This whimsical piece of art is called an audiokinetic sculp- ture. Balls are raised to a high point on the curved blue track. As the balls move down the track, they turn levers, spin rotors, and bounce off elastic membranes. The energy that each ball has—whether associated with the ball’s motion, the ball’s position above the ground, or the ball’: loss of mechanical energy due to fr tion —varies in a way that keeps the total energy of the system constant. WHAT TO EXPECT In this chapter, you will learn about work and different types of energy that are relevant to mechanics. Kinetic energy, which is associated with motion, and potential energy, which is related to an object's position, are two forms of energy that you will study. WHY IT MATTERS Work, energy, and power are related to one another. Everyday machines such as motors are usually described by the amount of work that they are capable of doing or by the amount of power that they produce. CHAPTER PREVIEW 1 Work Definition of Work 2 Energy Kinetic Energy Potential Energy 3 Conservation of Energy Conserved Quantities Mechanical Energy 4 Power Rate of Energy Transfer 159SECTION OBJECTIVES = Recognize the difference between the scientific and ordinary definitions of work. = Define work by relating it to force and displacement. 1 Identify where work is being performed in a variety of situations. 1 Calculate the net work done when many forces are applied to an object. work the product of the component of 1a force along the direction of dis- placement and the magnitude of the displacement Work 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 work 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 Ax 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 toa displacement Figure 1 ‘When the force on an object and the object’s displacement are in This person exerts a constant force on the car different directions, only the component of the force that is parallel and displaces it to the left. The work done on the ‘ car by the person is equal to the foree the person to the object's displacement does work. Components of the force ‘exerts times the displacement of the ear. 160 Chapter 5 perpendicular to a displacement do not do work.For 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 4, as in Figure 2, work can be expressed as follows: W= Facos@ If @=0°, then cos 0° = 1 and W= Fd, which is the definition of work given earlier. If @= 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. NET WORK DONE BY A CONSTANT NET FORCE Wret= Eyerdtcos 0 network = net force x displacement x cosine of the angle between them Work has dimensions of force times length. In the SI system, work has @ unit of newtons times meters (Nem), 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. W= Fdcos Figure 2 “The work done on this crate ie ‘equal to the force times the dis- placement times the cosine of the angle between them. Did you know? The joule is named for he British physicist James Prescott Joule (1818-1889), joule made major contributions tothe understanding ‘of energy heat and eleewriciy Sa PROBLEM How much work is done on a vacuum cleaner pulled 3.0 m by a force of 50.0 N at an angle of 30.0° above the horizontal? SOLUTION Given: F=50.0N @=300° d=3.0m ‘Unknown: = W=? ‘Use the equation for net work done by a constant force: W=Fdeos@ 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=130J Work and Energy 164Work 1. A tugboat pulls a ship with a constant net horizontal force of 5.00 x 10° 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. Aweight lifter lifts a set of weights a vertical distance of 2.00 m. Ifa 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. 1f 2.0 J of work is done in raising a 180 g apple, how far is it lifted? ‘The sign of works important Workis 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 Module § Work” you exert on the box is positive because that force is upward, in the same direc- provides an interactive lesson tion as the displacement. Work is negative when the force is in the direction with guided problem-colving practice to teach you about calculating net work. Negative (work Positive (+) work tenon Integrating Biology Visit go.hew.com for the activity Muscles and Work Lif Keyword HFEWRKX Figure 3 Depending on the angle, an applied force can either cause a moving car to slow down (left) which resulcs in negative work done on the car, or speed up (right), whieh resules in positive work done on the car. 162 Chapteropposite 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 fal in applying the equation for work, your answer will have the correct sign: Fora vadey of inks relatd 0 this cos Bis negative for angles greater than 90° but less than 270°. chapter goto wwwscilinks.org If the work done on an object results only in a change in the object's speed, Topic: Work the sign of the net work on the object tells you whether the object's speed is Scilinks Code: HFS1674 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 abject slows down and work is done by the object on something else. xz -~=~—~—é—‘“‘“‘;C;C*;~™ 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. Arope exerts a force on a bucket as the bucket is raised up a well €. Air exertsa force on a parachute as the parachutist falls to Earth. 