Compression Guide Work and Energydysoncentralne.pbworks.com/w/file/fetch/102939991/Physics Ch 5.pdf · • Recognize the difference between the scientific and ordinary defi- ... Work

CHAPTER REVIEW, ASSESSMENT, AND STANDARDIZED TEST PREPARATION

pp. 179 –181PACING • 45 min OSP Lesson Plans EXT Integrating Chemistry Chemical Reactions

b

ANC CBLTM Experiment Loss of Mechanical Energy*◆ aSection 4 Power

• Relate the concepts of energy, time, and power.• Calculate power in two different ways.• Explain the effect of machines on work and power.

Online and Technology Resources

Visit go.hrw.com to access online resources. Click Holt Online Learning for an online edition of this textbook, or enter the keyword HF6 Home for other resources. To access this chapter’s extensions, enter the keyword HF6WRKXT.

Planning Guide

Chapter Openerpp. 158 –160 CD Visual Concepts, Chapter 5 bANC Discovery Lab Exploring Work and Energy*◆ b

Section 1 WorkSection 1 Work• Recognize the difference between the scientific and ordinary defi-

nitions of work.• Define work by relating it to force and displacement.• Identify where work is being performed in a variety of situations.• Calculate the net work done when many forces are applied to an

object.

TE Demonstration Work, p.160 b TE Demonstration Quantifying Work, p.162 g

OSP Lesson Plans CD Interactive Tutor Module 6, Work–Kinetic

Energy Theorem gOSP Interactive Tutor Module 6, Worksheet gEXT Integrating Health Energy Costs of Walking

and Running b TR 17 Defining Potential Energy TR 18 Elastic Potential Energy

TE Demonstration Potential Energy, p.169 gANC Invention Lab Bungee Jumping: Energy*◆ a

158A Chapter 5 Work and Energy

Work and Energy To shorten instruction because of time limitations, omit the opener and abbrevi-ate the review.

Compression Guide

CHAPTER 5

pp. 164 –172 PACING • 90 min

PACING • 45 min

SE Chapter Highlights, p. 183 SE Chapter Review, pp. 184 –189 SE Graphing Calculator Practice, p. 188 g SE Alternative Assessment, p. 189 a SE Standardized Test Prep, pp. 190 –191 g SE Appendix D: Equations, p. 856 SE Appendix I: Additional Problems, pp. 883 – 884ANC Study Guide Worksheet Mixed Review* gANC Chapter Test A* gANC Chapter Test B* a OSP Test Generator

PACING • 90 min

Section 2 Energy• Identify several forms of energy.• Calculate kinetic energy for an object.• Apply the work–kinetic energy theorem to solve problems.• Distinguish between kinetic and potential energy.• Classify different types of potential energy.• Calculate the potential energy associated with an object’s position.

OSP Lesson Plans CD Interactive Tutor Module 5, Work gOSP Interactive Tutor Module 5, Worksheet gEXT Integrating Biology Muscles and Work b TR 15 Definition of Work TR 16 Positive and Negative Values of Work

OSP Lesson Plans TR 19 Friction and the Non-conservation of

Mechanical Energy TR 19A Classification of Energy TR 20A Energy of a Falling 75 g Egg

SE Quick Lab Mechanical Energy, p. 175 g SE Skills Practice Lab Conservation of Mechanical Energy

pp. 192–195 ◆ g

ANC Datasheet Conservation of Mechanical Energy* g TE Demonstration Mechanical Energy, p.173 a TE Demonstration Conservation of Energy, p.174 gANC CBLTM Experiment Conserv. Mechanical Energy*◆ g

pp. 173 –178 PACING • 90 minSection 3 Conservation of Energy• Identify situations in which conservation of mechanical

energy is valid.• Recognize the forms that conserved energy can take.• Solve problems using conservation of mechanical energy.

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• Lab Materials QuickList Software

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SE Sample Set F Power, pp. 180 –181 b TE Classroom Practice, p. 180 bANC Problem Workbook* and OSP Problem Bank Sample Set F b SE Conceptual Challenge, p. 179

SE Section Review, p. 181 gANC Study Guide Worksheet Section 4* gANC Quiz Section 4* b

UCP 1, 2, 3, 5ST 1, 2SPSP 5PS 5b

Maintained by the National Science Teachers Association. This CD-ROM consists of interactive activities that give students a fun way to extend their knowledge of physics concepts.

Topic: WorkSciLinks Code: HF61674Topic: Potential and Kinetic

EnergySciLinks Code: HF61196

Topic: Conservation of Energy

SciLinks Code: HF60345

This CD-ROM consists of multimedia presenta-tions of core physics concepts.

National ScienceEducation Standards

SE Sample Set A Work, pp. 161–162 b TE Classroom Practice, p. 161 bANC Problem Workbook* and OSP Problem Bank Sample Set A b


UCP 1, 2, 3SAI 1, 2


UCP 1, 2, 3HNS 1, 3PS 5b

Chapter 5 Planning Guide 158B

VisualConcepts

SE Sample Set B Kinetic Energy, pp. 165 –166 b TE Classroom Practice, p. 165 bANC Problem Workbook* and OSP Problem Bank Sample Set B b SE Sample Set C Work–Kinetic Energy Theorem, pp. 167–168 gANC Problem Workbook* and OSP Problem Bank Sample Set C g SE Sample Set D Potential Energy, pp. 171–172 g TE Classroom Practice, p. 171 gANC Problem Workbook* and OSP Problem Bank Sample Set D g


UCP 1, 2, 3SAI 1, 2PS 5a, 5b

SE Sample Set E Conservation of Mechanical Energy, pp. 176 –177 g TE Classroom Practice, p. 176 gANC Problem Workbook* and OSP Problem Bank Sample Set E g SE Appendix J: Advanced Topics The Equivalence of Mass and Energy,

pp. 918 – 919 a

SKILLS DEVELOPMENT RESOURCES REVIEW AND ASSESSMENT CORRELATIONS

KEY SE Student Edition TE Teacher Edition ANC Ancillary Worksheet

OSP One-Stop Planner CD CD or CD-ROM TR Teaching Transparencies

EXT Online Extension * Also on One-Stop Planner ◆ Requires advance prep

158

Section 1 introduces work andshows calculations of the workdone in a variety of situations.

Section 2 identifies and showscalculations using kinetic energy,the work–kinetic energy theo-rem, and different types ofpotential energy.

Section 3 explores the conditionsnecessary for conservation ofmechanical energy and appliesthis principle to problem solving.

Section 4 introduces the rela-tionships among work, time,power, force, and speed.

About the IllustrationThis audiokinetic sculpture wascreated by George Rhoads, whosesculptures can be seen at theBoston Museum of Science, at thePort Authority Bus Terminal inNew York City, and in variousshopping centers. After complet-ing the chapter, have studentsreturn to this photograph andapply the concepts of work andthe conservation of energy todescribe which balls probablyhave mostly potential energy andwhich have mostly kinetic energy.

Interactive Problem-Solving Tutor

See Module 5“Work” provides additional prac-tice calculating net work.

See Module 6“Work–Kinetic Energy Theorem”promotes additional develop-ment of problem-solving skillsinvolving work.

CHAPTER 5Overview

159159

This whimsical piece of art is called an audiokinetic sculp-ture. Balls are raised to a high point on the curved bluetrack. As the balls move down the track, they turn levers,spin rotors, and bounce off elastic membranes. The energythat each ball has—whether associated with the ball’smotion, the ball’s position above the ground, or the ball’sloss of mechanical energy due to friction—varies in a waythat keeps the total energy of the system constant.

CHAPTER 5

PE

PE+KE

KE

Work andEnergy

WHAT TO EXPECTIn this chapter, you will learn about work anddifferent types of energy that are relevant tomechanics. Kinetic energy, which is associatedwith motion, and potential energy, which isrelated to an object’s position, are two forms of energy that you will study.

WHY IT MATTERSWork, energy, and power are related to oneanother. Everyday machines such as motors areusually described by the amount of work thatthey are capable of doing or by the amount ofpower that they produce.

CHAPTER PREVIEW

1 WorkDefinition of Work

2 EnergyKinetic EnergyPotential Energy

3 Conservation of EnergyConserved QuantitiesMechanical Energy

4 PowerRate of Energy Transfer

Knowledge to Expect✔ “Students learn that energy

cannot be created ordestroyed, but only changedfrom one form to another.”(AAAS’s Benchmarks for Sci-ence Literacy, grades 6–8)

✔ “Energy is associated withheat, light, electricity,mechanical motion, sound,and the nature of a chemi-cal. Energy is transferred inmany ways.” (NRC’s Na-tional Science EducationStandards, grades 5–8)

✔ “Students tend to (a) associ-ate energy with living things;(b) believe that energy is afuel-like quantity; (c) thinkthat energy transformationsinvolve only one form ofenergy at a time.” (AAAS’sBenchmarks for Science Lit-eracy, The Research Base)

Knowledge to Review✔ Review the kinematic

equations.

✔ Newton’s second law statesthat force = mass × accelera-tion (F = ma).

✔ Kinetic friction is a resistiveforce exerted on a movingbody by its environment.

Items to Probe✔ Familiarity with phenomena

of energy transformation:Ask students to describe theaction of jumping up anddown on a trampoline interms of energy.

✔ Preconceptions about en-ergy dissipation: Ask students if energy is everlost in a process.

Tapping PriorKnowledge

160

WorkSECTION 1General Level SECTION 1

DEFINITION OF WORK

Many of the terms you have encountered so far in this book have meanings in

physics that are similar to their meanings in everyday life. In its everyday

sense, the term work means to do something that takes physical or mental

effort. But in physics, work has a distinctly different meaning. Consider the

following situations:

• A student holds a heavy chair at arm’s length for several minutes.

• A student carries a bucket of water along a horizontal path while walking

at constant velocity.

It might surprise you to know that as the term work is used in physics,

there is no work done on the chair or the bucket, even though effort is

required in both cases. We will return to these examples later.

Work is done on an object when a force causes a displacement of the object

Imagine that your car, like the car shown in Figure 1, has run out of gas and you

have to push it down the road to the gas station. If you push the car with a con-

stant horizontal force, the you do on the car is equal to the magnitude of

the force, F, times the magnitude of the displacement of the car. Using the sym-

bol d instead of Δx for displacement, we define work for a constant force as:

W = Fd

Work is not done on an object unless the object is moved with the

action of a force. The application of a force alone does not consti-

tute work. For this reason, no work is done on the chair when a stu-

dent holds the chair at arm’s length. Even though the student exerts

a force to support the chair, the chair does not move. The student’s

tired arms suggest that work is being done, which is indeed true.

The quivering muscles in the student’s arms go through many small

displacements and do work within the student’s body. However,

work is not done on the chair.

Work is done only when components of a force are parallel to a displacement

When the force on an object and the object’s displacement are in

different directions, only the component of the force that is parallel

to the object’s displacement does work. Components of the force

perpendicular to a displacement do not do work.

work

Chapter 5160

SECTION OBJECTIVES

■ Recognize the differencebetween the scientific andordinary definitions of work.

■ Define work by relating it toforce and displacement.

■ Identify where work is beingperformed in a variety ofsituations.

■ Calculate the net work donewhen many forces areapplied to an object.

Demonstration

WorkPurpose Determine whetherwork is done in various situations.Materials teacher’s text, springscale, stringProcedure Hang the textbookfrom the scale with the string.Hold the book stationary, have thestudents note the scale reading,and record the weight of the book(mg). Ask the students if thespring is exerting a force on thebook. (Yes, the spring exerts a forceon the book that is equal to andopposite the book’s weight.) Ask stu-dents if the spring is doing workon the book, which is being heldin a fixed position. (No, because thedisplacement of the book is zero.)

Now lift the book by pullingup on the spring at a constantvelocity and have the studentsnote the scale reading. Again askthe students if the spring is doingwork on the book. (Yes, the forceof the lifting, equal in magnitudeto the weight of the book, isupward, and the displacement isupward.) Have students calculatethe amount of work (m × g × h).

Hold the book at shoulderheight and carry it across theroom at a constant speed. Askstudents if work is being done on the book. (No, because theupward force is perpendicular tothe horizontal displacement.)

work

the product of the component ofa force along the direction of dis-placement and the magnitude ofthe displacement

Figure 1This person exerts a constant force on the car and displaces it to the left. The work done on thecar by the person is equal to the force the personexerts times the displacement of the car.

161

SECTION 1For example, imagine pushing a crate along the ground. If the force you

exert is horizontal, all of your effort moves the crate. If your force is at an

angle, only the horizontal component of your applied force causes a displace-

ment and contributes to the work. If the angle between the force and the

direction of the displacement is q, as in Figure 2, work can be expressed as

follows:

W = Fdcosq

If q = 0°, then cos 0° = 1 and W = Fd, which is the definition of work given

earlier. If q = 90°, however, then cos 90° = 0 and W = 0. So, no work is done on

a bucket of water being carried by a student walking horizontally. The upward

force exerted by the student to support the bucket is perpendicular to the dis-

placement of the bucket, which results in no work done on the bucket.

Finally, if many constant forces are acting on an object, you can find the net

work done on the object by first finding the net force on the object.

Work has dimensions of force times length. In the SI system, work has a unit

of newtons times meters (N•m), or joules (J). To give you an idea of how large a

joule is, consider that the work done in lifting an apple from your waist to the

top of your head is about 1 J.

NET WORK DONE BY A CONSTANT NET FORCE

Wnet = Fnetdcosq

net work = net force × displacement × cosine of the angle between them

Teaching TipPoint out that although the cratein Figure 2 is a large object, wethink of its mass as reduced to apoint at its center to simplify thesituation.

GENERAL

161Work and Energy

F

d

W = Fd cos

θ

θ Figure 2The work done on this crate isequal to the force times the dis-placement times the cosine of theangle between them.

The joule is named for the Britishphysicist James Prescott Joule( 18 18– 1889). Joule made majorcontributions to the understandingof energy, heat, and electricity.

Did you know?

SAMPLE PROBLEM A

Work

P R O B L E MHow much work is done on a vacuum cleaner pulled 3.0 m by a force of50.0 N at an angle of 30.0° above the horizontal?

S O L U T I O NGiven: F = 50.0 N q = 30.0° d = 3.0 m

Unknown: W = ?

Use the equation for net work done by a constant force:

W = Fd cosq

Only the horizontal component of the applied force is doing work on the

vacuum cleaner.

W = (50.0 N)(3.0 m)(cos 30.0°)

W = 130 J

WorkA 20.0 kg suitcase is raised 3.0 mabove a platform by a conveyorbelt. How much work is done onthe suitcase?

