is the distance away from equilibrium equilibrium that t he spring moves. moves. S ince

 work,  work, stretching (

) or compressing compressing (

is squared squared in the equation for for

) a spring by the same distance dis tance requires requires the t he same positive positive

amount of work.

Part C Now consider two springs A and B that are attached to a wall. Spring A has a spring constant that is four times that of the spring constant of spring B. If the same amount of energy is required to stretch both springs, what can be said about the distance each spring is stretched?

Hint 1. How to approach this problem The work done to stretch or compress a spring is given by ,  where  whe re

is the distance away from equilibrium equilibrium that t he spring is displaced.

Use this expression to relate the information provided about the work done on each spring and the spring constants to the distance each spring stretches.

r easoning to find a relationship between between the springs Hint 2. Use proportional reasoning From the problem statement you know that ,  where  whe re

and . Use this information to find an express expression ion for

.

ANSWER:

 = 0.25

ANSWER:

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Spring A must stretch 4 times as far as spring B Spring A must stretch 2 times as far as spring B. Spring A must stretch the same distance as spring B. Spring A must stretch half the distance spring B stretches. Spring A must stretch one-quarter of the distance spring B stretches.

Correct The The energy required to stretch st retch a spring is proportional proportional to that of of

and to

. If

is four times ti mes

,

must be half

, so the energy required is the same for both springs.

Part D Two Tw o identic al springs are att attached ached to two differ different ent mass masses, es,

and

, where

masses lie on a frictionless frictionless surf s urface. ace. Both springs are compressed the same distance,

is greater than

. The

, as shown in the figu figure. re.

Which of the following statements descibes the energy required to compress spring A and spring B?

ANSWER: Spring A requires more energy than spring B. Spring A requires the same amount of energy as spring B. Spring A requires less energy than spring B. Not enough information is provided to answer the question.

Correct Good job; you have have realiz realized ed an an important fact. The work done done on on a spring to compress it a distance dist ance by

is given given

. The amount of mass attached att ached to the spring does not affect affect the work required to stretch st retch or

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Exercise 7.15 A for force ce of of 70 700 0

stretche stre tches s a cer certain tain spr sprin ing g a dista distance nce of 0.4 0.400 00

.

Part A What is the potential potential ene energ rgy y of the sprin spring g when when it is stretched a distance distance of 0.400 0.400

?

ANSWER: = 140

Correct

Part B Whatt is its potentia Wha potentiall ener energy gy whe when n it is compr compresse essed d a distan distance ce of of 5.00 5.00

?

ANSWER: = 2.1 .19 9

Correct

Exercise 7.25 You ar are e asked asked to desig design n a spr spring ing tha thatt will will give a 13 1300 00 satellite sate llite a spee speed d of of 3.45 3.45 relati re lativ ve to an or orbiti biting ng spa space ce shuttle. Your Your spring is to giv give e the satellit satellite e a maximum acceleration of . The spring's mass, the recoil kinetic energy of the shuttle, and changes in gravitational potential energy will all be negligible.

Part A What must the force constant of the spring be? Tak ake e the free fal alll ac acc cel eler erat atio ion n to be

= 9.8 .80 0

.

ANSWER:  = 2.62×105

Correct

Part B What distance must the spring be compressed? http://sessi on.master ing physi cs.com/myct/assi g nmentPr i ntView?assi g nmentID= 2808817

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ANSWER: = 0.2 .24 43

Correct

Sliding In Socks Suppose that the coefficient of kinetic friction between Zak's feet and the floor, while wearing socks, is 0.250. Knowing this, Zak decides to get a running start and then slide across the floor.

Part A If Za Zak's k's speed is 3.00

when he st starts arts to sli slide, de, what dist distance ance

will he sli slide de befor before e st stopping? opping?

Express your answer in meters. ANSWER: 1.8 .84 4

Correct

Part B Now, suppose that Zak's younger cousin, Greta, sees him sliding and takes off her shoes so that she can slide as  well (assume her socks hav have e the same c oef oefficient ficient of kinetic fr fricti iction on as Zak's). Zak's). Instead of getting a running running start, she asks ask s Zak Zak to giv give e her a push. So, Zak Zak pushes pushes her with a force of 125 over ov er a dist distance ance of 1.00 . If her mass is 20.0

, what dist distance ance

does she sli slide de after Za Zak's k's push ends?

Remember that the frictional force acts on Greta during Zak's push and while she is sliding after the push. Express your answer in meters.