2. Ifa 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 x 10° N crate with a horizontal force of 345 Na 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 erate? €. What is the net work done on the erate? 4, 40.075 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 cach 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. ¢. 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 €. acrane lifting a car Work and Energy 163SECTION OBJECTIVES = Identify several forms of energy. = Calculate kinetic energy for an object. = Apply the work-kinetic ener- gy theorem to solve prob- lems. = Distinguish between ki and potential energy. = Classify different types of potential energy. = Calculate the potential energy associated with an object’ position. Kinetic energy the energy of an object that is due to the object's motion Integrating Health Visit go.hiwucom for the activity Energy Costs of Walking and Running, ff Keyword HFEWRKX ‘The work done on an object by a constant force equals the object mass times its acceleration times its dlsplacement. 164 Chapter 5 Energy 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 frictionless air track under the action of a constant net force, B 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 v toa final velocity ry. If the cart is displaced a distance of Ax, the work done by F during this displacement is Wret= FAx= maAx When you studied one-dimensional motion, you learned that the following relationship holds when an object undergoes constant acceleration: Pah 2adx Kinetic energy depends on speed and mass ‘The quantity } mv” has a special name in physics: kinetic energy. 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 kinetic energy =} mass x (speed)? 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 SS unit for work PHYSICS Kinetic energy depends on both an object's speed and its mass. If a bowling ee ball and a volleyball are traveling at the same speed, which do you think has Module 6 more kinetic energy? You may think that because they are moving with identi- Trane Kinetic Energy cal speeds they have exactly the same kinetic energy. However, the bowling _ provides an interactive lesson ball has mote kinetic energy than the volleyball traveling atthe same speed wit gudedprablem-soving because the bowling ball has more mass than the volleyball Sa Kinetic Energy PROBLEM SOLUTION 47.00 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 bowling ball? Is this speed reasonable for a table-tennis ball in play? Given: ‘The subscripts b and f indicate the bowling ball and the table-tennis ball, respectively Unknown: = 1,=7 7.00kg m4 =245g — vy=3.00 m/s First, calculate the kinetic energy of the bowling ball. KE, = mpv} = (7.00 keg)(3.00 m/s)*=31.5} ‘Then, solve for the speed of the table-tennis ball having the same kinetic energy as the bowling ball KE,= ty} = KEy= 31.5) /2KEp (G15) 2.45 x 107 kg ¥,= 1.60% 10? mis ‘This speed would be very fast for a table-tennis ball. Work and Energy 165Kinetic Energy 1. Calculate the speed of an 8,0 x 10° kg airliner with a kinetic energy of 1.1 1077, 2. What is the speed of a 0.145 kg baseball if its kinetic energy is 109 7? 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. Acarhas a kinetic energy of 4.32 x 10° J when traveling at a speed of 23 mis. What is its mass? ‘work-kinetic energy theorem the net work done by all the forces acting on an object is ‘equal to the change in the object's kinetic energy The net work done on a body equals its change in kinetic energy ‘The equation W,,.,= 5mv}—jmuyderived 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 ‘work-kinetic energy theorem, is often written as follows: WORK-KINETIC ENERGY THEOREM Wret= AKE net work = change in kinetic energy 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 moving hammer has kinetic the ring-the-bell game in Figure 5 has kinetic energy and can therefore do work energy and can do work on the puck, whieh can rise against gravity and ring the bell 166 Chapter ‘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.SAMPLE PROBLEM C Work-Kinetic