and Simple Machines Work, Energy, ,... ~ A-Not-So-Simple Machine CHAPTER Chapter Outline {concept Check

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1 CHAPTER 10 Work, Energy, and Simple Machines,... ~ A-Not-So-Simple Machine How does a multispeed bicycle let a rider ride over any kind of terrain with the least effort? Chapter Outline IO.! 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 What is energy? When you have a lot of energy you can run farther or faster; you can jump higher. Objects, as well as people, can have energy. A stone falling off a high ledge has enough energy to damage a car roof. One way to summarize the examples of energy above is to say that an object has energy if it can produce a change in itself or in its surroundings. This may not seem very exact, but there is no more precise definition. In this chapter, we will concentrate on ways of producing changes in objects or their environment. The human race has developed many tools and machines that make it easier to produce such changes. A 10-speed bicycle is a machine that allows the rider to select the speed that makes the bike easiest to ride whether it is going uphill or along a level path. You will discover the physical principles that make this kind of machine work. {concept Check The following terms or concepts from earlier chapters are important for a good understanding of this chapter. If you are not familiar with them, you should review them before studying 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

2 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 energy transferred. differentiate between work and power and correctly calculate power used WORK AND ENERGY Ifyou spend the morning lifting crates from the floor of a warehouse up 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 be transferred into the energy in the raised crates. We use the word work to indicate the amount of energy that was transferred from food to you to the crates. The word work has both an everyday and a scientific meaning. In the case of lifting crates, everyone will agree that work was done. In everyday life, however, we use the word when talking about other activities. For example, everyone says that learning physics is hard work! But in physics, we reserve the term work to mean a very special form of physical activity. FIGURE In physics, work is done only when a force causes an object to move a certain distance. Work is a product of a force and the displacement of an object in the direction of the force. POCKET LAB AN INCLINED MASS Attach a spring scale to a 1.0-kg mass with a string. Keep the string parallel to the table and measure the force needed to pull the mass at a steady speed. Increase the angle between the string and the table top, for example, to 30. Try to keep the angle constant as you pull the 1.0-kg mass along the table at a slow, steady speed. Notice the reading on the scale. How much force is in the direction of motion? How much work is done when the 1.0 kg moves 1.0 m? How does the work compare to the previous value? Work When 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 lifted higher. It seems reasonable to use the quantity force times distance to measure the amount of energy transferred when lifting. For cases where the force is constant, we define work as the product of the force exerted on an object and the distance the object moves in the direction of the force. In equation form, I W = Fd,l where W is the work, F is the magnitude of force, and d is the magnitude of displacement in the direction of the force. Note that work is a scalar quantity; it has no direction. The SI unit of work is the joule [JOOL]. The joule is named after James Prescott Joule, a nineteenth century English physicist and brewer. If a force of one newton moves an object 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 a heavy banner at the same height in the same place for an hour, you may get tired, but you do no work on the banner. Even if you carry the banner at constant velocity and at a constant height, you do no work on it. The force you exert is upward while the motion is sideways, so the banner gains no energy. Work is only done when the force and displacement are in the same direction. 198 Work, Energy, and Simple Machines

3 Force versus Displacement F FIGURE A force-displacement graph of a rock being pushed horizontally. ~ ,r---o Q) ~ ~ 10 o d 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 pushed horizontally. A net force of 20 N is needed to push the rock 1.5 m with constant velocity. The work done on the rock is the product of the force and the displacement, W = Fd = (20 N)(1.5 m) = 30 J. The shaded area under the curve of Figure 10-2 is equal to 20 N x 1.5 m, or 30 J. The area under the curve of a force-displacement graph represents the work done. If you increase either the width of the rectangle (displacement) or the height (force), you increase the work done. Work is done on an object only if it moves in the direction of the force. Example Problem Calculating Work A student Iifts a box of books that weighs 185 N. The box is lifted m. How much work does the student do? Given: mg = 185 N Unknown: work, W d = 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 Problems 1. 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 because the ground became soft. By what amount does the work done by the students change? 2. A delivery clerk carries a 34-N package from the ground to the fifth floor of an office building, a total height of 15 m. How much work is 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 the stairs, then down the hall. Your friend carries it down the hall, then up another stairwell. Who does more work? Direction of applied force Direction of motion FIGURE Work is done on the box only when it moves in the direction of the applied force Work and Energy 199

