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CHAPTER

10 ·Work, Energy,and SimpleMachines

, ............................~ A-Not-So-Simple Machine

How does a multispeed bicycle let a rider ride over anykind of terrain with the least effort?

What is energy? When you have a lot of energy you canrun farther or faster; you can jump higher. Objects, as

well as people, can have energy. A stone falling off a highledge has enough energy to damage a car roof. One way tosummarize the examples of energy above is to say that anobject has energy if it can produce a change in itself or inits surroundings. This may not seem very exact, but there isno more precise definition.

In this chapter, we will concentrate on ways of producingchanges in objects or their environment. The human racehas developed many tools and machines that make it easierto produce such changes. A 10-speed bicycle is a machinethat allows the rider to select the speed that makes the bikeeasiest to ride whether it is going uphill or along a levelpath. You will discover the physical principles that makethis kind of machine work.

Chapter OutlineIO.! WORK AND ENERGY· Work· Work and Direction of Force· Power

10.2 MACHINES· Simple and Complex Machines· Energy Conservation and

Mechanical Advantage· Compound Machines· The Human Walking Machine

{concept CheckThe following terms or conceptsfrom earlier chapters areimportant for a goodunderstanding of this chapter. Ifyou are not familiar with them,you should review them beforestudying this chapter.· finding area under graph,

Chapters 2, 3, 4· force and force components,

Chapter 5· Newton's second and third

laws, Chapter 5· friction, Chapter 5· cosine of an angle, Chapter 6

197

Objectives· display an ability to calculate work

done by a force.· identify the force that does the

work.• understand the relationship

between work done and energytransferred.

· differentiate between work andpower and correctly calculatepower used.

FIGURE 10-1. In physics, work isdone only when a force causes anobject to move a certain distance.

Work is a product of a force and thedisplacement of an object in thedirection of the force.

POCKETLAB

AN INCLINED MASSAttach a spring scale to a

1.0-kg mass with a string. Keepthe string parallel to the tableand measure the force neededto pull the mass at a steadyspeed. Increase the angle be-tween the string and the tabletop, for example, to 30°. Try tokeep the angle constant as youpull the 1.0-kg mass along thetable at a slow, steady speed.Notice the reading on the scale.How much force is in the direc-tion of motion? How much workis done when the 1.0 kg moves1.0 m? How does the work com-pare to the previous value?

198 Work, Energy, and Simple Machines

10.1 WORK AND ENERGY

Ifyou spend the morning lifting crates from the floor of a warehouseup onto a truck, you will get tired and hungry. You will need to eat

food to "get more energy." Somehow, the energy in the food will betransferred into the energy in the raised crates. We use the word workto indicate the amount of energy that was transferred from food to youto the crates. The word work has both an everyday and a scientificmeaning. In the case of lifting crates, everyone will agree that work wasdone. In everyday life, however, we use the word when talking aboutother activities. For example, everyone says that learning physics is hardwork! But in physics, we reserve the term work to mean a very specialform of physical activity.

WorkWhen lifting crates, or any other object, we do more work when the

crate is heavier. The task is even harder if the crate must be liftedhigher. It seems reasonable to use the quantity force times distance tomeasure the amount of energy transferred when lifting. For cases wherethe force is constant, we define work as the product of the force exertedon an object and the distance the object moves in the direction of theforce. In equation form,

I W = Fd,lwhere W is the work, F is the magnitude of force, and d is the magni-tude of displacement in the direction of the force. Note that work is ascalar quantity; it has no direction. The SI unit of work is the joule[JOOL]. The joule is named after James Prescott Joule, a nineteenth cen-tury English physicist and brewer. If a force of one newton moves anobject one meter, one joule of work is done.

1 joule (J) = 1 newton· meter (N . m)

Work is done on an object only if the object moves. If you hold aheavy banner at the same height in the same place for an hour, youmay get tired, but you do no work on the banner. Even if you carry thebanner at constant velocity and at a constant height, you do no workon it. The force you exert is upward while the motion is sideways, sothe banner gains no energy. Work is only done when the force anddisplacement are in the same direction.

Force versus Displacement

F

~ 20+--......,r---o.....---.--1Q)

~~ 10

do 1.0 2.0 3.0

Displacement (m)

A force-displacement graph can give you a picture of the work done.Figure 10-2 shows the force-displacement graph of a rock being pushedhorizontally. A net force of 20 N is needed to push the rock 1.5 m withconstant velocity. The work done on the rock is the product of the forceand the displacement, W = Fd = (20 N)(1.5 m) = 30 J. The shadedarea under the curve of Figure 10-2 is equal to 20 N x 1.5 m, or30 J. The area under the curve of a force-displacement graph representsthe work done. If you increase either the width of the rectangle (dis-placement) or the height (force), you increase the work done.

Example ProblemCalculating Work

A student Iifts a box of books that weighs 185 N. The box is lifted0.800 m. How much work does the student do?

Given: mg = 185 N Unknown: work, Wd = 0.800 m Basic equation: W = Fd

Solution: The student exerted enough force to lift the box, that is,enough to balance the weight of the box. Thus,

W = Fd = (185 N)(0.800 m) = 148 N . m = 148 J

Practice Problems1. A force of 825 N is needed to push a car across a lot. Two students

push the car 35 m.a. How much work is done?b. After a rainstorm, the force needed to push the car doubled be-

cause the ground became soft. By what amount does the workdone by the students change?