Energy Theorem SOLUTION 1. DEFINE 3. CALCULATE 4. EVALUATE Ona frozen pond, a person kicks a 10.0 kg sled, giving it an initial speed of 2.2 m/s. How far does the sled move if the coefficient of kinetic friction between the sled and the ice is 0.10? Given: m=100kg vj=22m/s vy=Omis 1=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. w, Inet = Fyepdicos @ ‘The net work done on the sled is provided by the force of kinetic friction. Wher Fydcos 0 = tigmgd cos ‘The force of kinetic friction is in the direction opposite d, so @ = 180°. Because the sled comes to rest, the final kinetic energy is zero. Woet= AKE= KEy - KE; =—}mv? Use the work-kinetie energy theorem, and solve for d. —} mv}=nymgdcos 8 2 2g cos 0 Substitute values into the equation: ~(2.2 mis)* ~ 2(0.10)(9.81 mi/s?)(cos 180" According to Newton's second law, the acceleration of the sled is about 1 m/s? and the time it takes the sled to stop is about 2s. Thus, the distance the sled trav- cled in the given amount of time should be less than the distance it would have traveled in the absence of friction. 2.5m<(2.2m/s)(2s)=4.4m Work and Energy 167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? 2. A2.0x 10° 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 mis. 3. A2.1x 10° 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 x 10° 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. A75kg 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. Tire fod tat you eat provides your body with energy. Your body needs this energy to move your muscles, to maintain a steady inter- nal temperature, and to carry out many other bodily processes, The energy in food is stored as a kind of potential energy in the chemical bonds within sugars and other organic molecules. ‘When you digest food, some of this energy is released. The energy is then stored again in sugar mole- cules, usually as glucose. When cells in your body need energy to carry out cellular processes, the cells break down the glucose molecules through a process called cellular respiration. The primary product of cellular respiration is a high- ‘energy molecule called adenosine triphosphate (ATP), which has asi nificant role in many chemical reac- tions in cells Nutritionists and food scientists use units of Calories to quantify the energy in food. A standard calorie (cal) is defined as the amount of energy required to increase the temperature of 1 mL ‘of water by 1°C, which equals 4.186 joules ()).A food Calorie is actualy 1 kilocalorie, or 4186 J People who are trying to lose ‘weight often monitor the number of Calories that they eat each day. ‘These people count Calories because the body stores unused energy as fat. Most food labels show the number of Calories in each serving of food. The amount of energy that your body needs each day depends on many factors, including your age, your weight, and ‘the amount of exercise that you ‘get A typically healthy and active person requires about 1500 to 2000 Calories 168 ChapterSPOTENTIAL 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 2 form of energy that is, due to the position of an object in relation to other objects or to a reference point. Potential energy 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 gravitational potential energy. 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 ftom the gravita- tional potential energy that is associated with the egg's initial position on the table rclative to the floor. Gravitational potential energy can be determined using the following equation: GRAVITATIONAL POTENTIAL ENERGY PE,= mgh gravitational potential energy =mass x free-fall acceleration x height ‘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 Farth’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. Figure 6 Energy is present in this example, but itis not kinetic energy because there is no motion, What kind of ‘energy ie i potential energy the energy associated with an object because of the position, shape, or condition of the object gravitational potential energy the potential energy stored in the gravitational fields of interacting bodies cha SciLinks Code: HF61196 Work and Energy Fora variety of lnk elated to this goto wuwwsciinks.org Topie: Potential and Kinetic Energy 189A ‘Suppose 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 balls 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. c Gravitational potential energy is a result of an object's position, so it must Figure 7 bbe measured relative to some zero level. The zero level is the vertical coordi- Feel potenthl every nate at which gravitational potential energy is defined to be zero. This zero converted to kinetic energy a8 level is