4 r, F«= F cos 25.0 = (125 N) (0.906) F h =113N a b FIGURE If a force is applied to the mower at an angle, the net force doing the work is the component that acts in the direction of the motion. Only the component in the direction of the motion does work. POCKET LAB WORKING OUT Attach a spring balance to a 1.0-kg mass with a string. Pull the mass along the table at a slow, steady speed keeping the balance parallelto the table top. Notice the reading on the spring balance. What are the physical factors that determine the amount of force? How much work is done in movingthe mass 1.0 m? Predictthe force and the work when a 2.0-kg mass is pulled along the table. Try it. Was your predictionaccurate? Work and Direction of Force Work 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 exerted perpendicular to the motion, no work is done. What if a force is exerted at some other angle to the motion? For example, if you push the lawn mower in Figure 10-4a, what work do you do? You know that any force can be replaced by its components. The 125-N force (F) you exert in the direction of the handle has two components, Figure 10-4b. The horizontal component (F h ) is 113 N; the vertical component (Fv) is - 53 N (downward). The vertical component is perpendicular to the motion. 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 is equal to the component of the force in the direction of the motion times the distance moved. The magnitude of the component of the force F acting in the direction of motion is found by multiplying the force F by the cosine of the angle between 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 objects do work? Earth's gravity acts downward, and the ground exerts the normal 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 horizontal component of the applied force is balanced by the force of friction, Ffriction' The angle between the force of friction and the direction of motion 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. The force is exerted in one direction, while the motion was in the opposite direction. Negative work indicates that work is being done on the grass by the mower. The positive sign of the work done by you exerting the force on the handle means you are doing work. 200 Work, Energy, and Simple Machines

5 What is the effect of doing work? When you lift a box of books onto a shelf, you give the box a certain property. If the box falls, it can do work; it might exert forces that crush another object. If the box is on a cart and you push on it, you will start it moving. Again, the box could exert forces that crush another object. In this case too, you have given the box energy-the ability to produce a change in itself or its surroundings. By doing work on the box, you have transferred energy from your body to the box. Thus, we say that work is the transfer of energy by mechanical means. You can think of work as energy transferred as the result 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. F. Y. I. In physics, work and energy have precise meanings, which must not be confused with their everyday meanings. Robert Oppenheimer wrote, "Often the very fact that the words of science are the same as those of our common life and tongue can be more misleading than enlightening." PROBLEM SOLVING STRATEGY When doing work problems, you should: 1. Carefully check the forces acting on the object. Draw a diagram indicating 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 it is positive. Example Work-Force Problem 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 How much work is done by the sailor if he exerts a force of 255 N on the rope and pulls the boat 30.0 m? Given: (J = 60.0 F = 255 N d = 30.0 m Solution: Unknown: work, W Basic equation: W = Fd cos (J 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 10 3 J FIGURE The force exerted by the sailor must exceed the force needed to move the boat because only the horizontal force changes the velocity Work and Energy 201

6 85N --_~L 4m FIGURE Use with Practice Problem 8. 3m Practice Problems 5. 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 displacement 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 is used. How much work does the force on the rope do? ~ 8. A worker pushes a crate weighing 93 N up an inclined plane, pushing horizontally, parallel to the ground, Figure 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 = How much work is done by friction? (Be careful of signs.) Power is the rate at which work is done. Power Until 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 the same whether the box is lifted in 2 seconds or if each book is lifted separately, so that it takes 20 minutes to put them all on the shelf. The work done is the same, but the power is different. Power is the rate of doing 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 calculated using FIGURE The same amount of work is being done as in Figure 10-4, but the power is greater here. Power is measured in watts (W). One watt is one joule of energy transferred in one second. A machine that does work at a rate of one joule per second has a power of one watt. A watt is a relatively small unit of power. For example, a glass of water weighs about 2 N. If you lift it 0.5 meter to your mouth, you do 1 joule of work. If you do it in one second, you are doing work at the rate of one watt. Becausea watt is such a small unit, power is often measured in kilowatts (kw). A kilowatt is 1000 watts. Example Problem Power An electric motor lifts an elevator that weighs 1.20 x 10 4 N a distance of 9.00 m in 15.0 s, Figure 10-8, a. What is the power of the motor in watts? b. What is the power in kilowatts? Given: F = 1.20 x 10 4 N Unknown: power, P d = 9.00 m.. W t = 15.0 s BaSICequation: P = t 202 Work, Energy, and Simple Machines