2. A delivery clerk carries a 34-N package from the ground to the fifthfloor of an office building, a total height of 15 m. How much workis done by the clerk?

3. What work is done by a forklift raising a 583-kg box 1.2 m?~ 4. You and a friend each carry identical boxes to a room one floor

above you and down the hall. You choose to carry it first up thestairs, then down the hall. Your friend carries it down the hall, thenup another stairwell. Who does more work?

FIGURE 10-2. A force-displacementgraph of a rock being pushedhorizontally.

Work is done on an object only if itmoves in the direction of the force.

Direction ofapplied force

Direction of motion

FIGURE 10-3. Work is done on thebox only when it moves in thedirection of the applied force.

10.1 Work and Energy 199

FIGURE 10-4. If a force is applied tothe mower at an angle, the net forcedoing the work is the componentthat acts in the direction of themotion.

Only the component in the direction ofthe motion does work.

POCKETLAB

WORKING OUTAttach a spring balance to a

1.0-kg mass with a string. Pullthe mass along the table at aslow, steady speed keeping thebalance parallel to the table top.Notice the readingon the springbalance. What are the physicalfactors that determine theamount of force? How muchwork is done in movingthe mass1.0 m? Predictthe force and thework when a 2.0-kg mass ispulled along the table. Try it.Was your predictionaccurate?


r,F« = F cos 25.0°

= (125 N) (0.906)Fh=113N

a b

Work and Direction of ForceWork is done only if a force is exerted in the direction of motion. The

person or object that exerts the force does the work. If a force is exertedperpendicular to the motion, no work is done. What if a force is exertedat some other angle to the motion? For example, if you push the lawnmower in Figure 10-4a, what work do you do? You know that anyforce can be replaced by its components. The 125-N force (F) you exertin the direction of the handle has two components, Figure 10-4b. Thehorizontal component (Fh) is 113 N; the vertical component (Fv) is- 53 N (downward). The vertical component is perpendicular to themotion. It does no work. Only the horizontal component does work.The work you do when you exert a force at an angle to a motion isequal to the component of the force in the direction of the motion timesthe distance moved.

The magnitude of the component of the force F acting in the directionof motion is found by multiplying the force F by the cosine of the anglebetween F and the direction of motion.

W = F(cos O)d= Fd cos 0

Other objects exert forces on the lawn mower. Which of these objectsdo work? Earth's gravity acts downward, and the ground exerts the nor-mal force upward. Both are perpendicular to the direction of motion.That is, the angle between the force and direction of motion is 90°.Since cos 90° = 0, no work is done.

The lawn exerts a force, friction, in the direction opposite the motion.In fact, if the lawn mower moves at a constant speed, the horizontalcomponent of the applied force is balanced by the force of friction,Ffriction' The angle between the force of friction and the direction of mo-tion is 180°. Since cos 180° = - 1, the work done by the grass is W =

- Ffrictiond. The work done by the friction of the grass is negative. Theforce is exerted in one direction, while the motion was in the oppositedirection. Negative work indicates that work is being done on the grassby the mower. The positive sign of the work done by you exerting theforce on the handle means you are doing work.

What is the effect of doing work? When you lift a box of books ontoa shelf, you give the box a certain property. If the box falls, it can dowork; it might exert forces that crush another object. If the box is on acart and you push on it, you will start it moving. Again, the box couldexert forces that crush another object. In this case too, you have giventhe box energy-the ability to produce a change in itself or its surround-ings.

By doing work on the box, you have transferred energy from yourbody to the box. Thus, we say that work is the transfer of energy bymechanical means. You can think of work as energy transferred as theresult of motion. When you lift a box, the work you do is positive.Energy is transferred from you to the box. When you lower the box,work is negative. The energy is transferred from the box to you.

PROBLEM SOLVING STRATEGYWhen doing work problems, you should:1. Carefully check the forces acting on the object. Draw a diagram in-

dicating all the force vectors.2. Ask, "What is the displacement? What is the angle between force and

displacement?"3. Check the sign of the work by determining the direction energy is

transferred. If the energy of the object increases, the work done on itis positive.

Example ProblemWork-Force and Displacement at an Angle

A sailor pulls a boat along a dock using a rope at an angle of 60.0°with the horizontal, Figure 10-5. How much work is done by thesailor if he exerts a force of 255 N on the rope and pulls the boat30.0 m?Given: (J = 60.0°

F = 255 Nd = 30.0 m

Solution: The work done by the sailor,W = Fd cos (J = (255 N)(30.0 m)(cos 60.0°)

= (255 N)(30.0 m)(0.500) = 3.83 x 103 J

Unknown: work, WBasic equation: W = Fd cos (J

F. Y. I.In physics, work and energy

have precise meanings, whichmust not be confused with theireveryday meanings. Robert Op-penheimer wrote, "Often thevery fact that the words of sci-ence are the same as those ofour common life and tongue canbe more misleading than enlight-ening."

FIGURE 10-5. The force exerted bythe sailor must exceed the forceneeded to move the boat becauseonly the horizontal force changes thevelocity.