arbitrary, and it is chosen to make a specific problem easier to solve. In ‘the bal fall from A to B. If Cis many cases, the statement of the problem suggests what to use as a zero level. the zero level, then only part of the total gravitational potential crerywomreesiolneic | _Elastic potential energy depends on distance compressed or stretched rey cane 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 sasti iat the tabletop. The kinetic energy of the block came from the stored energy in the clastic potential energy ‘compressed spring. This potential energy is called elastic potential energy. the energy available foruse when | Elastic potential energy is stored in any compressed or stretched object, such as a 2 deformed elastic object returns the stretched strings ofa tennis racket or guitar to its original configuration spring or the: es gu ‘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: ELASTIC POTENTIAL ENERGY PE estc= 3K" 2 distance: carl orstretched as total enrg = srng sont spring constant ‘The symbol kis called the spring constant, or force constant. For a flexible 2 parameter that is a measure spring, the spring constant is small, whereas for a stiff spring, the spring con- eae’ Fesistanee (2 Peing | stants large. Spring constants have units of newtons divided by meters (N/m). }-—_— Distance compressed. ————— vA Figure 8 ' ' “The distance to use inthe equation bw ' for elas potential energy ihe | Compressed lenath of spring ' distance the spring is compressed ' or stretched from its relaxed length }__Relaredtangthof spring. ——————| 170 Chapter 5Cate) Potential Energy SOLUTION DEFINE ‘3. CALCULATE ‘4, EVALUATE A70.0kg stuntman is attached to a bungee cord with an unstretched length of 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 the stuntman 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 the total potential energy relative to the water when the man stops falling? Given: m=70.0kg k=718N/m — g=9.81 m/s* h=50.0 m-44.0m=6.0m x= 44,0 m—15.0m=29.0m PE=0J at river level Unknown: PE gp Diagram: x er oT Relaxed length om Stretched length =44.0m ! L Foe, 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 clastic potential energy. 50.0 m PEt = PEt PEatastic PE,= mgh PE gtastc= 3K” Substitute the values into the equations and solve: PE,= (70.0 kg)(9.81 m/s?)(6.0 m) = 4.1 10° J Choose the zero potential energy location that makes the problem easiest to solve PE gastc= $(71.8 Nim)(29.0 m)? = 3.02 x 104] PE jy¢= 4.1 X 10? J + 3.02 x 10°] PE,y)= 3.43 X10} One way to evaluate the answer is to make an order-of-magnitude estimate. The gravitational potential energy is on the order of 10? kg x 10 m/s? 10 m= 10". ‘The elastic potential energy is on the order of 1 x 10” N/m x 10” m? = 10° J. ‘Thus, the total potential energy should be on the order of 2 x 10* J. This num- ber is close to the actual answer. Work and Energy 171Potential Energy 4. A spring with a force constant of 5.2 N/m has a relaxed length of 2.45 m. ‘When a massis 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. A40.0 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’ lowest position under the following conditions: ‘a, when the ropes are horizontal 'b. when the ropes make a 30.0° angle with the vertical ¢. at the bottom of the circular arc x= @ :«+X—@ °° 14 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 balls kinetic energy in joules? 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 . heating water «c. throwing a football 4. winding the mainspring of a clock 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. 172 Chapter5Conservation of Energy 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 itis 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, itis 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- Jum 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 ithas kinetic energy as well. Elastic potential energy is also present in the many springs that are part of the inner workings of the SECTION OBJECTIVES = Identify situations in which ‘conservation of mechanical ‘energy is valid. = Recognize the forms that conserved energy can take, = Solve problems using conser- vation of mechanical energy. Pred See “The Equivalence of Mass and Energy” in Appendix J: Advanced Topics to learn about Einstein's theory of relativity. ‘Total potential and kinetic energy must be taken ineo account in order to describe the total energy clock. The motion of the pendulum in a clock is shown in Figure 9. of the pendulum ina clock. Work and Energy 173mechanical energy the sum of kinetic energy and all forms of potential energy Energy can be classified in a number of ways. Figure 11 ‘The total mechanical energy, poten- tal energy plus kinetic energy is conserved as the egg falls 174 Chapter5 Analyzing situations involving kinetic, gravitational potential, and clastic potential energy is relatively simple. Unfortunately, analyzing situations involy- 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 wwe 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 mechanical energy. 