Answer5.9 × 102 J

W SE Sample, 1–3; Ch.Rvw. 7–10, 45–46, 50

PW 8–11PB 5–7

F SE Ch. Rvw. 50PW 5–7PB Sample, 1–4

d SE 4PW Sample, 1–4PB 8–10

PROBLEM GUIDE A

Solving for:

Use this guide to assign problems.SE = Student Edition TextbookPW = Problem WorkbookPB = Problem Bank on the

One-Stop Planner (OSP)

*Challenging ProblemConsult the printed Solutions Manual or the OSP for detailed solutions.


See Module 5“Work” provides additional prac-tice calculating net work.

162

SECTION 1

The sign of work is important

Work is a scalar quantity and can be positive or negative, as shown in Figure 3.Work is positive when the component of force is in the same direction as the

displacement. For example, when you lift a box, the work done by the force

you exert on the box is positive because that force is upward, in the same direc-

tion as the displacement. Work is negative when the force is in the direction

ANSWERS

Practice A1. 1.50 × 107 J

2. 7.0 × 102 J

3. 1.6 × 103 J

4. 1.1 m

Chapter 5162

Figure 3Depending on the angle, an appliedforce can either cause a moving carto slow down (left), which results innegative work done on the car, orspeed up (right), which results inpositive work done on the car.

F F

F F

d

dd

d

Negative (–) work Positive (+) work

PRACTICE A

Work

1. A tugboat pulls a ship with a constant net horizontal force of 5.00 × 103 N

and causes the ship to move through a harbor. How much work is done on

the ship if it moves a distance of 3.00 km?

2. A weight lifter lifts a set of weights a vertical distance of 2.00 m. If a con-

stant net force of 350 N is exerted on the weights, what is the net work done

on the weights?

3. A shopper in a supermarket pushes a cart with a force of 35 N directed at

an angle of 25° downward from the horizontal. Find the work done by the

shopper on the cart as the shopper moves along a 50.0 m length of aisle.

4. If 2.0 J of work is done in raising a 180 g apple, how far is it lifted?

PHYSICSPHYSICSModule 5“Work”provides an interactive lessonwith guided problem-solvingpractice to teach you about calculating net work.

Demonstration

Quantifying WorkPurpose Demonstrate the rela-tionship between the directionand the magnitude of a force.Materials plastic sled or piece ofcardboard, 3 m and 1 m lengthsof ropeProcedure Attach both ropes tothe sled, and ask a student volun-teer to sit on the sled. Ask stu-dents whether it will requiremore force to pull the sled acrossthe floor with the 1 m rope orwith the 3 m rope. (The 3 m ropewill require less force.) Have a student try to pull the sled witheach rope and report to the classwhich way is easier. Sketch bothsituations on the board, empha-sizing that the horizontal compo-nent of the force is smaller withthe short rope because it is heldat a greater angle above the horizontal.

GENERAL

Integrating BiologyVisit go.hrw.com for the activity“Muscles and Work.”

Keyword HF6WRKX

163

SECTION 1opposite the displacement. For example, the force of kinetic friction between

a sliding box and the floor is opposite to the displacement of the box, so the

work done by the force of friction on the box is negative. If you are very care-

ful in applying the equation for work, your answer will have the correct sign:

cos q is negative for angles greater than 90° but less than 270°.

If the work done on an object results only in a change in the object’s speed,

the sign of the net work on the object tells you whether the object’s speed is

increasing or decreasing. If the net work is positive, the object speeds up and

work is done on the object. If the net work is negative, the object slows down

and work is done by the object on something else.

Teaching TipWrite the following table on theboard to help students rememberthe various situations that affectthe sign of work.

Force is in positive workdirection ofmotion.

Force opposes negative workmotion.

Force is 90° to no workmotion.

Object is not no workin motion.

163Work and Energy

Developed and maintained by theNational Science Teachers Association

For a variety of links related to thischapter, go to www.scilinks.org

Topic: WorkSciLinks Code: HF61674

SECTION REVIEW

1. For each of the following cases, indicate whether the work done on the

second object in each example will have a positive or a negative value.

a. The road exerts a friction force on a speeding car skidding to a stop.

b. A rope exerts a force on a bucket as the bucket is raised up a well.

c. Air exerts a force on a parachute as the parachutist falls to Earth.

2. If a neighbor pushes a lawnmower four times as far as you do but exerts

only half the force, which one of you does more work and by how much?

3. A worker pushes a 1.50 × 103 N crate with a horizontal force of 345 N a

distance of 24.0 m. Assume the coefficient of kinetic friction between the

crate and the floor is 0.220.

a. How much work is done by the worker on the crate?

b. How much work is done by the floor on the crate?

c. What is the net work done on the crate?

4. A 0.075 kg ball in a kinetic sculpture moves at a constant speed along a

motorized vertical conveyor belt. The ball rises 1.32 m above the ground.

A constant frictional force of 0.350 N acts in the direction opposite the

conveyor belt’s motion. What is the net work done on the ball?

5. Critical Thinking For each of the following statements, identify

whether the everyday or the scientific meaning of work is intended.

a. Jack had to work against time as the deadline neared.

b. Jill had to work on her homework before she went to bed.

c. Jack did work carrying the pail of water up the hill.

6. Critical Thinking Determine whether work is being done in each of

the following examples:

a. a train engine pulling a loaded boxcar initially at rest

b. a tug of war that is evenly matched

c. a crane lifting a car

1. a. negativeb. positivec. negative

2. the neighbor; twice as much

3. a. 8.28 × 103 Jb. −7.92 × 103 Jc. 3.6 × 102 J

4. 0.00 J

5. a. everyday senseb. everyday sensec. scientific sense

6. a. yesb. noc. yes

SECTION REVIEWANSWERS

164

EnergySECTION 2General Level SECTION 2

KINETIC ENERGY

Kinetic energy is energy associated with an object in motion. Figure 4 shows a

cart of mass m moving to the right on a frictionless air track under the action

of a constant net force, F, acting to the right. Because the force is constant, we

know from Newton’s second law that the cart moves with a constant accelera-

tion, a. While the force is applied, the cart accelerates from an initial velocity vi

to a final velocity vf . If the cart is displaced a distance of ∆x, the work done by

F during this displacement is

Wnet = F∆x = ma∆x

When you studied one-dimensional motion, you learned that the following

relationship holds when an object undergoes constant acceleration:

vf2 = vi

2 + 2a∆x

a∆x =

Substituting this result into the equation Wnet = ma∆x gives

Wnet = m� �Wnet = 1

2mvf

2 − 12

mvi2

Kinetic energy depends on speed and mass

The quantity 12

mv2 has a special name in physics: The kinetic

energy of an object with mass m and speed v, when treated as a particle, is

given by the expression shown on the next page.

kinetic energy.

vf2 − vi

2

�2

vf2 − vi

2

�2

Chapter 5164

SECTION OBJECTIVES

■ Identify several forms ofenergy.

■ Calculate kinetic energy foran object.

■ Apply the work–kinetic ener-gy theorem to solve prob-lems.

■ Distinguish between kineticand potential energy.

■ Classify different types ofpotential energy.

■ Calculate the potential energy associated with anobject’s position.

MisconceptionAlert

Students may think that kineticenergy depends on the directionof motion. Ask them to comparethe kinetic energy of identicalcars traveling at the same speedin each of the following situa-tions: one driving north, one dri-ving south, one driving uphill,and one driving downhill. (Thekinetic energy is the same in eachcase because kinetic energydepends only on mass and speed,which are the same in each case.)

The Language of PhysicsThe symbol for kinetic energy,KE , may look like the product oftwo variables (K and E) to somestudents. Point out that the twoletters together designate kineticenergy. This symbol for kineticenergy is not universal. Somebooks use the letter K alone;others use E alone and specify the kind of energy in context.

STOP

kinetic energy

the energy of an object that isdue to the object’s motion

Figure 4The work done on an object by aconstant force equals the object’smass times its acceleration times itsdisplacement.

Integrating HealthVisit go.hrw.com for the activity“Energy Costs of Walking and Running.”

Keyword HF6WRKX

165

SECTION 2

Kinetic energy is a scalar quantity, and the SI unit for kinetic energy (and all

other forms of energy) is the joule. Recall that a joule is also used as the basic

unit for work.

Kinetic energy depends on both an object’s speed and its mass. If a bowling

ball and a volleyball are traveling at the same speed, which do you think has

more kinetic energy? You may think that because they are moving with identi-

cal speeds they have exactly the same kinetic energy. However, the bowling

ball has more kinetic energy than the volleyball traveling at the same speed

because the bowling ball has more mass than the volleyball.

KINETIC ENERGY

KE = 12

mv2

kinetic energy = 12

× mass × (speed)2

Teaching TipA joule is defined as a kilogram-meter squared per secondsquared. Point out to studentshow this equation for kineticenergy has the units of energy. Besure that they know that massmust be in kg and speed must bein m/s for the units of energy towork out properly.

GENERAL

165Work and Energy

PHYSICSPHYSICSModule 6“Work–Kinetic Energy Theorem”provides an interactive lessonwith guided problem-solvingpractice.

SAMPLE PROBLEM B

Kinetic Energy

P R O B L E MA 7.00 kg bowling ball moves at 3.00 m/s. How fast must a 2.45 g table-tennis ball move in order to have the same kinetic energy as the bowlingball? Is this speed reasonable for a table-tennis ball in play?

S O L U T I O NGiven: The subscripts b and t indicate the bowling ball and the

table-tennis ball, respectively.

mb = 7.00 kg mt = 2.45 g vb = 3.00 m/s

Unknown: vt = ?

First, calculate the kinetic energy of the bowling ball.

KEb = 12

mbvb2 = 1

2(7.00 kg)(3.00 m/s)2 = 31.5 J

Then, solve for the speed of the table-tennis ball having the same kinetic

energy as the bowling ball.

KEt = 12

mtv t2 = KEb = 31.5 J

vt =�2K�m�E

t�b� =��

This speed would be very fast for a table-tennis ball.

vt = 1.60 × 102 m/s

(2)(31.5 J)2.45 × 10−3 kg

Kinetic EnergyA 6.0 kg cat runs after a mouse at10.0 m/s. What is the cat’s kineticenergy?

Answer3.0 × 102 J

KE SE 3–4; Ch. Rvw. 14,19, 44

PW 6, 8–9PB Sample, 1–4

v SE Sample, 1–2; Ch.Rvw. 20, 37, 44, 48

PW 5–7PB 8–10

m SE 5PW Sample, 1–4PB 5–7

PROBLEM GUIDE B

Solving for:




166

SECTION 2

The net work done on a body equals its change in kinetic energy

The equation Wnet = 12

mv2f − 1

2mv2

i derived at the beginning of this section says

that the net work done by a net force acting on an object is equal to the change

in the kinetic energy of the object. This important relationship, known as the

is often written as follows:

When you use this theorem, you must include all the forces that do work

on the object in calculating the net work done. From this theorem, we see that

the speed of the object increases if the net work done on it is positive, because

the final kinetic energy is greater than the initial kinetic energy. The object’s

speed decreases if the net work is negative, because the final kinetic energy is

less than the initial kinetic energy.

The work–kinetic energy theorem allows us to think of kinetic energy as the

work that an object can do while the object changes speed or as the amount of

energy stored in the motion of an object. For example, the moving hammer in

the ring-the-bell game in Figure 5 has kinetic energy and can therefore do work

on the puck. The puck can do work against gravity by moving up and striking

the bell. When the bell is struck, part of the energy is converted into sound.

WORK–KINETIC ENERGY THEOREM

Wnet = ∆KE

net work = change in kinetic energy

work–kinetic energy theorem,

ANSWERS

Practice B1. 1.7 × 102 m/s

2. 38.8 m/s

3. the bullet with the greatermass; 2 to 1

4. 2.4 J, 9.6 J; the bullet with the greater speed; 1 to 4

5. 1.6 × 103 kg

The Languageof PhysicsThe symbol ∆ (the Greek letterdelta) is used to denote change.Students should be familiar withthis symbol from earlier chapters.Point out that although the con-text is different, the symbolmeans the same thing; namely, adifference between two quanti-ties. The subscripts i and f usedwith ME stand for initial andfinal amounts, respectively, ofmechanical energy. Thus, ∆ME isthe difference between MEf andMEi, or ∆ME = MEf − MEi.

Chapter 5166

PRACTICE B

Kinetic Energy

1. Calculate the speed of an 8.0 × 104 kg airliner with a kinetic energy of

1.1 × 109 J.

2. What is the speed of a 0.145 kg baseball if its kinetic energy is 109 J?

3. Two bullets have masses of 3.0 g and 6.0 g, respectively. Both are fired

with a speed of 40.0 m/s. Which bullet has more kinetic energy? What is

the ratio of their kinetic energies?

4. Two 3.0 g bullets are fired with speeds of 40.0 m/s and 80.0 m/s, respec-

tively. What are their kinetic energies? Which bullet has more kinetic

energy? What is the ratio of their kinetic energies?

5. A car has a kinetic energy of 4.32 × 105 J when traveling at a speed of

23 m/s. What is its mass?

Figure 5The moving hammer has kineticenergy and can do work on thepuck, which can rise against gravityand ring the bell.

work–kinetic energy theorem

the net work done by all theforces acting on an object isequal to the change in theobject’s kinetic energy

167

SECTION 2

167Work and Energy

SAMPLE PROBLEM C

Work–Kinetic Energy Theorem

P R O B L E MOn a frozen pond, a person kicks a 10.0 kg sled, giving it an initial speed of2.2 m/s. How far does the sled move if the coefficient of kinetic frictionbetween the sled and the ice is 0.10?

S O L U T I O NGiven: m = 10.0 kg vi = 2.2 m/s vf = 0 m/s mk = 0.10

Unknown: d = ?

Diagram:

Choose an equation or situation:This problem can be solved using the definition of work and the work–

kinetic energy theorem.

Wnet = Fnetdcosq

The net work done on the sled is provided by the force of kinetic friction.

Wnet = Fkdcosq = mkmgd cosq

The force of kinetic friction is in the direction opposite d, so q = 180°.Because the sled comes to rest, the final kinetic energy is zero.

Wnet = ΔKE = KEf − KEi = − ⎯12

⎯ mv2i

Use the work-kinetic energy theorem, and solve for d.

− ⎯12

⎯ mv2i = mkmgd cosq

d =

Substitute values into the equation:

d =

According to Newton’s second law, the acceleration of the sled is about −1 m/s2

and the time it takes the sled to stop is about 2 s. Thus, the distance the sled trav-

eled in the given amount of time should be less than the distance it would have

traveled in the absence of friction.