7 Solution: a. power of a motor, p = W = Fd = (1.20 x 10 4 N)(9.00 m) t t X 10 3 N. m/s = 7.20 x 10 3 J/s 7.20 X 10 3 W b. P = 7.20 X 10 3 W(,~o~WW) = 7.20 kw Practice Problems 9. 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 in watts and kilowatts? 10. A rock climber wears a 7.50-kg knapsack while scaling a cliff. After 30.0 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 elevator 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 only 35 kw of power at this speed. The other car needs its motor to produce 65 kw to move the car this fast. Forces exerted by friction from 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 directions? 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? FIGURE If the elevator took 20.0 s instead of 15.0 s, would the work change? Would the power change? A watt, one joule/second, is a relatively small unit. A kilowatt, 1000 watts, is commonly used. CONCEPT REVIEW 1.1 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 Work and Energy

8 Your Power Purpose To determine the work and power as you climb stairs. Materials. Each lab group will need a meter stick, a stopwatch, and a climber. Procedure 1. 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. Do NOT trip. 4. The timer will start the watch as the climber hits the first stair and stop the clock when the climber reaches the top. 5. Climbers and timers should rotate until all students have had the opportunity to climb. Observations and Dat 1. Calculate the work and the power for yourself. 2. Compare your work and power calculations to others. 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. Applications 1. Your local electric company supplies you 1 kw of power for 1 h for 8rt. Assume that you could climb stairs continuously for 1 h. How much money would this climb be worth? Th Work, Energy, and Simple Machines

9 10.2 MACHINES Everyone uses some machines every day. Some are simple tools like bottle openers and screwdrivers; others are complex objects such as bicycles and automobiles. Machines, whether powered by engines or people, make our tasks easier. A machine eases the load either by changing the magnitude or the direction of a force, but does not change the amount of work done. Simple and Complex Machines Consider the bottle opener in Figure 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. The work 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 bottle opener. 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 the cap cannot receive more energy than you put into the opener. Thus, the output work can never be larger than the input work. The machine simply aids in the transfer of energy from you to the bottle cap. Objectives demonstrate knowledge of why simple machines are useful. understand the concept of mechanical advantage in ideal and real machines and use it correctly in solving problems. recognize that complex machines are simple machines linked together; show an ability to calculate mechanical advantage for some complex machines. Energy Conservation and Mechanical Advantage The 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 ratio of resistance force to effort force, F/Fe, is called the mechanical advantage (MA) of the machine. That is, r, MA Fe Many machines, like the bottle opener, have a mechanical advantage greater 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 the definition of work. The input work is the product of the effort force you exert, Fe, and the displacement of your hand, de. In the same way, the output work is the product of the resistance force, F" and the displacement caused by the machine, d.. A machine can increase force, but it cannot increase energy. An ideal machine transfers all the energy, so the output work equals the input work, Wo = Wi, or Frdr = Fede. This equation can be rewritten Fr = dd e. We know that the mechanical Fe r advantage is given by MA = F/F e. For an ideal machine, we also have MA = de/dr. Because this equation is characteristic of an ideal machine, FIGURE A bottle opener is an example of a simple machine. It makes opening a bottle easier, but not less work Machines 205

10 a c FIGURE The simple machines pictured are the lever (a); pulley (b); wheel-and-axle (c); inclined plane (d); wedge (e); and screw (f). The mechanical advantage of a machine is the ratio of the force exerted by the machine to the force applied to the machine. the mechanical advantage is called the ideal mechanical advantage, IMA, de IMA = -d' r Note that you measure distances moved to calculate the ideal mechanical advantage, IMA, but you measure the forces exerted to find the actual mechanical advantage, MA. In a real machine, not all of the input work is available as output work. The efficiency of a machine is defined as the ratio of output work to input work. That is,,~ , efficiency Wo x 100%. Wi Efficiency is the ratio of the work done.by the machine to the work put into the machine. An ideal machine has equal output and input work, WjW i = 1, and the efficiency is 100%. All real machines have efficiencies less than 100%. We can express the efficiency in terms of the mechanical advantage and ideal mechanical advantage. F,.IFe efficiency = -d d x 100% e I r MA efficiency = --- x 100% IMA The IMA of most machines is fixed by the machine's design. An efficient machine has an MA almost equal to its IMA. A less efficient machine has a smaller MA. Lower efficiency means that a greater effort force is needed to exert the same resistance force. 206 Work, Energy, and Simple Machines