85N --_~L3m

4m

FIGURE 10-6. Use with PracticeProblem 8.

Power is the rate at which work isdone.

FIGURE 10-7. The same amount ofwork is being done as in Figure10-4, but the power is greater here.


Practice Problems5. How much work does the force of gravity do when a 25-N object

falls a distance of 3.5 m?6. An airplane passengercarries a 215-N suitcase up stairs, a displace-

ment of 4.20 m vertically and 4.60 m horizontally.a. How much work does the passengerdo?b. The same passenger carries the same suitcase back down the

same stairs. How much work does the passengerdo now?7. A rope is used to pull a metal box 15.0 m across the floor. The rope

is held at an angle of 46.0° with the floor and a force of 628 N isused. How much work does the force on the rope do?

~ 8. A worker pushes a crate weighing 93 N up an inclined plane, push-ing horizontally, parallel to the ground, Figure 10-6.a. The worker exerts a force of 85 N. How much work does he do?b. How much work is done by gravity? (Be careful of signs.)c. The coefficient of friction is J-t = 0.20. How much work is done

by friction? (Be careful of signs.)

PowerUntil now, none of the discussions of work have mentioned the time

it takes to move an object. The work done lifting a box of books is thesame whether the box is lifted in 2 seconds or if each book is liftedseparately, so that it takes 20 minutes to put them all on the shelf. Thework done is the same, but the power is different. Power is the rate ofdoing work. That is, power is the rate at which energy is transferred.Power is work done divided by the time it takes. Power can be calcu-lated using

Power is measured in watts (W). One watt is one joule of energytransferred in one second. A machine that does work at a rate of onejoule per second has a power of one watt. A watt is a relatively smallunit of power. For example, a glass of water weighs about 2 N. If youlift it 0.5 meter to your mouth, you do 1 joule of work. If you do it inone second, you are doing work at the rate of one watt. Becausea wattis such a small unit, power is often measured in kilowatts (kW). A kilo-watt is 1000 watts.

Example ProblemPower

An electric motor lifts an elevator that weighs 1.20 x 104 N adistance of 9.00 m in 15.0 s, Figure 10-8, a. What is the power ofthe motor in watts? b. What is the power in kilowatts?

Given: F = 1.20 x 104 N Unknown: power, Pd = 9.00 m .. Wt = 15.0 s BaSICequation: P = t

Solution:

a. power of a motor,

p = W = Fd = (1.20 x 104 N)(9.00 m)t t 15.05

7.20 X 103 N . m/s = 7.20 x 103 J/s

b. P = 7.20 X 103 W(,~o~WW) = 7.20 kW

7.20 X 103 W

Practice Problems9. A box that weighs 575 N is lifted a distance of 20.0 m straight up

by a rope. The job is done in 10.0 s. What power is developed inwatts and kilowatts?

10. A rock climber wears a 7.50-kg knapsack while scaling a cliff. After30.0 min, the climber is 8.2 m above the starting point.a. How much work does the climber do on the knapsack?b. If the climber weighs 645 N, how much work does she do lifting

herself and the knapsack?c. What is the average power developed by the climber?

11. An electric motor develops 65 kW of power as it lifts a loaded ele-vator 17.5 m in 35.0 s. How much force does the motor exert?

~ 12. Two cars travel the same speed, so that they move 105 km in 1 h.One car, a sleek sports car, has a motor that delivers only35 kW of power at this speed. The other car needs its motor toproduce 65 kW to move the car this fast. Forces exerted by frictionfrom the air resistance cause the difference.a. For each car, list the external horizontal forces exerted on it, and

give the cause of each force. Compare their magnitudes.b. By Newton's third law, the car exerts forces. What are their di-

rections?c. Calculate the magnitude of the forward frictional force exerted

by each car.d. The car engines did work. Where did the energy they transferred

come from?

CONCEPT REVIEW1.1 Explain in words, without the use of a formula, what work is.1.2 When a bowling ball rolls down a level alley, does Earth's gravity

do any work on the ball? Explain.1.3 As you walk, there is a static frictional force between your shoes

and the ground. Is any work done?1.4 Critical Thinking: If three objects exert forces on a body, can they

all do work at the same time? Explain.

----

FIGURE 10-8. If the elevator took20.0 s instead of 15.0 s, would thework change? Would the powerchange?

A watt, one joule/second, is a relativelysmall unit. A kilowatt, 1000 watts, iscommonly used.


PurposeTo determine the work and power as you climbstairs.

Materials. Each lab group will need a meter stick, a stop-

watch, and a climber.

Procedure1. Estimate the mass of the climber in kg. (Hint:

100 kg weighs approximately 220 lbs.)2. Measure the height (vertical distance) of the

stairs.3. The climber should approach the stairs with a

steady speed.

NOTE - Do NOT run. Do NOT skip stairs. DoNOT trip.

4. The timer will start the watch as the climber hitsthe first stair and stop the clock when theclimber reaches the top.

Your Power

5. Climbers and timers should rotate until all stu-dents have had the opportunity to climb.

1. Calculate the work and the power for yourself .2. Compare your work and power calculations to

others.

Observations and Dat

Th

1

Analqsl«1. Which students did the most work? Explain.2. Which students had the most power? Explain

with examples.3. Calculate your power in kilowatts.