2.5 m < (2.2 m/s)(2 s) = 4.4 m

d = 2.5 m

−(2.2 m/s)2

⎯⎯⎯⎯2(0.10)(9.81 m/s2)(cos 180°)

−v2i⎯⎯

2mkg cos q

vi

d

Fk

1. DEFINE

2. PLAN

3. CALCULATE

4. EVALUATE


d SE Sample, 1–3; Ch.Rvw. 21–22

PW 7PB 5

F PW 3PB 3–4

W SE 5; Ch. Rvw. 44, 46PW 5PB 6

KE SE 5; Ch. Rvw. 40, 44PW Sample, 1–2PB 7–8

v SE 5; Ch. Rvw. 38, 44PW 4PB 9–10

m PW 6PB Sample, 1–2

PROBLEM GUIDE C

Solving for:




See Module 6“Work–Kinetic Energy Theorem”promotes additional develop-ment of problem-solving skills.

168

SECTION 2

ANSWERS

Practice C1. 7.8 m

2. 21 m

3. 5.1 m

4. 3.0 × 102 N

Chapter 5168

PRACTICE C

Work–Kinetic Energy Theorem

1. A student wearing frictionless in-line skates on a horizontal surface is

pushed by a friend with a constant force of 45 N. How far must the stu-

dent be pushed, starting from rest, so that her final kinetic energy is 352 J?

2. A 2.0 × 103 kg car accelerates from rest under the actions of two forces.

One is a forward force of 1140 N provided by traction between the

wheels and the road. The other is a 950 N resistive force due to various

frictional forces. Use the work–kinetic energy theorem to determine how

far the car must travel for its speed to reach 2.0 m/s.

3. A 2.1 × 103 kg car starts from rest at the top of a driveway that is sloped

at an angle of 20.0° with the horizontal. An average friction force of

4.0 × 103 N impedes the car’s motion so that the car’s speed at the bot-

tom of the driveway is 3.8 m/s. What is the length of the driveway?

4. A 75 kg bobsled is pushed along a horizontal surface by two athletes.

After the bobsled is pushed a distance of 4.5 m starting from rest, its

speed is 6.0 m/s. Find the magnitude of the net force on the bobsled.

THE INSIDE STORYON THE ENERGY IN FOOD

This feature introduces students toanother form of potential energy:chemical energy. Chemical energy,like gravitational potential energyand elastic potential energy, is alatent, stored form of energy.However, chemical energy is notsimply or directly dependent onrelative position. Instead, chemicalenergy depends on the molecularstructure and the strength ofchemical bonds, and this strengthdepends on the relative affinities ofdifferent atoms in molecules.

THE INSIDE STORYON THE ENERGY IN FOOD

The food that you eat providesyour body with energy. Your bodyneeds this energy to move yourmuscles, to maintain a steady inter-nal temperature, and to carry outmany other bodily processes. Theenergy in food is stored as a kindof potential energy in the chemicalbonds within sugars and otherorganic molecules.

When you digest food, some ofthis energy is released. The energyis then stored again in sugar mole-cules, usually as glucose.When cellsin your body need energy to carryout cellular processes, the cells break down the glucose moleculesthrough a process called cellular

respiration. The primary product of cellular respiration is a high-energy molecule called adenosinetriphosphate (ATP), which has a sig-nificant role in many chemical reac-tions in cells.

Nutritionists and food scientistsuse units of Calories to quantifythe energy in food. A standardcalorie (cal) is defined as theamount of energy required toincrease the temperature of 1 mLof water by 1°C, which equals4.186 joules ( J). A food Calorie is actually 1 kilocalorie, or 4186 J.

People who are trying to loseweight often monitor the numberof Calories that they eat each day.

These people count Caloriesbecause the body stores unusedenergy as fat. Most food labelsshow the number of Calories ineach serving of food. The amountof energy that your body needseach day depends on many factors,including your age, your weight, andthe amount of exercise that youget. A typically healthy and activeperson requires about 1500 to2000 Calories per day.

169

SECTION 2POTENTIAL ENERGY

Consider the balanced boulder shown in Figure 6. As long as the boulder

remains balanced, it has no kinetic energy. If it becomes unbalanced, it will

fall vertically to the desert floor and will gain kinetic energy as it falls. A simi-

lar example is an arrow ready to be released on a bent bow. Once the arrow is

in flight, it will have kinetic energy.

Potential energy is stored energy

As we have seen, an object in motion has kinetic energy. But a system can have

other forms of energy. The examples above describe a form of energy that is

due to the position of an object in relation to other objects or to a reference

point. is associated with an object that has the potential to

move because of its position relative to some other location. Unlike kinetic

energy, potential energy depends not only on the properties of an object but

also on the object’s interaction with its environment.

Gravitational potential energy depends on height from a zero level

You learned earlier how gravitational forces influence the motion of a projec-

tile. If an object is thrown up in the air, the force of gravity will eventually

cause the object to fall back down, provided that the object was not thrown

too hard. Similarly, the force of gravity will cause the unbalanced boulder

in the previous example to fall. The energy associated with an object due to

the object’s position relative to a gravitational source is called

Imagine an egg falling off a table. As it falls, it gains kinetic energy. But

where does the egg’s kinetic energy come from? It comes from the gravita-

tional potential energy that is associated with the egg’s initial position on the

table relative to the floor. Gravitational potential energy can be determined

using the following equation:

The SI unit for gravitational potential energy, like for kinetic energy, is the

joule. Note that the definition for gravitational potential energy in this chap-

ter is valid only when the free-fall acceleration is constant over the entire

height, such as at any point near the Earth’s surface. Furthermore, gravita-

tional potential energy depends on both the height and the free-fall accelera-

tion, neither of which is a property of an object.

GRAVITATIONAL POTENTIAL ENERGY

PEg = mgh

gravitational potential energy = mass × free-fall acceleration × height

potential energy.gravitational

Potential energy

The Languageof PhysicsIn the symbol PEg, PE stands forpotential energy, and the sub-script g specifies that the sourceof this potential energy is gravity.Some texts use U rather than PEto represent potential energy.

GENERAL

169Work and Energy

Figure 6Energy is present in this example,but it is not kinetic energy becausethere is no motion. What kind ofenergy is it?

potential energy

the energy associated with anobject because of the position,shape, or condition of the object

gravitational potential energy

the potential energy stored in thegravitational fields of interactingbodies

Demonstration

Potential EnergyPurpose Show that potentialenergy is stored energy.Materials a racquetball cut in halfCaution Do not face the areawhere you drop the ball, because itmay rise up high enough to hit you.Procedure Pop the hollow hemisphere of the ball inside outand hold it hollow-side up.Before dropping it from a lowheight, ask students to predictwhether it will bounce back and,if so, approximately how high.

Release the ball. The half ballwill pop out on impact with thesurface and will bounce up to agreater height with its hollowside facing down. Ask studentswhere the additional gravitation-al potential energy came from.(Elastic potential energy wasstored in the half ball when it wasinverted inside out.)

GENERAL



Topic: Potential and Kinetic EnergySciLinks Code: HF61196

170

SECTION 2Suppose you drop a volleyball from a second-floor roof and it lands on the

first-floor roof of an adjacent building (see Figure 7). If the height is mea-

sured from the ground, the gravitational potential energy is not zero because

the ball is still above the ground. But if the height is measured from the first-

floor roof, the potential energy is zero when the ball lands on the roof.

Gravitational potential energy is a result of an object’s position, so it must

be measured relative to some zero level. The zero level is the vertical coordi-

nate at which gravitational potential energy is defined to be zero. This zero

level is arbitrary, and it is chosen to make a specific problem easier to solve. In

many cases, the statement of the problem suggests what to use as a zero level.

Elastic potential energy depends on distance compressed or stretched

Imagine you are playing with a spring on a tabletop. You push a block into the

spring, compressing the spring, and then release the block. The block slides across

the tabletop. The kinetic energy of the block came from the stored energy in the

compressed spring. This potential energy is called

Elastic potential energy is stored in any compressed or stretched object, such as a

spring or the stretched strings of a tennis racket or guitar.

The length of a spring when no external forces are acting on it is called the

relaxed length of the spring. When an external force compresses or stretches

the spring, elastic potential energy is stored in the spring. The amount of

energy depends on the distance the spring is compressed or stretched from its

relaxed length, as shown in Figure 8. Elastic potential energy can be deter-

mined using the following equation:

The symbol k is called the or force constant. For a flexible

spring, the spring constant is small, whereas for a stiff spring, the spring con-

stant is large. Spring constants have units of newtons divided by meters (N/m).

spring constant,

ELASTIC POTENTIAL ENERGY

PEelastic = 12

kx2

elastic potential energy = 12

× spring constant × �distance compressed�2

or stretched

elastic potential energy.

Chapter 5170

A

B

C

Figure 7If B is the zero level, then all the gravitational potential energy isconverted to kinetic energy as the ball falls from A to B. If C is the zero level, then only part of the total gravitational potentialenergy is converted to kinetic energy during the fall from A to B.

Distance compressed

Compressed length of spring

Relaxed length of spring

Figure 8The distance to use in the equationfor elastic potential energy is thedistance the spring is compressedor stretched from its relaxed length.

elastic potential energy

the energy available for use whena deformed elastic object returnsto its original configuration

spring constant

a parameter that is a measure of a spring’s resistance to beingcompressed or stretched

MisconceptionAlert

Some students do not realize thatthe potential energy of an object isrelative. Point out that the zero-level for measuring height is arbi-trarily defined in each problem.The potential energy is calculatedrelative to that level. Ask studentshow they would calculate thepotential energy of a book on theirdesk relative to the desk, to theclassroom floor, and to the roof.

Teaching TipAsk students whether it is pos-sible to have a negative potentialenergy. (Yes, a negative potentialenergy means that work must bedone to bring an object to thezero-level.) Then ask whether anobject can have a positive poten-tial energy relative to one refer-ence point and a negativepotential energy relative toanother reference point. Havestudents give examples to sup-port their answer. (Yes. Forexample, a book that is 0.5 mbelow a table and 0.5 m above theground has a positive potentialenergy relative to the ground buta negative potential energyrelative to the table.)

Visual Strategy

Figure 8Point out that the spring’s poten-tial energy depends on the differ-ence between the spring’s relaxedand compressed lengths.

GENERAL

STOP

171

SECTION 2

171Work and Energy

SAMPLE PROBLEM D

Potential Energy

P R O B L E MA 70.0 kg stuntman is attached to a bungee cord with an unstretched lengthof 15.0 m. He jumps off a bridge spanning a river from a height of 50.0 m.When he finally stops, the cord has a stretched length of 44.0 m. Treat thestuntman as a point mass, and disregard the weight of the bungee cord.Assuming the spring constant of the bungee cord is 71.8 N/m, what is thetotal potential energy relative to the water when the man stops falling?

S O L U T I O NGiven: m = 70.0 kg k = 71.8 N/m g = 9.81 m/s2

h = 50.0 m − 44.0 m = 6.0 m

x = 44.0 m − 15.0 m = 29.0 m

PE = 0 J at river level

Unknown: PEtot = ?

Diagram:

Choose an equation or situation:The zero level for gravitational potential energy is chosen to be at the surface

of the water. The total potential energy is the sum of the gravitational and

elastic potential energy.

PEtot = PEg + PEelastic

PEg = mgh

PEelastic = 12

kx2

Substitute the values into the equations and solve:

PEg = (70.0 kg)(9.81 m/s2)(6.0 m) = 4.1 × 103 J

PEelastic = 12

(71.8 N/m)(29.0 m)2 = 3.02 × 104 J

PEtot = 4.1 × 103 J + 3.02 × 104 J

One way to evaluate the answer is to make an order-of-magnitude estimate. The

gravitational potential energy is on the order of 102 kg × 10 m/s2 × 10 m = 104 J.

The elastic potential energy is on the order of 1 × 102 N/m × 102 m2 = 104 J.

Thus, the total potential energy should be on the order of 2 × 104 J. This num-

ber is close to the actual answer.

PEtot = 3.43 × 104 J

50.0 m Stretched length= 44.0 m

Relaxed length= 15.0 m

1. DEFINE

2. PLAN

3. CALCULATE

4. EVALUATE

Choose the zero potentialenergy location that makesthe problem easiest to solve.

Potential EnergyWhen a 2.00 kg mass is attachedto a vertical spring, the spring isstretched 10.0 cm such that themass is 50.0 cm above the table.

a. What is the gravitationalpotential energy associatedwith this mass relative to thetable?

b. What is the spring’s elasticpotential energy if the springconstant is 400.0 N/m?

c. What is the total potentialenergy of this system?

Answersa. 9.81 Jb. 2.00 Jc. 11.81 J

PE SE Sample, 1–3; Ch.Rvw. 23–25, 37

PW 7–9PB 9–10

k PW 10PB Sample, 1–3

h or d PW 4–6, 10PB 4–6

m PW Sample, 1–3PB 7–8

PROBLEM GUIDE D

Solving for:




172

SECTION 2

ANSWERS

Practice D1. 3.3 J

2. 3.1 × 10−2 J

3. a. 785 Jb. 105 Jc. 0.00 J

Chapter 5172

PRACTICE D

Potential Energy

1. A spring with a force constant of 5.2 N/m has a relaxed length of 2.45 m.

When a mass is attached to the end of the spring and allowed to come to

rest, the vertical length of the spring is 3.57 m. Calculate the elastic poten-

tial energy stored in the spring.

2. The staples inside a stapler are kept in place by a spring with a relaxed

length of 0.115 m. If the spring constant is 51.0 N/m, how much elastic

potential energy is stored in the spring when its length is 0.150 m?

3. A 40.0 kg child is in a swing that is attached to ropes 2.00 m long. Find

the gravitational potential energy associated with the child relative to the

child’s lowest position under the following conditions:

a. when the ropes are horizontal

b. when the ropes make a 30.0° angle with the vertical

c. at the bottom of the circular arc

SECTION REVIEW

1. A pinball bangs against a bumper, giving the ball a speed of 42 cm/s. If

the ball has a mass of 50.0 g, what is the ball’s kinetic energy in joules?

2. A student slides a 0.75 kg textbook across a table, and it comes to rest

after traveling 1.2 m. Given that the coefficient of kinetic friction

between the book and the table is 0.34, use the work–kinetic energy

theorem to find the book’s initial speed.

3. A spoon is raised 21.0 cm above a table. If the spoon and its contents

have a mass of 30.0 g, what is the gravitational potential energy associ-

ated with the spoon at that height relative to the surface of the table?

4. Critical Thinking What forms of energy are involved in the follow-

ing situations?

a. a bicycle coasting along a level road

b. heating water

c. throwing a football

d. winding the mainspring of a clock

5. Critical Thinking How do the forms of energy in item 4 differ

from one another? Be sure to discuss mechanical versus nonmechanical

energy, kinetic versus potential energy, and gravitational versus elastic

potential energy.