11 All machines, no matter how complex, are combinations of up to six simple machines shown in Figure They are the lever, pulley, wheel-and-axle, inclined plane, wedge, and screw. Gears, one of the simple machines used in a bicycle, are really a form of the wheel-andaxle. The IMA of all machines is the ratio of distances moved. Figure shows that for levers and wheel-and-axles this ratio can be replaced by the ratio of the distances between the places where the forces are applied and the pivot point. A common version of the wheel-andaxle is a pair of gears on a rotating shaft. The IMA is the ratio of the radii of the two gears. Compound Machines A compound machine consists of two or more simple machines linked so that the resistance force of one machine becomes the effort force 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 exert on the pedal, Fon pedal. The resistance is the force the sprocket exerts on the 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 rear wheel act like another wheel-and-axle. The resistance force is the force the wheel exerts on the road, Fon road. By Newton's third law, the ground exerts an equal forward force on the wheel. This force accelerates the bicycle forward. The mechanical advantage of a complex machine is the product of the mechanical advantages of the simple machines it is made up of. For example, in the case of the bicycle, MA = Fon chain Fon pedal Fon road Fon pedal' Fon road Fby chain CHIROPRACTOR Do you like to work on "the human machine?" A group practice in a small city desires a new partner to expand services. A belief in the holistic (non-drug, non-surgical) approach to medicine and a positive, professional demeanor are required for one to be a successful chiropractor. Completion of an accredited, 6- year Chiropractic College program with a Doctor of Chiropractic (D.C.) degree and a state license are re-. mat' t: Figure For levers and wheel and axles, the IMA is '!! r, r-r,--! I r; --~I I I I pivot point a 10.2 Machines 207

12 F. Y. I. A bicyclist in a road race like the Tour de France rides at about 20 milh (9 m/s) for over six hours a day. The power output of the racer is about one kilowatt. One-quarter of that power goes into moving the bike against the resistance of the air, gears, and tires. Three-quarters of the power is used to cool the racer's body. To generate this power, the racer eats food calories each day of the 23-day race. The IMA of each wheel-and-axle machine is the ratio of the distances moved. For the pedal sprocket, IMA For the rear wheel, we have For the bicycle, IMA then IMA pedal radius front sprocket radius = --' rear sprocket radius wheel radius = ----'----- pedal radius front sprocket radius rear sprocket radius front sprocket radius = ---' rear sprocket radius wheel radius pedal radius wheel radius Since both sprockets use the same chain and have teeth of the same size, you can simply count the number of teeth on the gears and find that IMA teeth on rear sprocket pedal arm length teeth on front sprocket wheel radius. = -----'----- On a multi-gear bike, the rider can change the mechanical advantage of the machine by choosing the size of one or both sprockets. When accelerating or climbing a hill, the rider increases the mechanical advantage to increase the force the wheel exerts on the road. On the other hand, when going at high speed on a level road, less force is needed, and the rider decreases the mechanical advantage to reduce the distance 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, no matter how much force downward you exert, the pedals will not turn. The force of your foot is most effective when the force is exerted perpendicular to the arm of the pedal. Whenever a force on a pedal is specified, you should assume that it is applied perpendicular to the arm. FIGURE Some of the most ingenious compound machines ever conceived came from the imagination of cartoonist Rube Goldberg, who was famous for devising complex ways for doing simple tasks. 208 Work, Energy, and Simple Machines