Applications1. Your local electric company supplies you 1 kW

of power for 1 h for 8rt. Assume that you couldclimb stairs continuously for 1 h. How muchmoney would this climb be worth?


10.2 MACHINES

Everyone uses some machines every day. Some are simple tools likebottle openers and screwdrivers; others are complex objects such as

bicycles and automobiles. Machines, whether powered by engines orpeople, make our tasks easier. A machine eases the load either bychanging the magnitude or the direction of a force, but does not changethe amount of work done.

Simple and Complex MachinesConsider the bottle opener in Figure 10-9. When you use the opener,

you lift the handle, doing work on the opener. The opener lifts the cap,doing work on it. The work you do is called the input work, Wi. Thework the machine does is called the output work, woo

Work, you remember, is the transfer of energy by mechanical means.You put work into the machine. That is, you transfer energy to the bottleopener. The machine does work on another object. The opener, in turn,transfers energy to the cap. The opener is not a source of energy, so thecap cannot receive more energy than you put into the opener. Thus,the output work can never be larger than the input work. The machinesimply aids in the transfer of energy from you to the bottle cap.

Energy Conservation and Mechanical AdvantageThe force you exert on a machine is called the effort force, Fe. The

force exerted by the machine is called the resistance force, Fr. The ratioof resistance force to effort force, F/Fe, is called the mechanical advan-tage (MA) of the machine. That is,

r,Fe

Many machines, like the bottle opener, have a mechanical advantagegreater than one. When the mechanical advantage is greater than one,the machine increases the force you apply.

We can calculate the mechanical advantage of a machine using thedefinition of work. The input work is the product of the effort force youexert, Fe, and the displacement of your hand, de. In the same way, theoutput work is the product of the resistance force, F" and the displace-ment caused by the machine, d.. A machine can increase force, but itcannot increase energy. An ideal machine transfers all the energy, sothe output work equals the input work,

MA

Wo = Wi, orFrdr = Fede .

This equation can be rewritten Fr = dde. We know that the mechanicalFe r

advantage is given by MA = F/Fe. For an ideal machine, we also haveMA = de/dr. Because this equation is characteristic of an ideal machine,

Objectives· demonstrate knowledge of why

simple machines are useful.· understand the concept of

mechanical advantage in ideal andreal machines and use it correctlyin solving problems.

· recognize that complex machinesare simple machines linkedtogether; show an ability tocalculate mechanical advantage forsome complex machines.

FIGURE 10-9. A bottle opener is anexample of a simple machine. Itmakes opening a bottle easier, butnot less work.

10.2 Machines 205

a

FIGURE 10-10. The simple machinespictured are the lever (a); pulley (b);wheel-and-axle (c); inclined plane(d); wedge (e); and screw (f).

The mechanical advantage of amachine is the ratio of the force exertedby the machine to the force applied tothe machine.

Efficiency is the ratio of the work done.by the machine to the work put into themachine.


c

the mechanical advantage is called the ideal mechanical advantage,IMA,

deIMA = -d'

r

Note that you measure distances moved to calculate the ideal mechan-ical advantage, IMA, but you measure the forces exerted to find theactual mechanical advantage, MA.

In a real machine, not all of the input work is available as outputwork. The efficiency of a machine is defined as the ratio of output workto input work. That is,

,~-------------------,

efficiency Wo x 100%.Wi

An ideal machine has equal output and input work, WjWi = 1, andthe efficiency is 100%. All real machines have efficiencies less than100%. We can express the efficiency in terms of the mechanical advan-tage and ideal mechanical advantage.

F,.IFeefficiency = -d d x 100%

e I r

MAefficiency = --- x 100%

IMA

The IMA of most machines is fixed by the machine's design. An effi-cient machine has an MA almost equal to its IMA. A less efficient ma-chine has a smaller MA. Lower efficiency means that a greater effortforce is needed to exert the same resistance force.

All machines, no matter how complex, are combinations of up to sixsimple machines shown in Figure 10-10. They are the lever, pulley,wheel-and-axle, inclined plane, wedge, and screw. Gears, one of thesimple machines used in a bicycle, are really a form of the wheel-and-axle. The IMA of all machines is the ratio of distances moved. Figure10-11 shows that for levers and wheel-and-axles this ratio can be re-placed by the ratio of the distances between the places where the forcesare applied and the pivot point. A common version of the wheel-and-axle is a pair of gears on a rotating shaft. The IMA is the ratio of theradii of the two gears.

Compound MachinesA compound machine consists of two or more simple machines

linked so that the resistance force of one machine becomes the effortforce of the second. For example, in the bicycle, the pedal and sprocket(or gear) act like a wheel-and-axle. The effort force is the force you exerton the pedal, Fon pedal. The resistance is the force the sprocket exerts onthe chain, Fon chain'

The chain exerts an effort force on the rear wheel sprocket, Fby chain,

equal to the force exerted on the chain. This sprocket and the rearwheel act like another wheel-and-axle. The resistance force is the forcethe wheel exerts on the road, Fon road. By Newton's third law, the groundexerts an equal forward force on the wheel. This force accelerates thebicycle forward.