1. 4.4 × 10−3 J

2. 2.8 m/s

3. 6.18 × 10−2 J

4. a. kinetic energyb. nonmechanical energyc. kinetic energy, gravita-

tional potential energyd. elastic potential energy

5. The heated water is aninstance of nonmechanicalenergy, because its mass isnot displaced with a velocityor with respect to a zero posi-tion, as would be the case forthe various types of mechan-ical energy. The bicycle andfootball both have masses inmotion, so they have kineticenergy. The wound springhas been displaced from itsrelaxed position and so haselastic potential energy, whilethe football is above theground and therefore has agravitational potential energyassociated with it.


SECTION OBJECTIVES

■ Identify situations in whichconservation of mechanicalenergy is valid.

■ Recognize the forms thatconserved energy can take.

■ Solve problems using conser-vation of mechanical energy.

173Work and Energy 173

SECTION 3General LevelConservation of Energy SECTION 3

The Language of PhysicsThe symbol ΣPE stands for “sumof the potential energies.” Just asthe Greek letter ∆ (delta) is used todenote difference, the Greek letterΣ (sigma) is used to denote sum.

CONSERVED QUANTITIES

When we say that something is conserved, we mean that it remains constant. If

we have a certain amount of a conserved quantity at some instant of time, we

will have the same amount of that quantity at a later time. This does not

mean that the quantity cannot change form during that time, but if we con-

sider all the forms that the quantity can take, we will find that we always have

the same amount.

For example, the amount of money you now have is not a conserved quan-

tity because it is likely to change over time. For the moment, however, let us

assume that you do not spend the money you have, so your money is con-

served. This means that if you have a dollar in your pocket, you will always

have that same amount, although it may change form. One day it may be in

the form of a bill. The next day you may have a hundred pennies, and the next

day you may have an assortment of dimes and nickels. But when you total the

change, you always have the equivalent of a dollar. It would be nice if

money were like this, but of course it isn’t. Because money is often

acquired and spent, it is not a conserved quantity.

An example of a conserved quantity that you are already familiar

with is mass. For instance, imagine that a light bulb is dropped on

the floor and shatters into many pieces. No matter how the bulb

shatters, the total mass of all of the pieces together is the same as the

mass of the intact light bulb because mass is conserved.

MECHANICAL ENERGY

We have seen examples of objects that have either kinetic or potential

energy. The description of the motion of many objects, however,

often involves a combination of kinetic and potential energy as well as

different forms of potential energy. Situations involving a combina-

tion of these different forms of energy can often be analyzed simply.

For example, consider the motion of the different parts of a pendu-

lum clock. The pendulum swings back and forth. At the highest point

of its swing, there is only gravitational potential energy associated

with its position. At other points in its swing, the pendulum is in

motion, so it has kinetic energy as well. Elastic potential energy is also

present in the many springs that are part of the inner workings of the

clock. The motion of the pendulum in a clock is shown in Figure 9.

Figure 9Total potential and kinetic energy must be takeninto account in order to describe the total energyof the pendulum in a clock.

ADVANCED TOPICS

See “The Equivalence of Massand Energy” in Appendix J:Advanced Topics to learn aboutEinstein’s theory of relativity.

Demonstration

Mechanical EnergyPurpose Show two kinds ofenergy in a mechanical system.Materials pendulum attached toa ring standProcedure As the pendulumswings to and fro, have studentsdescribe the motion in terms ofgravitational potential energyand kinetic energy when the bobis at different positions along itspath. (At maximum displacement,the gravitational potential energyis maximum and the bob’s kineticenergy is zero. The potential energyis gradually converted into kineticenergy. At the equilibrium posi-tion, the kinetic energy is maxi-mum and the gravitationalpotential energy is zero.)

174

SECTION 3Analyzing situations involving kinetic, gravitational potential, and elastic

potential energy is relatively simple. Unfortunately, analyzing situations involv-

ing other forms of energy—such as chemical potential energy—is not as easy.

We can ignore these other forms of energy if their influence is negligible or

if they are not relevant to the situation being analyzed. In most situations that

we are concerned with, these forms of energy are not involved in the motion

of objects. In ignoring these other forms of energy, we will find it useful to

define a quantity called The mechanical energy is the

sum of kinetic energy and all forms of potential energy associated with an

object or group of objects.

ME = KE + ΣPE

All energy, such as nuclear, chemical, internal, and electrical, that is not

mechanical energy is classified as nonmechanical energy. Do not be confused

by the term mechanical energy. It is not a unique form of energy. It is merely a

way of classifying energy, as shown in Figure 10. As you learn about new

forms of energy in this book, you will be able to add them to this chart.

mechanical energy.

Mechanical energy is often conserved

Imagine a 75 g egg located on a countertop 1.0 m above the ground, as shown

in Figure 11. The egg is knocked off the edge and falls to the ground. Because

the acceleration of the egg is constant as it falls, you can use the kinematic for-

mulas to determine the speed of the egg and the distance the egg has fallen at

any subsequent time. The distance fallen can then be subtracted from the ini-

tial height to find the height of the egg above the ground at any subsequent

time. For example, after 0.10 s, the egg has a speed of 0.98 m/s and has fallen a

distance of 0.05 m, corresponding to a height above the ground of 0.95 m.

Once the egg’s speed and its height above the ground are known as a function

of time, you can use what you have learned in this chapter to calculate both the

kinetic energy of the egg and the gravitational potential energy associated with

the position of the egg at any subsequent time. Adding the kinetic and poten-

tial energy gives the total mechanical energy at each position.

Chapter 5174

mechanical energy

the sum of kinetic energy and allforms of potential energy

Energy

Mechanical Nonmechanical

Kinetic Potential

Gravitational Elastic

Figure 10Energy can be classified in a numberof ways.

Figure 11The total mechanical energy, poten-tial energy plus kinetic energy, isconserved as the egg falls.

Demonstration

Conservation of EnergyPurpose Demonstrate the con-servation of mechanical energy.Materials steel ball, spring bal-ance, meterstickProcedure Measure the weight ofthe steel ball, and record this valueon the chalkboard. Drop the ballfrom shoulder height, and ask thestudents to describe the motion interms of potential energy andkinetic energy when the ball is atdifferent positions along its path.

Now measure the initial andfinal heights of the ball, and recordthese values on the chalkboard.Have students calculate the corre-sponding potential energy. Askthem to estimate how much kinet-ic energy the ball should have at itslowest point (same as its initialpotential energy) and at its midwaypoint, disregarding friction (half ofits initial potential energy). Explainthat if friction can be disregarded,the ball’s potential energy is con-verted into kinetic energy, whilethe total amount of mechanicalenergy remains constant.

GENERAL

175

SECTION 3

In the absence of friction, the total mechanical energy remains the same.

This principle is called conservation of mechanical energy. Although the

amount of mechanical energy is constant, mechanical energy itself can

change form. For instance, consider the forms of energy for the falling egg, as

shown in Table 1. As the egg falls, the potential energy is continuously con-

verted into kinetic energy. If the egg were thrown up in the air, kinetic energy

would be converted into gravitational potential energy. In either case,

mechanical energy is conserved. The conservation of mechanical energy can

be written symbolically as follows:

The mathematical expression for the conservation of mechanical energy

depends on the forms of potential energy in a given problem. For instance, if

the only force acting on an object is the force of gravity, as in the egg example,

the conservation law can be written as follows:

12

mv2i + mghi = 1

2mv2

f + mghf

If other forces (except friction) are present, simply add the appropriate poten-

tial energy terms associated with each force. For instance, if the egg happened

to compress or stretch a spring as it fell, the conservation law would also

include an elastic potential energy term on each side of the equation.

In situations in which frictional forces are present, the principle of

mechanical energy conservation no longer holds because kinetic energy is not

simply converted to a form of potential energy. This special situation will be

discussed more thoroughly later in this section.

CONSERVATION OF MECHANICAL ENERGY

MEi = MEf

initial mechanical energy = final mechanical energy(in the absence of friction)

175Work and Energy

Table 1 Energy of a Falling 75 g Egg

Time Height Speed PEg KE ME(s) (m) (m/s) (J) (J) (J)

0.00 1 .0 0.00 0.74 0.00 0.74

0. 10 0.95 0.98 0.70 0.036 0.74

0.20 0.80 2.0 0.59 0. 15 0.74

0.30 0.56 2.9 0.4 1 0.33 0.74

0.40 0.22 3.9 0. 16 0.58 0.74

Mechanical Energy

M A T E R I A L S L I S T

• medium-sized spring (spring balance)

• assortment of small balls, eachhaving a different mass

• ruler

• tape

• scale or balance

SAFETY CAUTION

Students should wear goggles toperform this lab.

First, determine the mass ofeach of the balls. Then, tape theruler to the side of a tabletop sothat the ruler is vertical. Place thespring vertically on the tabletopnear the ruler, and compress thespring by pressing down on one of the balls. Release the ball, andmeasure the maximum height itachieves in the air. Repeat thisprocess five times, and be sure tocompress the spring by the sameamount each time. Average theresults. From the data, can you pre-dict how high each of the otherballs will rise? Test your predictions.(Hint: Assume mechanical energy isconserved.)

12

34

56

78

910

TEACHER’S NOTESThis activity is meant to demon-strate energy transfer (from thespring to the ball) and the con-servation of mechanical energy.

The lab is most effective whenthe balls have significantly differ-ent masses and when the springis compressed the same amountin each case.

Because the system for all cases has the same MEi = 1

2kxi

2,which is converted into MEf = mghf , balls with a largermass will achieve a lower height.

Point out that if the measure-ments are reliable, they can be usedto determine the spring constant.

176

SECTION 3

Chapter 5176

SAMPLE PROBLEM E

Conservation of Mechanical Energy

P R O B L E MStarting from rest, a child zooms down a frictionless slidefrom an initial height of 3.00 m. What is her speed at thebottom of the slide? Assume she has a mass of 25.0 kg.

S O L U T I O NGiven: h = hi = 3.00 m m = 25.0 kg vi = 0.0 m/s

hf = 0 m

Unknown: vf = ?

Choose an equation or situation:The slide is frictionless, so mechanical energy is conserved.

Kinetic energy and gravitational potential energy are the

only forms of energy present.

KE = 12

mv2 PE = mgh

The zero level chosen for gravitational potential energy is the bottom of the

slide. Because the child ends at the zero level, the final gravitational potential

energy is zero.

PEg,f = 0

The initial gravitational potential energy at the top of the slide is

PEg,i = mghi = mgh

Because the child starts at rest, the initial kinetic energy at the top is zero.

KEi = 0

Therefore, the final kinetic energy is as follows:

KEf = 12

mv2f

Substitute values into the equations:

PEg,i = (25.0 kg)(9.81 m/s2)(3.00 m) = 736 J

KEf = (12

)(25.0 kg)v2f

Now use the calculated quantities to evaluate the final velocity.

MEi = MEf

PEi + KEi = PEf + KEf

736 J + 0 J = 0 J + (0.500)(25.0 kg)v2f

vf = 7.67 m/s

1. DEFINE

2. PLAN

3. CALCULATE

Conservation of Mechanical EnergyA small 10.0 g ball is held to a slingshot that is stretched 6.0 cm. The spring constant is 2.0 × 102 N/m.

a. What is the elastic potentialenergy of the slingshot beforeit is released?

b. What is the kinetic energy ofthe ball just after the slingshotis released?

c. What is the ball’s speed at thatinstant?

d. How high does the ball rise ifit is shot directly upward?

Answersa. 0.36 Jb. 0.36 Jc. 8.5 m/sd. 3.7 m

CALCULATOR SOLUTION

Your calculator should give an answerof 7.67333, but because the answeris limited to three significant figures,it should be rounded to 7.67.

v SE Sample, 1–3; Ch. Rvw.33–34, 47, 51–52

PW 4–5PB 8–10

h SE 4–5; Ch. Rvw. 39,41–42, 47, 52

PW Sample, 1–3PB 5–7

E SE Ch. Rvw. 37, 45, 48, 51PW Sample, 6–7PB Sample, 1–4

PROBLEM GUIDE E

Solving for:




177

SECTION 3

Energy conservation occurs even when acceleration varies

If the slope of the slide in Sample Problem E was constant, the acceleration

along the slide would also be constant and the one-dimensional kinematic

formulas could have been used to solve the problem. However, you do not

know the shape of the slide. Thus, the acceleration may not be constant, and

the kinematic formulas could not be used.

But now we can apply a new method to solve such a problem. Because the

slide is frictionless, mechanical energy is conserved. We simply equate the ini-

tial mechanical energy to the final mechanical energy and ignore all the details

in the middle. The shape of the slide is not a contributing factor to the sys-

tem’s mechanical energy as long as friction can be ignored.

Alternative Problem-Solving ApproachThe process shown in SampleProblem E can be reversed.Rather than calculating each typeof energy separately, begin withthe conservation of mechanicalenergy:

MEi = MEfNext determine what types ofenergy are involved and substi-tute the formulas for each type ofenergy into the equation.

In this case,

PEi = KEf

mghi = 12

mvf2

Solve for v in terms of the othervariables, and then substitute thegiven values into this equation.

v f2 = 2ghi

vf = �2g�h�i�vf = �2(�9.�81� m�/s�2)�(3�.0�0�m�)�vf = 7.67 m/s

ANSWERS

Practice E1. 20.7 m/s

2. 9.9 m/s; 14.0 m/s

3. 14.1 m/s

4. 0.25 m

5. 0.18 m

177Work and Energy

The expression for the square of the final speed can be written as follows:

v 2f =

2m

m

gh = 2gh

Notice that the masses cancel, so the final speed does not depend on the mass

of the child. This result makes sense because the acceleration of an object due

to gravity does not depend on the mass of the object.

4. EVALUATE

PRACTICE E


1. A bird is flying with a speed of 18.0 m/s over water when it accidentally

drops a 2.00 kg fish. If the altitude of the bird is 5.40 m and friction is

disregarded, what is the speed of the fish when it hits the water?

2. A 755 N diver drops from a board 10.0 m above the water’s surface. Find

the diver’s speed 5.00 m above the water’s surface. Then find the diver’s

speed just before striking the water.

3. If the diver in item 2 leaves the board with an initial upward speed of

2.00 m/s, find the diver’s speed when striking the water.

4. An Olympic runner leaps over a hurdle. If the runner’s initial vertical

speed is 2.2 m/s, how much will the runner’s center of mass be raised

during the jump?

5. A pendulum bob is released from some initial height such that the speed

of the bob at the bottom of the swing is 1.9 m/s. What is the initial height

of the bob?