13 ;;;;,---~ Example Problem The Bicycle Wheel FIGURE A series of simple machines combine to transmit the force the rider exerts on the pedal to the rear wheel of the bike. A student uses the bicycle wheel with gear radius 4.00 cm and wheel radius 35.6 cm. When a force of 155 N is exerted on the chain, the wheel rim moves 14.0 cm. Due to friction, its efficiency is 95.0%. a. What is the IMA of the wheel and gear? b. What is the MA of the wheel and gear? c. What force does the scale attached to the 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. MA wheel radius = 35.6 cm c. resistance force, F, efficiency = 95.0% d. effort displacement, de resistance displacement, d, = 14.0 cm Solution: de gear radius a.ima = -d = r wheel radius 4.00 cm 35.6 cm = MA b. Since efficiency = IMA x 100% IMA MA = eff % (95.0%)(0.112) F, c. MA = - Fe 100% = F, = (MA)(Fe) de d.ima = d, = (0.107)(155 N) de = (lma)(d,) = (0.112)(14.0 cm) 16.6 N 1.57 cm 10.2 Machines 209

14 Practice Problems 13. 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 separated a distance of 5.0 cm. A force of 1.9 x 10 4 N is needed to split the log, and the sledge exerts a force of 9.8 x 10 3 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. A force 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 N rock 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 force exerted on the chain and the distance the wheel rim moves remain the same, what quantities change, and by how much? BIOLOGY 1irrIrrrrrrr...,.. CONNECTION The Human Walking Machine Movement of the human body is explained by the same principles of force and work that describe all motion. Simple machines, in the form of levers, give us the ability to walk and run. Lever systems of the body are 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 (movable joints between bones), and 4) a resistance (the weight of the body or an object being lifted or moved), Figure Lever systems of the body are not very efficient, and mechanical advantages are low. This is why walking and jogging require energy (burn calories) and help individuals lose weight. When a person walks, the hip acts as a fulcrum and moves through the arc of a circle centered on the foot. The center of mass of the body moves as a resistance around the fulcrum in the same arc. The length of the radius of the circle is the length of the lever formed by the bones of the leg. Athletes in walking races increase their velocity by swinging their hips upward to increase this radius. A tall person has a lever with a lower MA than a short person. Although tall people can usually walk faster than short people, a tall person must apply a greater force to move the longer lever formed by the leg bones. Walking races are usually 20 or 50 km long. Because of the inefficiency of their lever systems and the length of a walking race, very tall people rarely have the stamina to win. FIGURE The human walking machine. CONCEPT REVIEW 2.1 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 210 Work, Energy, and Simple Machines

15 P hysicsand technology ARTIFICIAL JOINTS Great progress has been made in designing and substituting artificial joints for damaged or missing joints. Because of the tremendous stresses in the bones and joints of the arms and legs, the materials from which artificial parts are made and the means of attachment must be extremely strong. Titanium is a common material used to make artificial JOints. However, lightweight plastics and materials similar to bone are being developed and tested for strength. Permanent attachment of artificial joints is usually done by using special cement, by biologic fixation, or with "pressfit" systems. In biologic fixation, a porous material is used that allows bone growth into the artificial part. "Press-fit" bones are made so precisely that they snap into place around natural bones. Whatever method is used, the artificial joint must be able to withstand fairly normal stress loads. The hip and elbow joints are the most stressed areas. The ball and socket joint in the hip carries most of the body weight and is essential for walking. Though the elbow is not a weight-bearing joint, it is the fulcrum of the forearm lever and must endure significant stress. For example, holding a 10-N weight in the palm of the hand with the elbow at 90 places a force of about 90 N on the elbow.. What is another major consideration in the design and use of artificial limb and joint materials? 2.2 If you increase the efficiency of a simple machine, does the a. 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 the amount of work. b. On the same graph, draw the force supplied by the machine as a function of resistance distance. Shade in the area representing the work done by the machine. Calculate this work and compare to your answer above. 2.4 Critical Thinking: The mechanical advantage of a multi-gear bike is changed by moving the chain so that it moves to the correct back sprocket. 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 small or large sprocket? b. As you reach your traveling speed, you want to rotate the pedals as few times as possible. Do you choose a small or large sprocket? c. Many bikes also let you choose the size of the front sprocket. If you want even more force to accelerate while climbing a hill, would you move to a larger or smaller front sprocket? 10.2 Machines 211