Hint 1. How to approach the problem This problem can be solved using work and energy. Pick the moment just before the push starts as the initial time, and and pick the point at which she stops sliding as the fina finall time. What is the change in energy between these two times? Express your answer in joules. ANSWER:  = 0

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= 1.5 .55 5

Correct

Exercise 7.32 While a roof roofer er is workin working g on a roo rooff that that slan s lants ts at 40 40.0 .0 ab abov ove e the horizon horizontal, tal, he accidentally accidentally nudge nudges s his 90.0 causing it to start sliding downward, starting from rest.

toolbox, toolb ox,

Part A If it starts s tarts 4.25

from fro m the lower edge of the roof roof,, how fast fast will the toolbox be moving moving just as it rea reaches ches the edge of of

the roof roof if the the kinetic kinetic friction friction for force ce on it is 18.0

?

ANSWER: = 6.07

Correct

Exercise 7.36 An object object moving moving in the

-plane is acted on by a conserva conservative tive force described by the potenti potential-energy al-energy function , whe where re is a positiv positive e constant.

Part A Derive Deriv e an express expression ion for the force

expressed express ed in terms of the unit vect ectors ors

and .

Express your answer in terms of the given quantities. ANSWER:

 =

Correct

Potential Energy Graphs and Motion Learning Goal: To be able to interpret potential energy diagrams and predict the corresponding motion of a particle. http://sessi on.master ing physi cs.com/myct/assi g nmentPr i ntView?assi g nmentID= 2808817

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Potential energy diagrams for a particle are useful in predicting the motion of that particle. These diagrams allow one to determine the direction of the force acting on the particle at any point, the points of stable and unstable equilibrium, the particle's kinetic energy, etc. Consider the potential energy diagram shown. The curve represents the value of potenti potential al energy as a function of the particle's particl e's coordinate . The The horizontal line abov above e the curve curve represents the constant value of the total energy of the particl par ticle e . The total ener energy gy is the sum of kinetic ( ) and potential potenti al ( ) energies of the partic particle. le. The key idea in interpreting the graph can be expressed in the equation

 where  where is the the x   x  component   component of the net force as function of the partic particle's le's coordinate . Note the negative negative sign: It means that the x  the x  component  component of the net force is negative when the derivative is positive and vice versa. For instance, if the particle is moving to the right, and its potential energy is increasing, the net force would be pulling the particle to the left.

If you are still having trouble visualizing this, consider the following: If a massive particle is increasing its gravitational potential energy (that is, moving upward), the force of gravity is pulling in the opposite opposite direction  direction (that is, downward). If the x  the x  component  component of the net force is zero, the particle is said to be in equilibrium equilibrium.. There are two kinds of equilibrium: Stable equilibrium means equilibrium means that small deviations from the equilibrium point create a net force that accelerates the particle back toward the equilibrium point (think of a ball rolling between two hills). Unstable equilibrium means equilibrium means that small deviations from the equilibrium point create a net force that accelerates the particle further away from the equilibrium point (think of a ball on top of a hill).

In answering the following following questi questions, ons, we will assume that there is a single varying varying force force acting act ing on the partic particle le along the x  the  x  axis.   axis. Therefore, we will use the term force force instead  instead of the cumbersome x cumbersome x component component of the net force f orce..

Part A The force acting on the particle at point A is __________.

Hint 1. Sign of the derivative If a funct function ion increases (as

increases)) in a certai increases certain n region, region, then the deriv derivative ative of the funct function ion in that region is

positive.

Hint 2. Sign of the component If

increases to the right, right, as in the graph shown, then a (one(one-dimensi dimensional) onal) vect ector or with a posit positiv ive e x 

component points to the right, and vice versa.

ANSWER:

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directed to the right directed to the left equal to zero

Correct Consider the graph in the region of point A. If the particle is moving to the right, it would be "climbing the hill," and the force would "pull it down," that is, pull the particle back to the left. Another, more abstract way of thinking about this is to say that the slope of the graph at point A is positive is positive;; therefore, therefore, the direction direct ion of

is

negative. negative.

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

Hint 1. Sign of the derivative If a funct function ion increases (as

increases)) in a certai increases certain n region, region, then the deriv derivative ative of the funct function ion in that region is

positive, and vice versa.

Hint 2. Sign of the component If

increases to the right, right, as in the graph shown, then a (one(one-dimensi dimensional) onal) vect ector or with a posit positiv ive e x 

component points to the right, and vice versa.

ANSWER: directed to the right directed to the left equal to zero

Correct

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

Hint 1. Derivative of a function at a local maximum At a local maximum, the derivative of a function is equal to zero.

ANSWER:

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directed to the right directed to the left equal to zero

Correct The The slope of the graph is zero; therefore, the derivative derivative

, and

.