Topic: Conservation of EnergySciLinks Code: HF60345

178

SECTION 3Mechanical energy is not conserved in thepresence of friction

If you have ever used a sanding block to sand a rough

surface, such as in Figure 12, you may have noticed

that you had to keep applying a force to keep the block

moving. The reason is that kinetic friction between the

moving block and the surface causes the kinetic energy

of the block to be converted into a nonmechanical

form of energy. As you continue to exert a force on the

block, you are replacing the kinetic energy that is lost

because of kinetic friction. The observable result of

this energy dissipation is that the sanding block and

the tabletop become warmer.

In the presence of kinetic friction, nonmechanical

energy is no longer negligible and mechanical energy

is no longer conserved. This does not mean that en-

ergy in general is not conserved—total energy is

always conserved. However, the mechanical energy is

converted into forms of energy that are much more

difficult to account for, and the mechanical energy is

therefore considered to be “lost.”

MisconceptionAlert

Some students may confuse theconservation of mechanical energywith the general energy conserva-tion law. Point out that althoughmechanical energy is not alwaysconserved, the total energy isalways conserved. For example, asthe sanding block’s kinetic energydecreases, energy is transferred tothe rough surface in the form ofinternal energy (this topic will bediscussed in the chapter on heatand temperature). As a result, thetemperatures of the block and surface increase slightly. The totalenergy in the system remains con-stant, although the mechanicalenergy decreases.

STOP

Chapter 5178

d

Figure 12(a) As the block slides, its kinetic energy tends to decreasebecause of friction. The force from the hand keeps it moving.(b) Kinetic energy is dissipated into the block and surface.

(a)

(b)

SECTION REVIEW

1. 2.93 m/s

2. No, the roller coaster will notreach the top of the secondhill. If the total mechanicalenergy is constant, the rollercoaster will reach its initialheight and then begin rollingback down the hill.

3. a. yesb. noc. yes, if air resistance is

disregarded

4. Answers may vary. Thedownward-sloping trackconverts potential energy tokinetic energy. Levers employkinetic energy to increasepotential energy. Springs andelastic membranes convertkinetic energy to elasticpotential energy and backagain. Mechanical energy isnot conserved; some energyis lost because of kinetic friction.


1. If the spring of a jack-in-the-box is compressed a distance of 8.00 cm from

its relaxed length and then released, what is the speed of the toy head when

the spring returns to its natural length? Assume the mass of the toy head is

50.0 g, the spring constant is 80.0 N/m, and the toy head moves only in the

vertical direction. Also disregard the mass of the spring. (Hint: Remember

that there are two forms of potential energy in the problem.)

2. You are designing a roller coaster in which a car will be pulled to the top of

a hill of height h and then, starting from a momentary rest, will be released

to roll freely down the hill and toward the peak of the next hill, which is

1.1 times as high. Will your design be successful? Explain your answer.

3. Is conservation of mechanical energy likely to hold in these situations?

a. a hockey puck sliding on a frictionless surface of ice

b. a toy car rolling on a carpeted floor

c. a baseball being thrown into the air

4. Critical Thinking What parts of the kinetic sculpture on the open-

ing pages of this chapter involve the conversion of one form of energy to

another? Is mechanical energy conserved in these processes?

SECTION OBJECTIVES

■ Relate the concepts of energy, time, and power.

■ Calculate power in two different ways.

■ Explain the effect ofmachines on work andpower.

179

SECTION 4General LevelPower SECTION 4

Teaching TipTo help students understand therelationship P = Fv, have themcalculate power both ways in asimple example. For example,ask students to calculate thework done, the power using thefirst equation, the speed, and thepower using the second equationwhen a 100.0 N force moves anobject 20.0 m in 5.0 s at constantspeed (2.00 × 10 3 J, 4.0 × 10 2 W,4.0 m/s, 4.0 × 10 2 W).

ANSWERS

Conceptual Challenge1. Assuming mechanical energy

is conserved, the sameamount of energy is neededto reach the top in bothcases. Because the sameamount of work must bedone, the path with a longerdistance takes more time andhence requires less power.

2. Light bulbs don’t have theenergy stored within them;energy is transferred to themin the form of electricity at arate of 60 J/s.

GENERAL

RATE OF ENERGY TRANSFER

The rate at which work is done is called More generally, power is the

rate of energy transfer by any method. Like the concepts of energy and work,

power has a specific meaning in science that differs from its everyday meaning.

Imagine you are producing a play and you need to raise and lower the cur-

tain between scenes in a specific amount of time. You decide to use a motor

that will pull on a rope connected to the top of the curtain rod. Your assistant

finds three motors but doesn’t know which one to use. One way to decide is to

consider the power output of each motor.

If the work done on an object is W in a time interval ∆t, then the average

power delivered to the object over this time interval is written as follows:

It is sometimes useful to rewrite this equation in an alternative form by

substituting the definition of work into the definition of power.

W = Fd

P = ∆W

t = F

∆d

t

The distance moved per unit time is just the speed of the object.

POWER

P = ∆W

t

power = work ÷ time interval

power.

power

a quantity that measures the rateat which work is done or energyis transformed

1. Mountain Roads Many mountain roads arebuilt so that they zigzag up the mountain rather thango straight up toward the peak. Discuss the advan-tages of such a design from the viewpoint of energyconservation and power.

2. Light Bulbs A light bulb is described as having 60 watts.What’s wrong with this statement?

Integrating ChemistryVisit go.hrw.com for the activity“Chemical Reactions.”

Keyword HF6WRKX

180

SECTION 4

The SI unit of power is the watt, W, which is defined to be one joule per

second. The horsepower, hp, is another unit of power that is sometimes used.

One horsepower is equal to 746 watts.

The watt is perhaps most familiar to you from your everyday experience

with light bulbs (see Figure 13). A dim light bulb uses about 40 W of power,

while a bright bulb can use up to 500 W. Decorative lights use about 0.7 W

each for indoor lights and 7.0 W each for outdoor lights.

In Sample Problem F, the three motors would lift the curtain at different

rates because the power output for each motor is different. So each motor

would do work on the curtain at different rates and would thus transfer ener-

gy to the curtain at different rates.

POWER (ALTERNATIVE FORM)

P = Fv

power = force × speed

Chapter 5180

PowerTwo horses pull a cart. Eachexerts a force of 250.0 N at aspeed of 2.0 m/s for 10.0 min.

a. Calculate the power deliveredby the horses.

b. How much work is done bythe two horses?

Answersa. 1.0 × 103 Wb. 6.0 × 105 J

Alternative Problem-Solving ApproachCalculate the time it would takeeach motor to do the same work:

W = Fd = mgd = 14 × 103 J

∆t = W/P

∆t1 = 14 × 103 J/1.0 × 103 W = 14 s

∆t2 = 14 × 103 J/3.5 × 103 W = 4.0 s

∆t3 = 14 × 103 J/5. 5 × 103 W = 2.5 s

This approach shows that the sec-ond motor comes closest to 5.0 s.

GENERAL

Figure 13The power of each of these bulbstells you the rate at which energy isconverted by the bulb.The bulbs inthis photo have power ratings thatrange from 0.7 W to 200 W.

SAMPLE PROBLEM F

Power

P R O B L E MA 193 kg curtain needs to be raised 7.5 m, at constant speed, in as close to5.0 s as possible. The power ratings for three motors are listed as 1.0 kW,3.5 kW, and 5.5 kW. Which motor is best for the job?

S O L U T I O NGiven: m = 193 kg ∆t = 5.0 s d = 7.5 m

Unknown: P = ?

Use the definition of power. Substitute the equation for work.

P = ∆W

t =

F

∆d

t =

m

∆g

t

d

=

P = 2.8 × 103 W = 2.8 kW

The best motor to use is the 3.5 kW motor. The 1.0 kW

motor will not lift the curtain fast enough, and the

5.5 kW motor will lift the curtain too fast.

(193 kg)(9.81 m/s2)(7.5 m)

5.0 s

181

SECTION 4

ANSWERS

Practice F1. 66 kW

2. 2.38 × 104 W (23.8 kW)

3. 2.61 × 108 s (8.27 years)

4. 3.6 × 103 s (1.0 h)

5. a. 7.50 × 104 Jb. 2.50 × 104 W

181Work and Energy

PRACTICE F

Power

1. A 1.0 × 103 kg elevator carries a maximum load of 800.0 kg. A constant

frictional force of 4.0 × 103 N retards the elevator’s motion upward.

What minimum power, in kilowatts, must the motor deliver to lift the

fully loaded elevator at a constant speed of 3.00 m/s?

2. A car with a mass of 1.50 × 103 kg starts from rest and accelerates to a

speed of 18.0 m/s in 12.0 s. Assume that the force of resistance remains

constant at 400.0 N during this time. What is the average power devel-

oped by the car’s engine?

3. A rain cloud contains 2.66 × 107 kg of water vapor. How long would it

take for a 2.00 kW pump to raise the same amount of water to the

cloud’s altitude, 2.00 km?

4. How long does it take a 19 kW steam engine to do 6.8 × 107 J of work?

5. A 1.50 × 103 kg car accelerates uniformly from rest to 10.0 m/s in 3.00 s.

a. What is the work done on the car in this time interval?

b. What is the power delivered by the engine in this time interval?

SECTION REVIEW

1. A 50.0 kg student climbs 5.00 m up a rope at a constant speed. If the stu-

dent’s power output is 200.0 W, how long does it take the student to

climb the rope? How much work does the student do?

2. A motor-driven winch pulls the 50.0 kg student in the previous item

5.00 m up the rope at a constant speed of 1.25 m/s. How much power

does the motor use in raising the student? How much work does the

motor do on the student?

3. Critical Thinking How are energy, time, and power related?

4. Critical Thinking People often use the word powerful to describe

the engines in some automobiles. In this context, how does the word

relate to the definition of power? How does this word relate to the alter-

native definition of power?

P SE Sample, 1–2, 5;Ch. Rvw. 36

PW 5–6PB 8–10

�t SE 3–4; Ch. Rvw. 35PW 3–4PB Sample, 1–3

W SE 5PW Sample, 1–2PB 4–7

PROBLEM GUIDE F

Solving for:




1. 12.3 s; 2.45 × 103 J

2. 613 W; 2.45 × 103 J

3. Power equals energy trans-ferred divided by time oftransfer.

4. A powerful engine is capableof doing more work in agiven time. The force andspeed delivered by a power-ful engine is large relative toless powerful engines.


Roller Coaster DesignerTwo of Steve’s first roller coastersare the Ninjas at Six Flags OverMid-America and at Six FlagsMagic Mountain. His WestCoaster, built on Santa MonicaPier, towers five stories above thePacific Ocean. The cars on theSteel Force at Dorney Park inPennsylvania reach speeds ofover 75 mi/h and drop more than200 ft to disappear into a 120 fttunnel. The Mamba at Worlds ofFun in Missouri features twogiant back-to-back hills, a fastspiral, and five camelbackhumps. The camelbacks aredesigned to pull your seat outfrom under you, so that you feellike you’re floating. Roller coasterfans call this feeling airtime.

CHAPTER 5

PHYSICS CAREERS

182182182

Roller CoasterDesigner

As the name states, the cars of a rollercoaster really do coast along the tracks. Amotor pulls the cars up a high hill at thebeginning of the ride. After the hill, howev-er, the motion of the car is a result of gravi-ty and inertia. As the cars roll down the hill,they must pick up the speed that they needto whiz through the rest of the curves,loops, twists, and bumps in the track. Tolearn more about designing roller coasters,read the interview with Steve Okamoto.

How did you become a roller coaster designer?I have been fascinated with roller coasters eversince my first ride on one. I remember goingto Disneyland as a kid. My mother wasalways upset with me because I kept look-ing over the sides of the rides, trying to fig-ure out how they worked. My interest infinding out how things worked led me tostudy mechanical engineering.

What sort of training do youhave?

I earned a degree in productdesign. For this degree, I studiedmechanical engineering and studioart. Product designers consider anobject’s form as well as its function.They also take into account theinterests and abilities of the product’sconsumer. Most rides and parkshave some kind of theme, so I mustconsider marketing goals and con-cerns in my designs.

What is the nature of yourwork?

To design a roller coaster, I studysite maps of the location. Then, I goto the amusement park to look atthe actual site. Because most ridesI design are for older parks (few

CAREERSPHYSICS

parks are built from scratch), fitting a coasteraround, above, and in between existing rides andbuildings is one of my biggest challenges. I alsohave to design how the parts of the ride will worktogether. The towers and structures that support

the ride have to be strong enough to hold upa track and speeding cars that are full of

people. The cars themselves need spe-cial wheels to keep them locked ontothe track and seat belts or bars to

keep the passengers safely inside. It’slike putting together a puzzle, exceptthe pieces haven’t been cut out yet.

What advice do you have fora student who is interested indesigning roller coasters?

Studying math and science is very impor-tant. To design a successful coaster, I haveto understand how energy is converted

from one form to another as the cars movealong the track. I have to calculate speedsand accelerations of the cars on each partof the track. They have to go fast enough tomake it up the next hill! I rely on my knowl-edge of geometry and physics to create theroller coaster’s curves, loops, and dips.

Chapter 5

The roller coaster pictured here is named WildThing and is located in Minnesota. The highestpoint on the track is 63 m off the ground andthe cars’ maximum speed is 118 km/h.

KEY IDEAS

Section 1 Work• Work is done on an object only when a net force acts on the object to dis-

place it in the direction of a component of the net force.

• The amount of work done on an object by a force is equal to the compo-

nent of the force along the direction of motion times the distance the

object moves.

Section 2 Energy• Objects in motion have kinetic energy because of their mass and speed.

• The net work done on or by an object is equal to the change in the kinetic

energy of the object.

• Potential energy is energy associated with an object’s position. Two forms

of potential energy discussed in this chapter are gravitational potential

energy and elastic potential energy.

Section 3 Conservation of Energy• Energy can change form but can never be created or destroyed.

• Mechanical energy is the total kinetic and potential energy present in a

given situation.

• In the absence of friction, mechanical energy is conserved, so the amount

of mechanical energy remains constant.

Section 4 Power• Power is the rate at which work is done or the rate of energy transfer.

• Machines with different power ratings do the same amount of work in dif-

ferent time intervals.

KEY TERMS

work (p. 160)

kinetic energy (p. 164)

work–kinetic energy theorem (p. 166)

potential energy (p. 169)

gravitational potential energy (p. 169)

elastic potential energy (p. 170)

spring constant (p. 170)

mechanical energy (p. 174)

power (p. 179)

183

CHAPTER 5

Highlights

Teaching TipExplaining concepts in writtenform helps solidify students’understanding of difficult con-cepts and helps enforce goodcommunication skills. Have stu-dents summarize the differencesbetween mechanical and non-mechanical energy and betweenkinetic energy, gravitationalpotential energy, and elasticpotential energy. Essays shouldalso include a thorough discus-sion of work and its link to kinet-ic and potential energy. Be surestudents explain concepts clearlyand correctly and use good sen-tence structure.