16 CHAPTER SUMMARY 10.1 Work and Energy 10 REVIEW Work is the product of the force exerted on an object and the distance the object moves in the direction of the force. Work is the transfer of energy by mechanical means. Power is the rate of doing work. That is, power is the rate at which energy is transferred. It is measured in watts Machines Machines, whether powered by engines or humans, make work easier. A machine eases the load either by changing the magnitude or the direction of the force exerted to do work. The mechanical advantage, MA, is the ratio of resistance force to effort force. The ideal mechanical advantage, IMA, is the ratio of the displacements. In all real machines, MA is less than IMA. KEY TERMS work joule energy power watt machine REVIEWING effort force resistance force mechanical advantage ideal mechanical advantage efficiency compound machine CONCEPTS 1. 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 frictionless surface. What forces act on the object? What work is done?... ' 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 machine than you put in? 7. How are the pedals of a bicycle a simple machine? APPLYING CONCEPTS 1. Which requires more work, carrying a 420-N knapsack up a 200 m high hill or carrying a 210-N knapsack up a 400 m high hill? Why? 2. You slowly lift a box of books from the floor and put it on a table. Earth's gravity exerts a force, magnitude mg, downward, and you exert a force, magnitude mg, upward. The two forces have equal magnitude and opposite direction. It appears no work is done, but you know you did work. Explain what work is done. 3. Guy has to get a piano onto a 2.0 m high platform. He can use a 3.0 m long, frictionless ramp or a 4.0 m long, frictionless ramp. Which ramp will Guy use if he wants to do the least amount of work? 4. Grace has an after-school job carrying cartons of new copy paper up a flight of stairs, and then carrying used paper back down the stairs. The mass of the paper does not change. Grace's physics teacher suggests that Grace does no work all day, so she should not be paid. In what sense is the physics teacher correct? What arrangement of payments might Grace make to ensure compensation? FIGURE Work, Energy, and Simple Machines

17 5. Grace now carries the copy paper boxes down a level, 15.0 m long hall. Is Grace working now? Explain. 6. Two people of the same mass climb the same flight of stairs. The first person climbs the stairs in 25 s; the second person takes 35 s. a. Which person does more work? Explain your answer. b. Which person produces more power? Explain 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 hand on the handle and locate the nail in the claw to make the effort force as small as possible? 9. How could you increase the mechanical advantage of a wedge without changing the ideal mechanical advantage? PROBLEMS 10.1 Work and Energy 1. Lee pushes horizontally with a 80-N force on a 20-kg mass 10m across a floor. Calculate the amount of work Lee did. 2. The third floor of a house is 8.0 m above street level. How much work is needed to move a 150-kg refrigerator to the third floor? 3. Stan does 176 J of work lifting himself m. What is Stan's mass? ~ 4. A crane lifts a 2.25 x 103_N bucket containing 1.15 m 3 of soil (density = 2.00 x 10 3 kg/m 3 ) to a height of 7.50 m. Calculate the work the crane performs. 5. The graph in Figure shows the force needed to stretch a spring. Find the work needed to stretch it from 0.12 m to 0.28 m ~ 4.0,f 2.0 o~~~~~~~~--~~~~~ Elongation FIGURE Use with Problems 5 and 6. (m) ~ 6. In Figure 10-16, the magnitude of the force necessary to stretch a spring is plotted against the distance the spring is stretched. a. Calculate the slope of the graph and show that F = 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 calculating the area under the curve from 0.00 m to 0.20 m in Figure c. Show that the answer to part b can be calculated using the formula W = 1J2kd 2, where W is the work, k = 25 N/m (the slope of the graph), and d is the distance the spring is stretched (0.20 m). ~ 7. John pushes a crate across the floor of a factory with a horizontal force. The roughness of the floor changes, and John must exert a force 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 distance. b. Find the work John does pushing the crate. 8. Sally applies a horizontal force of 462 N with a rope to drag a wooden crate across a floor with a constant speed. The rope tied to the crate is pulled at an angle of 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 the crate? 9. Mike pulls a sled across level snow with a force of 225 N along a rope that is 35.0 above the horizontal. If the sled moved a distance 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 10 4 J of work and is done by pulling on a rope with a force of 125 N. At what angle is the rope held? 11. Karen has a mass of 57.0 kg and she rides the up escalator at Woodley Park Station of the Washington D.C. Metro. Karen rode a distance of 65 m, the longest escalator in the free world. How much work did the escalator do on Karen if it has an inclination of 30? Chapter 10 Review 213