The mechanical advantage of a complex machine is the product ofthe mechanical advantagesof the simple machines it is made up of. Forexample, in the case of the bicycle,

MA = Fon chain

Fon pedal

Fon road

Fon pedal'

Fon road

Fby chain

r-r,--!I1+1-- r; --~IIII

pivot point

a

CHIROPRACTORDo you like to work on "the human

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. mat' t:

Figure 10-11. For levers and wheel

and axles, the IMA is '!!r,

10.2 Machines 207

F. Y. I.A bicyclist in a road race like

the Tour de France rides atabout 20 milh (9 m/s) for over sixhours a day. The power output ofthe racer is about one kilowatt.One-quarter of that power goesinto moving the bike against theresistance of the air, gears, andtires. Three-quarters of thepower is used to cool the racer'sbody. To generate this power,the racer eats 7000-8000 foodcalories each day of the 23-dayrace.

FIGURE 10-12. Some of the mostingenious compound machines everconceived came from theimagination of cartoonist RubeGoldberg, who was famous fordevising complex ways for doingsimple tasks.


The IMA of each wheel-and-axle machine is the ratio of the distancesmoved. For the pedal sprocket,

pedal radiusIMA = --'------

front sprocket radIusFor the rear wheel, we have

rear sprocket radiusIMA = ----'-----wheel radius

For the bicycle, then

pedal radiusIMA = ---'------front sprocket radiusrear sprocket radiusfront sprocket radius

rear sprocket radiuswheel radius

pedal radiuswheel radius·

Since both sprockets use the same chain and have teeth of the samesize, you can simply count the number of teeth on the gears and findthat

teeth on rear sprocket pedal arm lengthIMA = -----'-----teeth on front sprocket wheel radius .

On a multi-gear bike, the rider can change the mechanical advantageof the machine by choosing the size of one or both sprockets. Whenaccelerating or climbing a hill, the rider increases the mechanical ad-vantage to increase the force the wheel exerts on the road. On the otherhand, when going at high speed on a level road, less force is needed,and the rider decreases the mechanical advantage to reduce the dis-tance the pedals must move for each revolution of the wheel.

You know that if the pedal is at the top or bottom of its circle, nomatter how much force downward you exert, the pedals will not turn.The force of your foot is most effective when the force is exerted per-pendicular to the arm of the pedal. Whenever a force on a pedal isspecified, you should assume that it is applied perpendicular to the arm.

- ------...;;;;,---~

Example ProblemFIGURE 10-13. A series of simplemachines combine to transmit theforce the rider exerts on the pedal tothe rear wheel of the bike.The Bicycle Wheel

A student uses the bicycle wheel with gear radius 4.00 cm andwheel radius 35.6 cm. When a force of 155 N is exerted on thechain, the wheel rim moves 14.0 cm. Due to friction, its efficiencyis 95.0%. a. What is the IMA of the wheel and gear? b. What is theMA of the wheel and gear? c. What force does the scale attached tothe wheel read? d. How far did the student pull the chain?Given: effort force, Fe = 155 N Unknowns: a. IMA

gear radius = 4.00 cm b. MAwheel radius = 35.6 cm c. resistance force, F,efficiency = 95.0% d. effort displacement, deresistance displacement,d, = 14.0 cm

Solution:

de gear radiusa.IMA = -d =

r wheel radius4.00 cm35.6 cm

= 0.112

MAb. Since efficiency = IMA x 100%

IMAMA = eff·--

100%(95.0%)(0.112)

100%= 0.107

F,c. MA = -Fe

F, = (MA)(Fe)

= (0.107)(155 N)

ded.IMA = d,

de = (lMA)(d,)= (0.112)(14.0 cm)

16.6 N

1.57 cm

10.2 Machines 209

• ••••••••••••••••• 1irrIrrrrrrr...

BIOLOGY ,..CONNECTION

FIGURE 10-14. The human walkingmachine.

Practice Problems13. A sledge hammer is used to drive a wedge into a log to split it.

When the wedge is driven 20 cm into the log, the log is separateda distance of 5.0 cm. A force of 1.9 x 104 N is needed to split thelog, and the sledge exerts a force of 9.8 x 103 N.a. What is the IMA of the wedge?b. Find the MA of the wedge.c. Calculate the efficiency of the wedge as a machine.

14. A worker uses a pulley system to raise a 225-N carton 16.5 m. Aforce of 129 N is exerted and the rope is pulled 33.0 m.a. What is the mechanical advantage of the pulley system?b. What is the efficiency of the system?

15. A boy exerts a force of 225 N on a lever to raise a 1.25 x 103-Nrock a distance of 0.13 m. If the efficiency of the lever is 88.7%,how far did the boy move his end of the lever?

~ 16. If the gear radius is doubled in the example above, while the forceexerted on the chain and the distance the wheel rim moves remainthe same, what quantities change, and by how much?

The Human Walking MachineMovement of the human body is explained by the same principles of

force and work that describe all motion. Simple machines, in the formof levers, give us the ability to walk and run. Lever systems of the bodyare complex, but each system has four basic parts: 1) a rigid bar (bone),2) a source of force (muscle contraction), 3) a fulcrum or pivot (movablejoints between bones), and 4) a resistance (the weight of the body or anobject being lifted or moved), Figure 10-14. Lever systems of the bodyare not very efficient, and mechanical advantages are low. This is whywalking and jogging require energy (burn calories) and help individualslose weight.