HighlightsCHAPTER 5

183Work and Energy

Variable Symbols

Quantities Units Conversions

W work J joule = N•m= kg•m2/s2

KE kinetic energy J joule

PEg gravitational potential energy J joule

PEelastic elastic potential energy J joule

P power W watt = J/s

PROBLEM SOLVING

See Appendix D: Equations for a summary of the equationsintroduced in this chapter. Ifyou need more problem-solvingpractice, see Appendix I: Additional Problems.

CHAPTER 5

184

ReviewReview

CHAPTER 5

ANSWERS1. No, a change in speed corre-

sponds to a change in kineticenergy, which cannot occurwithout work (either positiveor negative) being done on theobject.

2. a. yes, positiveb. noc. yes, positived. yes, negative

3. No, force would decrease, butdistance would increase, whichwould keep work constant.

4. The tension is perpendicularto the bob’s motion, so it doesnot do work on the bob. Thecomponent of the bob’sweight that is perpendicular tothe bob’s motion does not dowork on the bob, but the com-ponent that is in the directionof its motion does.

5. The car leaving longer skidmarks was moving faster.

6. yes; no; yes, the ball’s weightand air resistance

7. 53 J, −53 J

8. 2.4 × 105 J

9. 47.5 J

10. a. 6230 Jb. −6230 Jc. 0.640

11. a. nob. yesc. yes

WORK

Review Questions

1. Can the speed of an object change if the net workdone on it is zero?

2. Discuss whether any work is being done by each ofthe following agents and, if so, whether the work ispositive or negative.

a. a chicken scratching the groundb. a person reading a signc. a crane lifting a bucket of concreted. the force of gravity on the bucket in (c)

3. Furniture movers wish to load a truck using a rampfrom the ground to the rear of the truck. One of themovers claims that less work would be required ifthe ramp’s length were increased, reducing its anglewith the horizontal. Is this claim valid? Explain.

Conceptual Questions

4. A pendulum swings back andforth, as shown at right. Doesthe tension force in the stringdo work on the pendulumbob? Does the force of gravitydo work on the bob? Explainyour answers.

5. The drivers of two identicalcars heading toward each other apply the brakes atthe same instant. The skid marks of one of the carsare twice as long as the skid marks of the other vehi-cle. Assuming that the brakes of both cars apply thesame force, what conclusions can you draw aboutthe motion of the cars?

6. When a punter kicks a football, is he doing work onthe ball while his toe is in contact with it? Is hedoing work on the ball after the ball loses contactwith his toe? Are any forces doing work on the ballwhile the ball is in flight?

Practice Problems

For problems 7–10, see Sample Problem A.

7. A person lifts a 4.5 kg cement block a vertical dis-tance of 1.2 m and then carries the block horizontal-ly a distance of 7.3 m. Determine the work done bythe person and by the force of gravity in this process.

8. A plane designed for vertical takeoff has a mass of8.0 × 103 kg. Find the net work done by all forces onthe plane as it accelerates upward at 1.0 m/s2

through a distance of 30.0 m after starting from rest.

9. When catching a baseball, a catcher’s glove movesby 10 cm along the line of motion of the ball. If thebaseball exerts a force of 475 N on the glove, howmuch work is done by the ball?

10. A flight attendant pulls her 70.0 N flight bag a dis-tance of 253 m along a level airport floor at a constantvelocity. The force she exerts is 40.0 N at an angle of52.0° above the horizontal. Find the following:

a. the work she does on the flight bagb. the work done by the force of friction on the

flight bagc. the coefficient of kinetic friction between the

flight bag and the floor

ENERGY

Review Questions

11. A person drops a ball from the top of a buildingwhile another person on the ground observes theball’s motion. Each observer chooses his or her ownlocation as the level for zero potential energy. Willthey calculate the same values for:

a. the potential energy associated with the ball?b. the change in potential energy associated with

the ball?c. the ball’s kinetic energy?

Chapter 5184

185

5 REVIEW

12. No, kinetic energy cannot benegative because mass is alwayspositive and the speed term ofthe equation is squared.

13. yes, because potential energydepends on the distance to anarbitrary zero level, which canbe above or below the object

14. 1 to 25

15. The gravitational force does notdo work on the satellite becausethe force of gravity is alwaysperpendicular to the path ofthe motion. The satellite’s speedmust be constant.

16. The work required to stop thecar equals the car’s initialkinetic energy. If speed is dou-bled, work is quadrupled.Thus, the car will travel 140 m.Its kinetic energy is changedinto internal energy.

17. Work must be done againstgravity in order to climb astaircase at a constant speed.Walking on a horizontal sur-face does not require work tobe done against gravity.

18. The work done by frictionequals the change in mechani-cal energy, so the particle’sspeed decreases.

19. 7.6 × 104 J

20. 1.7 × 104 m/s

21. 2.0 × 101 m

22. 1.4 m

23. a. 5400 J, 0 J; 5400 Jb. 0 J, –5400 J; 5400 Jc. 2700 J, –2700 J; 5400 J

24. a. −19.6 Jb. 39.2 Jc. 0 J

25. a. (0.5)(500.0 N/m)(4.00 × 10−2 m)2 = 0.400 J

b. (12

)(500.0 N/m)

(−3.00 × 10−2 m)2 = 0.225 Jc. (0.5)(500.0 N/m)(0 m)2 = 0 J

12. Can the kinetic energy of an object be negative?Explain your answer.

13. Can the gravitational potential energy associatedwith an object be negative? Explain your answer.

14. Two identical objects move with speeds of 5.0 m/sand 25.0 m/s. What is the ratio of their kinetic energies?


15. A satellite is in a circular orbit above Earth’s surface.Why is the work done on the satellite by the gravita-tional force zero? What does the work–kinetic energy theorem predict about the satellite’s speed?

16. A car traveling at 50.0 km/h skids a distance of 35 mafter its brakes lock. Estimate how far it will skid ifits brakes lock when its initial speed is 100.0 km/h.What happens to the car’s kinetic energy as it comesto rest?

17. Explain why more energy is needed to walk up stairsthan to walk horizontally at the same speed.

18. How can the work–kinetic energy theorem explainwhy the force of sliding friction reduces the kineticenergy of a particle?

Practice Problems

For problems 19–20, see Sample Problem B.

19. What is the kinetic energy of an automobile with amass of 1250 kg traveling at a speed of 11 m/s?

20. What speed would a fly with a mass of 0.55 g needin order to have the same kinetic energy as the auto-mobile in item 19?

For problems 21–22, see Sample Problem C.

21. A 50.0 kg diver steps off a diving board and dropsstraight down into the water. The water provides anupward average net force of 1500 N. If the diver comesto rest 5.0 m below the water’s surface, what is the totaldistance between the diving board and the diver’s stop-ping point underwater?

22. In a circus performance, a monkey on a sled is givenan initial speed of 4.0 m/s up a 25° incline. The com-bined mass of the monkey and the sled is 20.0 kg,and the coefficient of kinetic friction between thesled and the incline is 0.20. How far up the inclinedoes the sled move?

For problems 23–25, see Sample Problem D.

23. A 55 kg skier is at the top of a slope, as shown in theillustration below. At the initial point A, the skier is10.0 m vertically above the final point B.

a. Set the zero level for gravitational potentialenergy at B, and find the gravitational potentialenergy associated with the skier at A and at B.Then find the difference in potential energybetween these two points.

b. Repeat this problem with the zero level atpoint A.

c. Repeat this problem with the zero level mid-way down the slope, at a height of 5.0 m.

24. A 2.00 kg ball is attached to a ceiling by a string. Thedistance from the ceiling to the center of the ball is1.00 m, and the height of the room is 3.00 m. What isthe gravitational potential energy associated with theball relative to each of the following?

a. the ceilingb. the floorc. a point at the same elevation as the ball

25. A spring has a force constant of 500.0 N/m. Showthat the potential energy stored in the spring is asfollows:

a. 0.400 J when the spring is stretched 4.00 cmfrom equilibrium

b. 0.225 J when the spring is compressed 3.00 cmfrom equilibrium

c. zero when the spring is unstretched

10.0 m

A

B

185Work and Energy

186

5 REVIEW

26. a. nonmechanicalb. mechanicalc. mechanicald. mechanicale. both

27. As the athlete runs faster, KEincreases. As he is lifted abovethe ground, KE decreases as PEgand PEelastic increase (PEelasticcomes from the bent pole). Atthe highest point, KE = 0 andPEg is at its maximum value. Asthe athlete falls, KE increasesand PEg decreases. When theathlete lands, KE is at its maxi-mum value and PEg = 0.

28. The ball will not hit the lectur-er because, according to theprinciple of energy conserva-tion, it would need an input ofenergy to reach a heightgreater than its initial height.If the ball were given a push,the lecturer would be in danger.

29. a. Athlete does work on the weight. PEg increases.

b. No work done on the weight. PEg is constant.

c. Athlete does negative workon the weight. PEg decreases.

30. at the ball’s lowest height; atits maximum height

31. no, because energy wouldn’tbe conserved

32. two, gravitational potentialenergy and elastic potentialenergy; yes, because totalmechanical energy is conserved if there is no dissipation of energy

33. 12.0 m/s

34. a. 10.9 m/sb. 11.6 m/s

35. 17.2 s

36. 5.9 × 108 W

CONSERVATION OF MECHANICALENERGY

Review Questions

26. Each of the following objects possesses energy.Which forms of energy are mechanical, which arenonmechanical, and which are a combination?

a. glowing embers in a campfireb. a strong windc. a swinging pendulumd. a person sitting on a mattresse. a rocket being launched into space

27. Discuss the energy transformations that occur duringthe pole-vault event shown in the photograph below.Disregard rotational motion and air resistance.

28. A strong cord suspends a bowling ball from the cen-ter of a lecture hall’s ceiling, forming a pendulum.The ball is pulled to the tip of a lecturer’s nose at thefront of the room and is then released. If the lectur-er remains stationary, explain why the lecturer isnot struck by the ball on its return swing. Wouldthis person be safe if the ball were given a slightpush from its starting position at the person’s nose?


29. Discuss the work done and change in mechanicalenergy as an athlete does the following:

a. lifts a weightb. holds the weight up in a fixed positionc. lowers the weight slowly

30. A ball is thrown straight up. At what position is itskinetic energy at its maximum? At what position isgravitational potential energy at its maximum?

31. Advertisements for a toy ball once stated that itwould rebound to a height greater than the heightfrom which it was dropped. Is this possible?

32. A weight is connected to a spring that is suspendedvertically from the ceiling. If the weight is displaceddownward from its equilibrium position andreleased, it will oscillate up and down. How manyforms of potential energy are involved? If air resis-tance and friction are disregarded, will the totalmechanical energy be conserved? Explain.

Practice Problems

For problems 33–34, see Sample Problem E.

33. A child and sled with a combined mass of 50.0 kgslide down a frictionless hill that is 7.34 m high. If thesled starts from rest, what is its speed at the bottom ofthe hill?

34. Tarzan swings on a 30.0 m long vine initiallyinclined at an angle of 37.0° with the vertical. Whatis his speed at the bottom of the swing if he does thefollowing?

a. starts from restb. starts with an initial speed of 4.00 m/s

POWER

Practice Problems

For problems 35–36, see Sample Problem F.

35. If an automobile engine delivers 50.0 hp of power,how much time will it take for the engine to do 6.40 ×105 J of work? (Hint: Note that one horsepower, 1 hp,is equal to 746 watts.)

36. Water flows over a section of Niagara Falls at therate of 1.2 × 106 kg/s and falls 50.0 m. How muchpower is generated by the falling water?

Chapter 5186

187

5 REVIEW

37. a. 0.633 Jb. 0.633 Jc. 2.43 m/sd. 0.422 J, 0.211 J

38. 0.265 m/s

39. 5.0 m

40. 1.2 × 103 J

41. 2.5 m

42. 10.2 m

43. Although the total distancetraveled by each ball is differ-ent, the displacements are thesame, so the change in poten-tial energy for each ball is thesame. Also, each ball has thesame initial kinetic energy, sothe final kinetic energy of eachball (and thus the speed ofeach) will be the same.

44. a. 1.2 Jb. 5.0 m/sc. 6.3 J

45. a. 61 Jb. −45 Jc. 0 J

46. 2.4 × 104 J

47. a. 28.0 m/sb. 30.0 m above the ground

48. a. 5.42 m/sb. 0.300c. −147 J

49. 0.107

MIXED REVIEW

37. A 215 g particle isreleased from rest atpoint A inside asmooth hemispheri-cal bowl of radius30.0 cm, as shown atright. Calculate thefollowing:

a. the gravitationalpotential energy atA relative to B

b. the particle’s kinetic energy at Bc. the particle’s speed at Bd. the potential energy and kinetic energy at C

38. A person doing a chin-up weighs 700.0 N, disregard-ing the weight of the arms. During the first 25.0 cmof the lift, each arm exerts an upward force of 355 Non the torso. If the upward movement starts fromrest, what is the person’s speed at this point?

39. A 50.0 kg pole vaulter running at 10.0 m/s vaultsover the bar. If the vaulter’s horizontal componentof velocity over the bar is 1.0 m/s and air resistanceis disregarded, how high was the jump?

40. An 80.0 N box of clothes is pulled 20.0 m up a 30.0°ramp by a force of 115 N that points along theramp. If the coefficient of kinetic friction betweenthe box and ramp is 0.22, calculate the change in thebox’s kinetic energy.

41. Tarzan and Jane, whose total mass is 130.0 kg, starttheir swing on a 5.0 m long vine when the vine is at anangle of 30.0° with the horizontal. At the bottom ofthe arc, Jane, whose mass is 50.0 kg, releases the vine.What is the maximum height at which Tarzan can landon a branch after his swing continues? (Hint: TreatTarzan’s and Jane’s energies as separate quantities.)

42. A 0.250 kg block on a vertical spring with a springconstant of 5.00 × 103 N/m is pushed downward,compressing the spring 0.100 m. When released, theblock leaves the spring and travels upward vertical-ly. How high does it rise above the point of release?

43. Three identical balls, all with the same initial speed,are thrown by a juggling clown on a tightrope. Thefirst ball is thrown horizontally, the second is

A

R

B

C

R23–

thrown at some angle above the horizontal, and thethird is thrown at some angle below the horizontal.Disregarding air resistance, describe the motions ofthe three balls, and compare the speeds of the ballsas they reach the ground.