18 12. Chris carried a carton of milk, weight 10.0 N, along a level hall to the kitchen, a distance of 3.50 m. How much work did Chris do? 13. A student librarian picks up a 22-N book from the floor to a height of 1.25 m. He carries the book 8.0 m to the stacks and places the book on a shelf that is 0.35 m high. How much work does he do on the book? ~ 14. Pete slides a crate up a ramp at an angle of 30.0 by exerting a 225-N force parallel to the ramp. The crate moves at constant speed. The coefficient of friction is How much work has been done when the crate is raised a vertical distance of 1.15 m? ~ 15. A 4200-N piano is to be slid up a 3.5-m frictionless plank that makes an angle of 30.0 with the horizontal. Calculate the work done in sliding the piano up the plank. ~ 16. A 60-kg crate is slid up an inclined ramp 2.0 m long onto a platform 1.0 m above floor level. A 400-N force, parallel to the ramp, is needed to slide the crate up the ramp at a constant 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 floor to the platform? 17. Brutus, a champion weightlifter, raises 240 kg a 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-kg mass 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 a 145-N force horizontally. Robin moves it 60.0 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 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 Displacement 60.0 ~ 40.0 Ql ~ ~ 20.0 ~~--~~--r-~~r-~~d o Displacement (m) FIGURE Use with Problem 20. ~ 21. In 35.0 s, a pump delivers drn" of oil into barrels on a platform 25.0 m above the pump intake pipe. The density of the oil is 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 constant speed. Pete drags the crate using a rope 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 with the ground. Wayne pulls with a force of 42.3 N. The sled moves 16 m in 3.0 s. What is Wayne's power? 24. A lawn roller is rolled across a lawn by a force of 115 N along the direction of the handle, which is 22Sabove the horizontal. If George develops 64.6 W of power for 90.0 s, what distance is the roller pushed? ~ 25. A 12.0 m long conveyor belt, inclined at 30.0, is used to transport bundles of newspapers from the mail room up to the cargo bay to be loaded on delivery trucks. Each newspaper has a mass of 1.00 kg and there are 25 newspapers per bundle. Determine the useful power of the conveyor if it delivers 15 bundles per minute. 214 Work, Energy, and Simple Machines

19 ~ 26. An engine moves a boat through the water at a constant speed of 15 m/s. The engine must exert a force of 6.0 x 10 3 N to balance the force that water exerts against the hull. What power 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 can the 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 delivers 48 kw of power. Calculate the average force of friction that is resisting the motion of the car Machines ) 29. Stan raises a 1000-N piano a distance of 5.00 m using a set of pulleys. Stan pulls in 20.0 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 refrigerator up a ramp into a house. The refrigerator has a mass of 115 kg. The ramp is 2.10 m long and rises m. The mover pulls the dolly with a force of 496 N up the ramp. The dolly 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 distance of m. Paul pulls the rope a distance 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 is 18 m long and 4.5 m high. a. What force parallel to the ramp (F 11 is required to slide a 25-kg box to the top of the ramp 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 actually required? FIGURE Use with Problem Because there is very little friction, the lever is an extremely efficient simple machine. Using a 90.0% efficient lever, what input work is required to lift an 18.0-kg mass through a distance of 0.50 m? 34. What work is required to lift a 215-kg mass a distance of 5.65 m using a machine that is 72.5% efficient? ~ 35. A motor having an efficiency of 88% operates a crane having an efficiency of 42%. With what constant speed does the crane lift a 410-kg crate of machine parts if the power supplied to the motor is 5.5 kw? ~ 36. A complex machine is constructed by attaching the lever to the pulley system. Consider an ideal complex machine consisting of a lever with an IMA of 3.0 and a pulley system with an IMA of 2.0. a. Show that the IMA of this complex machine is 6.0. b. If the complex machine is 60.0% efficient, how much effort must be applied to the lever to lift a 540-N box? c. If you move the effort side of the lever 12.0 cm, how far is the box lifted? THINKING PHYSIC-LY A powerful rifle is one that propels a bullet with a high muzzle velocity. Does this use of the word "powerful" agree with the physics definition of power? Support your answers. Chapter 10 Review 215