When a person walks, the hip acts as a fulcrum and moves throughthe arc of a circle centered on the foot. The center of mass of the bodymoves as a resistance around the fulcrum in the same arc. The lengthof the radius of the circle is the length of the lever formed by the bonesof the leg. Athletes in walking races increase their velocity by swingingtheir hips upward to increase this radius.

A tall person has a lever with a lower MA than a short person. Al-though tall people can usually walk faster than short people, a tall per-son must apply a greater force to move the longer lever formed by theleg bones. Walking races are usually 20 or 50 km long. Because of theinefficiency of their lever systems and the length of a walking race, verytall people rarely have the stamina to win.

CONCEPT REVIEW2.1 Many hand tools are simple machines. Classify the tools below as

levers, wheel-and-axles, inclined planes, wedges, or pulleys.a. screwdriver, b. pliers, c. chisel, d. nail puller, e. wrench


Physicsandtechnology

ARTIFICIAL JOINTS

Great progress has beenmade in designing and sub-

stituting artificial joints for dam-aged or missing joints. Becauseof the tremendous stresses inthe bones and joints of the armsand legs, the materials fromwhich artificial parts are madeand the means of attachmentmust be extremely strong.

Titanium is a common mate-rial used to make artificialJOints. However, lightweightplastics and materials similar tobone are being developed andtested for strength.

Permanent attachment of ar-tificial joints is usually done byusing special cement, by bio-logic fixation, or with "press-fit" systems. In biologic fixa-tion, a porous material is usedthat allows bone growth intothe artificial part. "Press-fit"bones are made so preciselythat they snap into placearound natural bones. What-ever method is used, the artifi-cial joint must be able to with-stand fairly normal stress loads.

The hip and elbow joints arethe most stressed areas. Theball and socket joint in the hipcarries most of the body weightand is essential for walking.Though the elbow is not aweight-bearing joint, it is thefulcrum of the forearm leverand must endure significant

2.2 If you increase the efficiency of a simple machine, does thea. MA increase, decrease, or remain the same?b. IMA increase, decrease, or remain the same?

2.3 A worker exerts a force of 20 N on a machine with IMA = 2.0,moving it 10 cm.a. Draw a graph of the force as a function of distance. Shade in the

area representing the work done by this force and calculate theamount of work.

b. On the same graph, draw the force supplied by the machine asa function of resistance distance. Shade in the area representingthe work done by the machine. Calculate this work and compareto your answer above.

2.4 Critical Thinking: The mechanical advantage of a multi-gear bike ischanged by moving the chain so that it moves to the correct backsprocket.a. To start out, you must accelerate the bike, so you want to have

the bike exert the largest possible force. Do you choose a smallor large sprocket?

b. As you reach your traveling speed, you want to rotate the pedalsas few times as possible. Do you choose a small or largesprocket?

c. Many bikes also let you choose the size of the front sprocket. Ifyou want even more force to accelerate while climbing a hill,would you move to a larger or smaller front sprocket?

stress. For example, holding a10-N weight in the palm of thehand with the elbow at 90°places a force of about 90 N onthe elbow.. What is another major consid-eration in the design and use ofartificial limb and joint materi-als?

10.2 Machines 211

CHAPTER

10 REVIEW ......... '. . . . . . ... . . . . . . ....

SUMMARY

10.1 Work and Energy

· Work is the product of the force exerted on anobject and the distance the object moves in thedirection of the force.

· Work is the transfer of energy by mechanicalmeans.

· Power is the rate of doing work. That is, poweris the rate at which energy is transferred. It ismeasured in watts.

10.2 Machines

· Machines, whether powered by engines or hu-mans, make work easier. A machine eases theload either by changing the magnitude or the di-rection of the force exerted to do work.

· The mechanical advantage, MA, is the ratio ofresistance force to effort force.

· The ideal mechanical advantage, IMA, is the ra-tio of the displacements. In all real machines,MA is less than IMA.

KEY TERMSworkjouleenergypowerwattmachine

effort forceresistance forcemechanical advantageideal mechanical advantageefficiencycompound machine

REVIEWING CONCEPTS1. In what units is work measured?2. A satellite orbits Earth in a circular orbit. Does

Earth's gravity do any work on the satellite?3. An object slides at constant speed on a fric-

tionless surface. What forces act on the ob-ject? What work is done?


4. Define work and power.5. What is a watt equivalent to in terms of kg, m,

and s?6. Is it possible to get more work out of a ma-

chine than you put in?7. How are the pedals of a bicycle a simple ma-

chine?

APPLYING CONCEPTS1. Which requires more work, carrying a 420-N

knapsack up a 200 m high hill or carrying a210-N knapsack up a 400 m high hill? Why?

2. You slowly lift a box of books from the floorand put it on a table. Earth's gravity exerts aforce, magnitude mg, downward, and you ex-ert a force, magnitude mg, upward. The twoforces have equal magnitude and opposite di-rection. It appears no work is done, but youknow you did work. Explain what work isdone.

3. Guy has to get a piano onto a 2.0 m high plat-form. He can use a 3.0 m long, frictionlessramp or a 4.0 m long, frictionless ramp. Whichramp will Guy use if he wants to do the leastamount of work?