44. A 0.60 kg rubber ball has a speed of 2.0 m/s at pointA and kinetic energy of 7.5 J at point B. Determinethe following:

a. the ball’s kinetic energy at Ab. the ball’s speed at Bc. the total work done on the ball from A to B

45. Starting from rest, a 5.0 kg block slides 2.5 m down arough 30.0° incline in 2.0 s. Determine the following:

a. the work done by the force of gravityb. the mechanical energy lost due to frictionc. the work done by the normal force between

the block and the incline

46. A skier of mass 70.0 kg is pulled up a slope by amotor-driven cable. How much work is required topull the skier 60.0 m up a 35° slope (assumed to befrictionless) at a constant speed of 2.0 m/s?

47. An acrobat on skis starts from rest 50.0 m above theground on a frictionless track and flies off the trackat a 45.0° angle above the horizontal and at a heightof 10.0 m. Disregard air resistance.

a. What is the skier’s speed when leaving thetrack?

b. What is the maximum height attained?

48. Starting from rest, a 10.0 kg suitcase slides 3.00 mdown a frictionless ramp inclined at 30.0° from thefloor. The suitcase then slides an additional 5.00 malong the floor before coming to a stop. Determinethe following:

a. the suitcase’s speed at the bottom of the rampb. the coefficient of kinetic friction between the

suitcase and the floorc. the change in mechanical energy due to friction

49. A light horizontal spring has a spring constant of105 N/m. A 2.00 kg block is pressed against one endof the spring, compressing the spring 0.100 m. Afterthe block is released, the block moves 0.250 m to theright before coming to rest. What is the coefficientof kinetic friction between the horizontal surfaceand the block?

187Work and Energy

188

5 REVIEW

50. a. 310 Jb. −150 Jc. 180 N

51. a. 66 Jb. 2.3 m/sc. 66 Jd. −16 J

50. A 5.0 kg block is pushed 3.0 m at aconstant velocity up a vertical wall by aconstant force applied at an angle of30.0° with the horizontal, as shown atright. If the coefficient of kinetic fric-tion between the block and the wall is0.30, determine the following:

a. the work done by the force on the blockb. the work done by gravity on the blockc. the magnitude of the normal force between the

block and the wall

51. A 25 kg child on a 2.0 m long swing is released fromrest when the swing supports make an angle of30.0° with the vertical.

a. What is the maximum potential energy asso-ciated with the child?

b. Disregarding friction, find the child’s speed atthe lowest position.

c. What is the child’s total mechanical energy?d. If the speed of the child at the lowest position

is 2.00 m/s, what is the change in mechanicalenergy due to friction?

Chapter 5188

F

θ

d

In this activity, you will use this equation and

your graphing calculator to produce a table of

results for various values of q. Column one of the

table will be the displacement (X) in meters, and

column two will be the work done (Y1) in joules.

Visit go.hrw.com and enter the keyword

HF6WRKX to find this graphing calculator activity.

Refer to Appendix B for instructions on download-

ing the program for this activity.

Work of DisplacementWork done, as you learned earlier in this chapter, is

a result of the net applied force, the distance of the

displacement, and the angle of the applied force rel-

ative to the direction of displacement. Work done is

described by in the following equation:

Wnet = Fnetdcosq

The equation for work done can be represented on a

graphing calculator as follows:

Y1 = FXCOS(θ)

Graphing Calculator PracticeVisit go.hrw.com for answers to thisGraphing Calculator activity.

Keyword HF6WRKXT

189

5 REVIEW

52. a. 1.45 mb. 1.98 m/sc. 5.33 m/s

52. A ball of mass 522 g starts at rest and slides down africtionless track, as shown at right. It leaves thetrack horizontally, striking the ground.

a. At what height above the ground does the ballstart to move?

b. What is the speed of the ball when it leaves thetrack?

c. What is the speed of the ball when it hits theground?

h

1.00 m

1.25 m

m = 522 g

189Work and Energy

1. Design experiments for measuring your power out-

put when doing push-ups, running up a flight of

stairs, pushing a car, loading boxes onto a truck,

throwing a baseball, or performing other energy-

transferring activities. What data do you need to

measure or calculate? Form groups to present and

discuss your plans. If your teacher approves your

plans, perform the experiments.

2. Investigate the amount of kinetic energy involved

when your car’s speed is 60 km/h, 50 km/h, 40 km/h,

30 km/h, 20 km/h, and 10 km/h. (Hint: Find your

car’s mass in the owner’s manual.) How much work

does the brake system have to do to stop the car at

each speed?

If the owner’s manual includes a table of braking

distances at different speeds, determine the force

the braking system must exert. Organize your find-

ings in charts and graphs to study the questions and

to present your conclusions.

3. Investigate the energy transformations of your body

as you swing on a swing set. Working with a partner,

measure the height of the swing at the high and low

points of your motion. What points involve a maxi-

mum gravitational potential energy? What points

involve a maximum kinetic energy? For three other

points in the path of the swing, calculate the gravita-

tional potential energy, the kinetic energy, and the

velocity. Organize your findings in bar graphs.

4. In order to save fuel, an airline executive recom-

mended the following changes in the airlines’

largest jet flights:

a. restrict the weight of personal luggage

b. remove pillows, blankets, and magazines from

the cabin

c. lower flight altitudes by 5 percent

d. reduce flying speeds by 5 percent

Research the information necessary to calculate

the approximate kinetic and potential energy of a

large passenger aircraft. Which of the measures

described above would result in significant savings?

What might be their other consequences? Summa-

rize your conclusions in a presentation or report.

5. Make a chart of the kinetic energies your body can

have. Measure your mass and speed when walking,

running, sprinting, riding a bicycle, and driving a car.

Make a poster graphically comparing these findings.

6. You are trying to find a way to bring electricity to a

remote village in order to run a water-purifying

device. A donor is willing to provide battery charg-

ers that connect to bicycles. Assuming the water-

purification device requires 18.6 kW•h daily, how

many bicycles would a village need if a person can

average 100 W while riding a bicycle? Is this a useful

way to help the village? Evaluate your findings for

strengths and weaknesses. Summarize your com-

ments and suggestions in a letter to the donor.

Alternative Assessment

Alternative AssessmentANSWERS

1. Student plans should be safeand should include measuringwork and the time intervals.

2. Students should recognizethat all of the car’s KE mustbe brought to zero, becausevf = 0 m/s. Therefore, thebrake system must do asmuch work as the car’s KE (ifair resistance and friction areneglected).

3. Student plans should be safeand should include measure-ments of height, mass, andspeed. Kinetic energy is high-est at the bottom of the swing.

4. Students will need to researchinformation about altitude,friction, speed, and massesinvolved to evaluate the plans.

5. Student posters should indi-cate that increasing speedcauses their KE to increase.

6. Students’ letters will vary but should acknowledge that 186 h of bicycling areneeded for a day of use.Thus, at least eight bicycleswould be required.

CHAPTER 5

190

ANSWERS

1. D

2. H

3. C

4. F

5. D

6. J

7. B

8. J

MULTIPLE CHOICE

1. In which of the following situations is work notbeing done?

A. A chair is lifted vertically with respect to the floor.B. A bookcase is slid across carpeting.C. A table is dropped onto the ground.D. A stack of books is carried at waist level across

a room.

2. Which of the following equations correctly describesthe relation between power, work, and time?

F. W =

G. W =

H. P =

J. P =

Use the graph below to answer questions 3–5. Thegraph shows the energy of a 75 g yo-yo at differenttimes as the yo-yo moves up and down on its string.

3. By what amount does the mechanical energy ofthe yo-yo change after 6.0 s?

A. 500 mJB. 0 mJC. −100 mJD. −600 mJ

600

400

200

0

0 1 2 3

Potential energyKinetic energyMechanical energy

4 5 6 7 8

Ener

gy (m

J)

Time (s)

tW

Wt

tP

Pt

4. What is the speed of the yo-yo after 4.5 s?

F. 3.1 m/sG. 2.3 m/sH. 3.6 m/sJ. 1.6 m/s

5. What is the maximum height of the yo-yo?

A. 0.27 mB. 0.54 mC. 0.75 mD. 0.82 m

6. A car with mass m requires 5.0 kJ of work to movefrom rest to a final speed v. If this same amount ofwork is performed during the same amount oftime on a car with a mass of 2m, what is the finalspeed of the second car?

F. 2vG. �2�v

H.

J.

Use the passage below to answer questions 7–8.

A 70.0 kg base runner moving at a speed of 4.0 m/sbegins his slide into second base. The coefficient offriction between his clothes and Earth is 0.70. His slidelowers his speed to zero just as he reaches the base.

7. How much mechanical energy is lost because offriction acting on the runner?

A. 1100 JB. 560 JC. 140 JD. 0 J

8. How far does the runner slide?

F. 0.29 mG. 0.57 mH. 0.86 mJ. 1.2 m

v�2�

v2

Chapter 5190

Standardized Test PrepStandardized Test Prep

191

9. A

10. G

11. 206 W

12. v = �2gh�13. 4.4 m/s

14. 1200 J

15. 1200 J

16. 1900 J

17. 290 m

Use the passage below to answer questions 9–10.

A spring scale has a spring with a force constant of250 N/m and a weighing pan with a mass of 0.075 kg.During one weighing, the spring is stretched a distanceof 12 cm from equilibrium. During a second weighing,the spring is stretched a distance of 18 cm.

9. How much greater is the elastic potential energy ofthe stretched spring during the second weighingthan during the first weighing?

A.

B.

C.

D.

10. If the spring is suddenly released after each weigh-ing, the weighing pan moves back and forththrough the equilibrium position. What is theratio of the pan’s maximum speed after the secondweighing to the pan’s maximum speed after thefirst weighing? Consider the force of gravity onthe pan to be negligible.

F. H.

G. J.

SHORT RESPONSE

11. A student with a mass of 66.0 kg climbs a staircasein 44.0 s. If the distance between the base and thetop of the staircase is 14.0 m, how much powerwill the student deliver by climbing the stairs?

Base your answers to questions 12–13 on the informa-tion below.

A 75.0 kg man jumps from a window that is 1.00 mabove a sidewalk.

12. Write the equation for the man’s speed when hestrikes the ground.

49

32

23

94

49

23

32

94

13. Calculate the man’s speed when he strikes theground.

EXTENDED RESPONSE

Base your answers to questions 14–16 on the informa-tion below.

A projectile with a mass of 5.0 kg is shot horizontallyfrom a height of 25.0 m above a flat desert surface. Theprojectile’s initial speed is 17 m/s. Calculate the follow-ing for the instant before the projectile hits the surface:

14. The work done on the projectile by gravity.

15. The change in kinetic energy since the projectilewas fired.

16. The final kinetic energy of the projectile.

17. A skier starts from rest at the top of a hill that isinclined at 10.5° with the horizontal. The hillsideis 200.0 m long, and the coefficient of frictionbetween the snow and the skis is 0.075. At the bot-tom of the hill, the snow is level and the coefficientof friction is unchanged. How far does the skiermove along the horizontal portion of the snowbefore coming to rest? Show all of your work.

Skier

10.5 °

200.0 m

191Work and Energy

When solving a mathematicalproblem, you must first decide which equation orequations you need to answer the question.

Chapter 5192

PROCEDURE

Preparation

1. Read the entire lab procedure, and plan the steps you will take.

2. If you are not using a datasheet provided by your teacher, prepare a data

table in your lab notebook with four columns and seven rows. In the first

row, label the first through fourth columns Trial, Mass (kg), Stretched

Spring (m), and Force (N). In the first column, label the second through

seventh rows 1, 2, 3, 4, 5, and 6. Above or below the data table, make a

space to enter the value for Initial Spring (m).

3. If you are not using a datasheet provided by your teacher, prepare a sec-

ond data table in your lab notebook with three columns and seven rows.

In the first row, label the first through third columns Trial, Highest Point

A mass on a spring will oscillate vertically when it is lifted to the length of the

relaxed spring and released. The gravitational potential energy increases from a

minimum at the lowest point to a maximum at the highest point. The elastic

potential energy in the spring increases from a minimum at the highest point,

where the spring is relaxed, to a maximum at the lowest point, where the spring

is stretched. Because the mass is temporarily at rest, the kinetic energy of the

mass is zero at the highest and lowest points. Thus, the total mechanical energy

at those points is the sum of the elastic potential energy and the gravitational

potential energy.

A Hooke’s law apparatus combines a stand for mounting a hanging spring

and a vertical ruler for measuring the displacement of a mass attached to the

spring. In this lab, you will use a Hooke’s law apparatus to determine the

spring constant of a spring. You will also collect data during the oscillation of

a mass on the spring and use your data to calculate gravitational potential

energy and elastic potential energy at different points in the oscillation.

OBJECTIVES

•Determine the springconstant of a spring.

•Calculate elastic poten-tial energy.

•Calculate gravitationalpotential energy.

•Determine whethermechanical energy isconserved in an oscil-lating spring.

MATERIALS LIST• Hooke’s law apparatus

• meterstick

• rubber bands

• set of masses

• support stand and clamp

192

Lab PlanningBeginning on page T34 arepreparation notes and teachingtips to assist you in planning.

Blank data tables (as well as some sample data) appear on the One-Stop Planner.

No Books in the Lab?See the Datasheets for In-Text Labs workbook for areproducible master copy ofthis experiment.

CBL™ OptionA CBL™ version of this labappears in the CBL™Experiments workbook.

Safety CautionRemind students to attach massessecurely and to make sure thearea is clear before allowingmasses to oscillate. Remind stu-dents not to pull too hard on the spring because it will notreturn to the correct equilibriumposition. Also, do not add toomuch mass (which will stretchthe spring to the point ofdeforming it).

Skills Practice LabCHAPTER 5Skills Practice Lab

CHAPTER 5

Conservation ofMechanical Energy

SAFETY

• Tie back long hair, secure loose clothing, and remove loose jewelry toprevent their getting caught in moving or rotating parts. Put on goggles.

• Attach masses securely. Perform this experiment in a clear area.Swinging or dropped masses can cause serious injury.

193

CHAPTER 5 LAB

193Work and Energy

Figure 1Step 5: If the scale is adjusted toread 0.0 cm, record 0.00 m as theinitial spring length in your datatable.Step 7: In this part of the lab, youwill collect data to find the springconstant of the spring.Step 10: In this part of the lab,you will oscillate a mass on thespring to find out whether mechani-cal energy is conserved.

(m), and Lowest Point (m). In the first column, label the second through

seventh rows 1, 2, 3, 4, 5, and 6. Above or below the data table, make a

space to enter the value for Initial Distance (m).