Part D The acceleration of the particle at point B is __________.

between acceleration and force Hint 1. Relation between The relation between acceleration and force is given by Newton's 2nd law, .

ANSWER: directed to the right directed to the left equal to zero

Correct If the net force is zero, so is the acceleration. The particle is said to be in a state of equilibrium. equilibrium.

Part E If the particle is located slightly to the left of point B, its acceleration is __________.

Hint 1. The force on such a par ticle To the left of B,

is an increasi increasing ng function and so its deriv derivative ative is posit positiv ive. e. Th This is impli implies es that the the x   x 

component of the force on a particle at this location is negative, or that the force is directed to the left, just like at A. What can you say now about the acceleration?

ANSWER:

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directed to the right directed to the left equal to zero

Correct

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

Hint 1. The force on such a par ticle To the right of B,

is a decreasing function and so its deriv derivative ative is negativ negative. e. Th This is impli implies es that the the x   x 

component of the force on a particle at this location is positive, or that the force is directed to the right, just like at C. What can you now say about the acceleration?

ANSWER: directed to the right directed to the left equal to zero

Correct As you can see, small deviations from equilibrium at point B cause a force that accelerates the particle further away; hence the particle is in unstable equilibrium. equilibrium.

Part G Name all labeled points on the graph corresponding to unstable unstable equilibrium.  equilibrium. List your choices alphabetically, with no commas or spaces; for instance, if you choose points B, D, and E, type your answer as BDE .

Definition ion of unstable equilibrium Hint 1. Definit Unstable equilibrium means that small deviations from the equilibrium point create a net force that accelerates the particle further away from the equilibrium point (think of a ball on top of a hill).

ANSWER: BF

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Correct

Part H Name all labeled points on the graph corresponding to stable stable equilibrium.  equilibrium. List your choices alphabetically, with no commas or spaces; for instance, if you choose points B, D, and E, type your answer as BDE .

Definition ion of stable equilibrium Hint 1. Definit Stable equilibrium means equilibrium means that small deviations from the equilibrium point create a net force that accelerates the particle back toward the equilibrium point. (Think of a ball rolling between two hills.)

ANSWER: DH

Correct

Part I Name all labeled points on the graph where the acceleration of the particle is zero. List your choices alphabetically, with no commas or spaces; for instance, if you choose points B, D, and E, type your answer as BDE .

between acceleration and force Hint 1. Relation between The relation between acceleration and force is given by Newton's 2nd law, .

ANSWER: BDFH

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

Part J Name all labeled points such that when a particle is released from rest there, it would accelerate to the left. List your choices alphabetically, with no commas or spaces; for instance, if you choose points B, D, and E, type your answer as BDE . http://sessi on.master ing physi cs.com/myct/assi g nmentPr i ntView?assi g nmentID= 2808817

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Determine ine the sign of the the x   x  component  component of force Hint 1. Determ If the acceleration is to the left, so is the force. This means that the x  the  x  component  component of the force is __________. ANSWER: positive negative

Hint 2. What is the behavior of

?

If the x  the x component component of the force at a point is negativ negative, e, then the deriv derivative ative of means that in the region around the point

at that point is posit positiv ive. e. This

is ________ __________. __.

ANSWER: increasing decreasing

ANSWER: AE

Correct

Part K Consider points A, E, and G. Of these three points, which one corresponds to the greatest magnitude of acceleration of the particle?

Acceleration eration and force Hint 1. Accel The greatest acceleration corresponds to the greatest magnitude of the net force, represented on the graph by the magnitude of the slope.

ANSWER: A E G

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Correct

Kinetic Kineti c energy ener gy If the tot total al energy of the partic particle le is known, one can also use the graph of energy ener gy of the particle particl e since

to draw concl conclusions usions about the kinet kinetic ic

. As a remind reminder, er, on this graph, graph, the total ener energy gy

is shown by the hor horizontal izontal line.

Part L What point on the graph corresponds to the maximum kinetic energy of the moving particle?

Hint 1.

,

, and

Since the total energy does not change, the maximum kinetic energy corresponds to the minimum potential energy.

ANSWER: D

Correct It makes sense that the kinetic energy of the particle is maximum at one of the (force) equilibrium points. For example, think of a pendulum (which has only one force equilibrium point--at the very bottom).

Part M At what point on the graph does the particle have the lowest speed? ANSWER: B

Correct As you can see, many different conclusions can be made about the particle's motion merely by looking at the graph. It is helpful to understand the character of motion qualitatively before you attempt quantitative problems. This problem should prove useful in improving such an understanding.

Score Summary: Your score on this assignment is 101%. You received 14.07 out of a possible total of 14 points.

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