4. Grace has an after-school job carrying car-tons of new copy paper up a flight of stairs,and then carrying used paper back down thestairs. The mass of the paper does notchange. Grace's physics teacher suggeststhat Grace does no work all day, so sheshould not be paid. In what sense is the phys-ics teacher correct? What arrangement ofpayments might Grace make to ensure com-pensation?

FIGURE 10-15.

5. Grace now carries the copy paper boxesdown a level, 15.0 m long hall. Is Grace work-ing now? Explain.

6. Two people of the same mass climb the sameflight of stairs. The first person climbs thestairs in 25 s; the second person takes 35 s.a. Which person does more work? Explain

your answer.b. Which person produces more power? Ex-

plain your answer.7. How can one increase the ideal mechanical

advantage of a machine?8. A claw hammer is used to pull a nail from a

piece of wood. How can you place your handon the handle and locate the nail in the clawto make the effort force as small as possible?

9. How could you increase the mechanical ad-vantage of a wedge without changing theideal mechanical advantage?

PROBLEMS

10.1 Work and Energy1. Lee pushes horizontally with a 80-N force on

a 20-kg mass 10m across a floor. Calculatethe amount of work Lee did.

2. The third floor of a house is 8.0 m abovestreet level. How much work is needed tomove a 150-kg refrigerator to the third floor?

3. Stan does 176 J of work lifting himself0.300 m. What is Stan's mass?

~ 4. A crane lifts a 2.25 x 103_Nbucket containing1.15 m3 of soil (density = 2.00 x 103 kg/m3)

to a height of 7.50 m. Calculate the work thecrane performs.

5. The graph in Figure 10-16 shows the forceneeded to stretch a spring. Find the workneeded to stretch it from 0.12 m to 0.28 m.

8.0

6.0

~ 4.0,f

2.0

o~~~~~~~~--~~~~~0.10 0.20 0.30

Elongation (m)

FIGURE 10-16. Use with Problems 5and 6.

~ 6. In Figure 10-16, the magnitude of the forcenecessary to stretch a spring is plottedagainst the distance the spring is stretched.a. Calculate the slope of the graph and show

thatF = kd,

where k = 25 N/m.b. Find the amount of work done in stretching

the spring from 0.00 m to 0.20 m by cal-culating the area under the curve from0.00 m to 0.20 m in Figure 10-16.

c. Show that the answer to part b can be cal-culated using the formula

W = 1J2kd2,

where W is the work, k = 25 N/m (theslope of the graph), and d is the distancethe spring is stretched (0.20 m).

~ 7. John pushes a crate across the floor of a fac-tory with a horizontal force. The roughness ofthe floor changes, and John must exert aforce of 20 N for 5 m, then 35 N for 12 m,then 10 N for 8 m.a. Draw a graph of force as a function of dis-

tance.b. Find the work John does pushing the crate.

8. Sally applies a horizontal force of 462 N witha rope to drag a wooden crate across a floorwith a constant speed. The rope tied to thecrate is pulled at an angle of 56.0°.a. How much force is exerted by the rope on

the crate?b. What work is done by Sally if the crate is

moved 24.5 m?c. What work is done by the floor through

force of friction between the floor and thecrate?

9. Mike pulls a sled across level snow with aforce of 225 N along a rope that is 35.0°above the horizontal. If the sled moved a dis-tance of 65.3 m, how much work did Mike do?

10. An 845-N sled is pulled a distance of 185 m.The task requires 1.20 x 104 J of work andis done by pulling on a rope with a force of125 N. At what angle is the rope held?

11. Karen has a mass of 57.0 kg and she ridesthe up escalator at Woodley Park Station ofthe Washington D.C. Metro. Karen rode a dis-tance of 65 m, the longest escalator in thefree world. How much work did the escalatordo on Karen if it has an inclination of 30°?

Chapter 10 Review 213

12. Chris carried a carton of milk, weight 10.0 N,along a level hall to the kitchen, a distance of3.50 m. How much work did Chris do?

13. A student librarian picks up a 22-N book fromthe floor to a height of 1.25 m. He carries thebook 8.0 m to the stacks and places the bookon a shelf that is 0.35 m high. How muchwork does he do on the book?

~ 14. Pete slides a crate up a ramp at an angle of30.0° by exerting a 225-N force parallel to theramp. The crate moves at constant speed.The coefficient of friction is 0.28. How muchwork has been done when the crate is raiseda vertical distance of 1.15 m?

~ 15. A 4200-N piano is to be slid up a 3.5-m fric-tionless plank that makes an angle of 30.0°with the horizontal. Calculate the work donein sliding the piano up the plank.

~ 16. A 60-kg crate is slid up an inclined ramp2.0 m long onto a platform 1.0 m above floorlevel. A 400-N force, parallel to the ramp, isneeded to slide the crate up the ramp at aconstant speed.a. How much work is done in sliding the crate

up the ramp?b. How much work would be done if the crate

were simply lifted straight up from the floorto the platform?

17. Brutus, a champion weightlifter, raises 240 kga distance of 2.35 m.a. How much work is done by Brutus lifting

the weights?b. How much work is done holding the

weights above his head?c. How much work is done lowering them

back to the ground?d. Does Brutus do work if the weights are let

go and fall back to the ground?e. If Brutus completes the lift in 2.5 s, how

much power is developed?