Spring Constant

4. Set up the Hooke’s law apparatus as shown in Figure 1.

5. Place a rubber band around the scale at the initial resting position of the

pointer, or adjust the scale or pan to read 0.0 cm. Record this position of the

pointer as Initial Spring (m). If you have set the scale at 0.0 cm, record

0.00 m as the initial spring position.

6. Measure the distance from the floor to the rubber band on the scale.

Record this measurement in the second data table under Initial Distance

(m). This distance must remain constant throughout the lab.

7. Find a mass that will stretch the spring so that the pointer moves approx-

imately one-quarter of the way down the scale.

8. Record the value of the mass. Also record the position of the pointer

under Stretched Spring in the data table.

9. Perform several trials with increasing masses until the spring stretches to

the bottom of the scale. Record the mass and the position of the pointer

for each trial.


10. Find a mass that will stretch the spring to about twice its original length.

Record the mass in the second data table. Leave the mass in place on

the pan.

Tips and Tricks• For best results, use weights of

less than 1.0 N for steps 10–14.

• Show students how to read the scale on the Hooke’s lawapparatus.

• Demonstrate releasing themass hanger so it will oscillatevertically without twisting.

• Draw a diagram of the appa-ratus on the chalkboard, andlabel the distances studentswill be measuring in the lab:Initial Distance, Initial Spring,Stretched Spring, Highest Point,and Lowest Point. Show stu-dents how to refer to the dia-gram to find the elongation ofthe spring and the height ofthe mass at each point.

CheckpointsStep 5: Students should adjustthe scale to zero at the initialposition of the spring if possible.

Step 6: Make sure students aremeasuring the vertical distancefrom the floor to the initial posi-tion of the spring.

Step 13: Students may need topractice a technique to identifythe highest and lowest pointswhile the mass is oscillating.Without disturbing the appara-tus, they might use pencils aspointers to mark the place untilthey can place their rubber bands.

HAPTER XX LABCHAPTER 5 LAB

Chapter 5194

11. Raise the pan until the pointer is at the zero position, the position where

you measured the Initial Spring measurement.

12. Gently release the pan to let the pan drop. Watch closely to identify the

high and low points of the oscillation.

13. Use a rubber band to mark the lowest position to which the pan falls, as

indicated by the pointer. This point is the lowest point of the oscillation.

Record the values as Highest Point and Lowest Point in your data table.

14. Perform several more trials, using a different mass for each trial. Record

all data in your data table.

15. Clean up your work area. Put equipment away safely so that it is ready to

be used again.

ANALYSIS

1. Organizing Data Use your data from the first data table to calculate the

elongation of the spring. Use the equation elongation = initial spring −stretched spring.

2. Organizing Data For each trial, convert the masses used to measure

the spring constant to their force equivalents. Use the equation Fg = mag.

3. Organizing Data For each trial, calculate the spring constant using the

equation k = ⎯elon

fo

g

r

a

c

t

e

ion⎯. Take the average of all trials, and use this value

as the spring constant.

4. Organizing Data Using your data from the second data table, calculate

the elongation of the spring at the highest point of each trial. Use the

equation elongation = highest point − initial spring. Refer to Figure 2.

5. Organizing Data Calculate the elongation of the spring at the lowest

point of each trial. Use the equation elongation = lowest point − initial

spring. Refer to Figure 2.

6. Organizing Data For each trial, calculate the elastic potential energy,

PEelastic = ⎯12

⎯ kx2, at the highest point of the oscillation.

7. Organizing Data For each trial, calculate the elastic potential energy at

the lowest point of the oscillation.

8. Analyzing Results Based on your calculations in items 6 and 7, where

is the elastic potential energy greatest? Where is it the least? Explain these

results in terms of the energy stored in the spring.

9. Organizing Data Calculate the height of the mass at the highest point

of each trial. Use the equation highest = initial distance − elongation.

194

ANSWERSAnalysis1. Student answers will vary.Typical values range from0.022 m to 0.118 m.

2. For sample data, values rangefrom F = 0.050 N to F = 2.02 N.

3. Student answers will vary. Forsample data, the value for kavgwas 19.6 N/m.

4. Student answers will vary.Typical values range from0.000 m to 0.006 m.

5. Typical values range from0.021 m to 0.117 m.

6. For sample data, values range from 0.00 J to 3.52 × 10−4 J.

7. For sample data, values range from 4.31 × 10−3 J to 1.34 × 10−1 J.

8. PEelastic is greatest at the low-est point and least at the highestpoint because the elongation isgreatest at the lowest point andbecause PEelastic depends on theelongation squared.

9. Student answers will vary.Typical values range from0.234 m to 0.240 m.

10. Typical values range from0.123 m to 219 m.

Initial spring (m)

Initi

al d

ista

nce

(m)

Floor

Highest point (m)

Lowest point (m)

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

Hooke s law apparatus,

Figure 2

195

CHAPTER 5 LAB

11. Typical values range from 0.119 J to 0.475 J.

12. Typical values range from 0.073 J to 0.389 J.

13. Gravitational PE is greatest atthe highest point because it is proportional to the height of themass.

14. Student answers will vary.Make sure students use the rela-tionship PEtotal = PEg + PEelastic .For sample data, values rangefrom 0.119 J to 0.475 J at the highpoint and 0.161 J to 0.393 J at thelow point.

Conclusions15. Mechanical energy is con-served; the sum of the elastic andgravitational potential energies isthe same at the top and bottom ofthe oscillation.

16. A stiffer spring would givegreater values for the spring con-stant. The elastic potential energywould be greater, but the gravita-tional potential energy would notchange.

Extensions17. Student answers will vary. Forsample data, PEg at the midpointranges from 0.096 J to 0.408 J, andPEelastic at the midpoint rangesfrom 1.08 × 10−3 J to 3.70 × 10−2 J.Values for KE at the midpointrange from 0.001 J to 0.030 J, andvalues for the speed at the mid-point range from 0.11 m/s to0.90 m/s.

18. Student designs and resultswill vary. Students should recog-nize that some elastic materialsact as a spring only whenstretched, not when compressed.If you wish students to carry outtheir designed experiments, youmay provide them with varioussprings and elastic items, or havethem find items at home.195Work and Energy

10. Organizing Data Calculate the height of the mass at the lowest point

of each trial. Use the equation lowest = initial distance − elongation.

11. Organizing Data For each trial, calculate the gravitational potential

energy, PEg = magh, at the highest point of the oscillation.

12. Organizing Data For each trial, calculate the gravitational potential

energy at the lowest point of the oscillation.

13. Analyzing Results According to your calculations in items 11 and 12,

where is the gravitational potential energy the greatest? Where is it the

least? Explain these results in terms of gravity and the height of the mass

and the spring.

14. Organizing Data Find the total potential energy at the top of the oscil-

lation and at the bottom of the oscillation.

CONCLUSIONS

15. Drawing Conclusions Based on your data, is mechanical energy con-

served in the oscillating mass on the spring? Explain how your data sup-

port your answers.

16. Making Predictions How would using a stiffer spring affect the value

for the spring constant? How would this change affect the values for the

elastic and gravitational potential energies?

EXTENSIONS

17. Extending Ideas Use your data to find the midpoint of the oscillation

for each trial. Calculate the gravitational potential energy and the elastic

potential energy at the midpoint. Use the principle of the conservation of

mechanical energy to find the kinetic energy and the speed of the mass at

the midpoint.

18. Designing Experiments Based on what you have learned in this lab,

design an experiment to measure the spring constants of springs and

other elastic materials in common products, such as the springs inside

ball point pens, rubber bands, or even elastic waistbands. Include in your

plan a way to determine how well each spring or elastic material con-

serves mechanical energy. If you have time and your teacher approves

your plan, carry out the experiment on several items, and make a table

comparing your results for the various items.

CHAPTER REVIEW, ASSESSMENT, AND STANDARDIZED TEST PREPARATION

Online and Technology Resources

Visit go.hrw.com to access online resources. Click Holt Online Learning for an online edition of this textbook, or enter the keyword HF6 Home for other resources. To access this chapter’s extensions, enter the keyword HF6MOMXT.

Planning Guide

Chapter Openerpp. 196 – 197 CD Visual Concepts, Chapter 6 b

Section 1 Momentum and Impulse• Compare the momentum of different moving objects.• Compare the momentum of the same object moving with dif-

ferent velocities.• Identify examples of change in the momentum of an object.• Describe changes in momentum in terms of force and time.

OSP Lesson Plans TR 20 Impulse-Momentum Theorem TR 21A Stopping Distances

TE Demonstration Impulse, p.200 bANC CBL™ Experiment Impulse and Momentum*◆ g

OSP Lesson Plans CD Interactive Tutor Module 7, Conservation of

Momentum gOSP Interactive Tutor Module 7, Worksheet

g

TR 21 Force and Change in Momentum TR 22A Momentum in a Collision

SE Inquiry Lab Conservation of Momentum, pp. 230 – 231 ◆ g

ANC Datasheet Inquiry Lab, Conservation of Momentum* g

ANC Datasheet Skills Practice Lab, Conservation of Momentum* g

ANC CBLTM Experiment Conservation of Momentum*◆ g

196A Chapter 6 Momentum and Collisions

Momentum and Collisions To shorten instruction because of time limitations, omit the opener and Section 3 and abbreviate the review.

Compression Guide

CHAPTER 6

pp. 205 – 211 PACING • 90 min

PACING • 45 min

SE Chapter Highlights, p. 222 SE Chapter Review, pp. 223 – 227 SE Graphing Calculator Practice, p. 226 g SE Alternative Assessment, p. 227 a SE Standardized Test Prep, pp. 228 –229 g SE Appendix D: Equations, pp. 856 – 857 SE Appendix I: Additional Problems, pp. 884 – 886ANC Study Guide Worksheet Mixed Review* gANC Chapter Test A* gANC Chapter Test B* a OSP Test Generator

PACING • 90 min

Section 2 Conservation of Momentum• Describe the interaction between two objects in terms of the

change in momentum of each object.• Compare the total momentum of two objects before and after

they interact.• State the law of conservation of momentum.• Predict the final velocities of objects after collisions, given the

initial velocities.

OSP Lesson Plans TR 22 Types of Collisions TR 23A Inelastic Collision TR 24A Elastic Collision

SE Quick Lab Elastic and Inelastic Collisions, p. 217 g

TE Demonstration Inelastic Collisions, p.212 g

pp. 212 – 220 Advanced Level

Section 3 Elastic and Inelastic Conditions• Identify different types of collisions.• Determine the changes in kinetic energy during perfectly

inelastic collisions.• Compare conservation of momentum and conservation of

kinetic energy in perfectly inelastic and elastic collisions.• Find the final velocity of an object in perfectly inelastic and

elastic collisions.

PACING • 45 min

OBJECTIVES LABS, DEMONSTRATIONS, AND ACTIVITIES TECHNOLOGY RESOURCES

• Lab Materials QuickList Software

• Holt Calendar Planner• Customizable Lesson Plans• Printable Worksheets

• ExamView ® Test Generator• Interactive Teacher Edition• Holt PuzzlePro ®

• Holt PowerPoint ® Resources

This CD-ROM package includes:

pp. 198 – 204 PACING • 45 min

www.scilinks.orgMaintained by the National Science Teachers Association. This CD-ROM consists of

interactive activities that give students a fun way to extend their knowledge of physics concepts.

Topic: MomentumSciLinks Code: HF60988Topic: RocketrySciLinks Code: HF61324

Topic: CollisionsSciLinks Code: HF60311

Each video segment is accompanied by a Critical Thinking Worksheet.Segment 6Egg Drop Contest

CNN Science in the NewsThis CD-ROM consists of multimedia presenta-tions of core physics concepts.

National ScienceEducation Standards

SE Sample Set A Momentum, pg. 199 bANC Problem Workbook* and OSP Problem Bank Sample Set A b SE Sample Set B Force and Impulse, pg. 201 b TE Classroom Practice, p. 201 bANC Problem Workbook* and OSP Problem Bank Sample Set B b SE Sample Set C Stopping Distance, pp. 202 – 203 TE Classroom Practice, p. 202 bANC Problem Workbook* and OSP Problem Bank Sample Set C b


UCP 1,2,3HNS 3

SE Sample Set D Conservation of Momentum, pp. 208 – 209 g TE Classroom Practice, p. 208 gANC Problem Workbook* and OSP Problem Bank Sample Set D g SE Conceptual Challenge, p. 206


UCP 1,2,3,5SAI 1,2ST 1,2SPSP 1,4,5PS 5a

Chapter 6 Planning Guide 196B

SE Sample Set E Perfectly Inelastic Collisions, pp. 213 – 214 g TE Classroom Practice, p. 213 gANC Problem Workbook* and OSP Problem Bank Sample Set E g SE Sample Set F Kinetic Energy in Perfectly Inelastic Collisions,

pp. 215 – 216 g TE Classroom Practice, p. 215 gANC Problem Workbook* and OSP Problem Bank Sample Set F g SE Sample Set G Elastic Collisions, pp. 218 – 219 a TE Classroom Practice, p. 218 aANC Problem Workbook* and OSP Problem Bank Sample Set G a

SE Section Review, p. 220 aANC Study Guide Worksheet Section 3* aANC Quiz Section 3* g

UCP 1,2,3SAI 1,2PS 5a

VisualConcepts

SKILLS DEVELOPMENT RESOURCES REVIEW AND ASSESSMENT CORRELATIONS

KEY SE Student Edition TE Teacher Edition ANC Ancillary Worksheet

OSP One-Stop Planner CD CD or CD-ROM TR Teaching Transparencies

EXT Online Extension * Also on One-Stop Planner ◆ Requires advance prep

CHAPTER XCHAPTER 6Overview

196

Section 1 defines momentum in terms of mass and velocity,introduces the concept ofimpulse, and relates impulse andmomentum.

Section 2 explores the law ofconservation of momentum anduses this law to predict the finalvelocity of an object after a collision.

Section 3 distinguishes betweenelastic, perfectly inelastic, andinelastic collisions and discusseswhether kinetic energy is con-served in each type of collision.

About the IllustrationSoccer is a good example to helpstudents understand the conceptof momentum and distinguish itfrom force, velocity, and kineticenergy. This photograph is a dra-matic example of a player collid-ing with a ball and changing themomentum of the ball. Use thisexample to illustrate the vectornature of momentum; the photo-graph can open a discussionabout how the direction as wellas the magnitude of momentumis affected by the collision.


See Module 7“Conservation of Momentum”promotes additional developmentof problem-solving skills for thischapter.

Documents

Compression Guide Work and Energydysoncentralne.pbworks.com/w/file/fetch/102939991/Physics Ch 5.pdf · • Recognize the difference between the scientific and ordinary defi- ... Work