18. A force of 300 N is used to push a 145-kgmass 30.0 m horizontally in 3.00 s.a. Calculate the work done on the mass.b. Calculate the power.

19. Robin pushes a wheelbarrow by exerting a145-N force horizontally. Robin moves it60.0 m at a constant speed for 25.0 s.a. What power does Robin develop?b. If Robin moves the wheelbarrow twice as

fast, how much power is developed?


~ 20. Use the graph in Figure 10-17.a. Calculate the work done to pull the object

7.0 m.b. Calculate the power if the work were done

in 2.0 s.

F Force versus Displacement60.0

~ 40.0Ql

~~ 20.0

~~--~~--r-~~r-~~do 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0Displacement (m)

FIGURE 10-17. Use with Problem 20.

~ 21. In 35.0 s, a pump delivers 0.550 drn" of oilinto barrels on a platform 25.0 m above thepump intake pipe. The density of the oil is0.820 q/crn".a. Calculate the work done by the pump.b. Calculate the power produced by the

pump.22. A horizontal force of 805 N is needed to drag

a crate across a horizontal floor with a con-stant speed. Pete drags the crate using arope held at an angle of 32°.a. What force does Pete exert on the rope?b. How much work does Pete do on the crate

when moving it 22 m?c. If Pete completes the job in 8.0 s, what

power is developed?23. Wayne pulls a 305-N sled along a snowy path

using a rope that makes a 45.0° angle withthe ground. Wayne pulls with a force of42.3 N. The sled moves 16 m in 3.0 s. Whatis Wayne's power?

24. A lawn roller is rolled across a lawn by a forceof 115 N along the direction of the handle,which is 22Sabove the horizontal. If Georgedevelops 64.6 W of power for 90.0 s, whatdistance is the roller pushed?

~ 25. A 12.0 m long conveyor belt, inclined at 30.0°,is used to transport bundles of newspapersfrom the mail room up to the cargo bay to beloaded on delivery trucks. Each newspaperhas a mass of 1.00 kg and there are 25 news-papers per bundle. Determine the usefulpower of the conveyor if it delivers 15 bundlesper minute.

~ 26. An engine moves a boat through the water ata constant speed of 15 m/s. The engine mustexert a force of 6.0 x 103 N to balance theforce that water exerts against the hull. Whatpower does the engine develop?

~ 27. A 188-W motor will lift a load at the rate(speed) of 6.50 cm/s. How great a load canthe motor lift at this speed?

28. A car is driven at a constant speed of 21 m/s(76 km/h) down a road. The car's engine de-livers 48 kW of power. Calculate the averageforce of friction that is resisting the motion ofthe car.

10.2 Machines

29. Stan raises a 1000-N piano a distance of5.00 m using a set of pulleys. Stan pulls in20.0 m of rope.a. How much effort force did Stan apply if this

was an ideal machine?b. What force is used to overcome friction if

the actual effort is 300 N?c. What is the work output?d. What is the work input?e. What is the ideal mechanical advantage?

30. A mover's dolly is used to deliver a refrigera-tor up a ramp into a house. The refrigeratorhas a mass of 115 kg. The ramp is 2.10 mlong and rises 0.850 m. The mover pulls thedolly with a force of 496 N up the ramp. Thedolly and ramp constitute a machine.a. What work does the mover do?b. What is the work done on the refrigerator

by the machine?c. What is the efficiency of the machine?

31. A pulley system lifts a 1345-N weight a dis-tance of 0.975 m. Paul pulls the rope a dis-tance of 3.90 m, exerting a force of 375 N.a. What is the ideal mechanical advantage of

the system?b. What is the mechanical advantage?c. How efficient is the system?

~ 32. The ramp in Figure 10-18 is 18 m long and4.5 m high.a. What force parallel to the ramp (F

11) is re-

quired to slide a 25-kg box to the top of theramp if friction is neglected?

b. What is the IMA of the ramp?C. What are the real MA and the efficiency of

the ramp if a parallel force of 75 N is ac-tually required?

FIGURE 10-18. Use with Problem 32.

33. Because there is very little friction, the leveris an extremely efficient simple machine. Us-ing a 90.0% efficient lever, what input work isrequired to lift an 18.0-kg mass through a dis-tance of 0.50 m?

34. What work is required to lift a 215-kg mass adistance of 5.65 m using a machine that is72.5% efficient?

~ 35. A motor having an efficiency of 88% operatesa crane having an efficiency of 42%. Withwhat constant speed does the crane lift a410-kg crate of machine parts if the powersupplied to the motor is 5.5 kW?

~ 36. A complex machine is constructed by attach-ing the lever to the pulley system. Consideran ideal complex machine consisting of a le-ver with an IMA of 3.0 and a pulley systemwith an IMA of 2.0.a. Show that the IMA of this complex ma-

chine is 6.0.b. If the complex machine is 60.0% efficient,

how much effort must be applied to the le-ver to lift a 540-N box?

c. If you move the effort side of the lever12.0 cm, how far is the box lifted?

THINKING PHYSIC-LY

A powerful rifle is one that propels a bullet witha high muzzle velocity. Does this use of theword "powerful" agree with the physics defini-tion of power? Support your answers.

Chapter 10 Review 215

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