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+3PE 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. Eneray Mechanical Nonmechanical Kin Gravitational Elastic 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 5, 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.Energy 759 Egg Time Height Speed PE, KE ME @ @™ (rnls) dg) 0) dg) 0.00 10 0.00 074 0.00 074 0.10 095 098 070 0.036 074 0.20 0.80 20 0.59 ous 74 030 056 29 oAl 033 074 0.40 022 39 0.16 0.58 074 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: CONSERVATION OF MECHANICAL ENERGY (ME; = ME, initial mechanical energy = final mechanical energy (in the absence of friction) 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: ? mu}+ mgh;=2mv} + mghy 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. Quick Lab Mechanical Energy MATERIALS LIST medium-sized spring (spring balance) assortment of small balls, each having a different mass ruler tape seale or balance & sarery caution Seudents should wear goggles to perform this Ib, @ jaaabedadyo Firs determine che mass of ‘each of the balls. Then, tape the ruler to the side of a tabletop 20 thatthe ruler is vertical. Place the spring vertically on the tabletop near the ruler and compress the spring by pressing down on one ‘of he balls, Release the balland ‘measure the maximum height i achieves inthe ar. Repeat this process five mes, and be sure to ‘compress the spring by the same amount each time. Average the results From the data can you pre- dict how high each of the other balls will rise? Test your predictions (Hine: Assume mechanical energy is conserved) Work and Energy 175SAMPLE PROBLEM E Conservation of Mechanical Energy PROBLEM Starting from rest, a child zooms down frictionless slide from an initial height of 3.00 m. What is her speed at the bottom of the slide? Assume she has a mass of 25.0 kg. SOLUTION 1.DEFINE Given: h=hj=3.00m m=25.0kg v,=0.0 mis hy=Om Unknown: = ¥j=? 2.PLAN 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, 2 KE=4mv’ PE= igh ‘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. PE,p=0 ‘The initial gravitational potential energy at the top of the slide is PE,;= mgh;= mgh Because the child starts at res, the initial kinetic energy at the top is zero, "Therefore, the final kinetic energy is as follows: KEp= mi} S.CALCULATE Substitute values into the equations: PE,;= (25.0 kg)(9.81 m/s*)(3.00 m) = 736] KE,= (2)(25.0 kg)v} Now use the calculated quantities to evaluate the final velocity ME,= ME, PE;-+ KE,= PE y+ KEy CALCULATOR SOLUTION 3 Yourcalculator should give an answer 736 }+0J=0]+ (0.500)(25.0 kg)v} of 7.67333, but because the answer is limited to three significant figures, y= 7.67 mis it should be roundes to 7.67 176 ChapterEVALUATE ‘The expression for the square of the final speed can be written as follows: 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. Conservation of Mechanical Energy 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. A755 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 mis, 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? Energy conservation occurs even when acceleration varies NTA IF the slope ofthe slid in Sample Problem £ was constant, the aceleration (°/nyacg, 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 For avatey of inks related his know the shape of the slide. Thus, the acceleration may not be constant, and shaper goto wwnscilinks org the kinematic formulas could not be used. Topic: Conservation of Energy But now we can apply a new method to solve such a problem. Because the SciLinks Code: HFe034s 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. Work and Energy 177@ of Figure 12 (a) As the block slides, it kinetic energy tends to decrease because of friction, The force from the hand keeps it moving. (©) Kinetic energy is dissipated into the block and surface Mechanical energy is not conserved in the presence 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.” x= -°+~« ° ° °° |= 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 178 Chapter 5 ahill of height h and then, starting from a momentary rest, wll 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. Is conservation of mechanical energy likely to hold in these situations? ‘a, ahockey puck sliding on a frictionless surface of ice b. a toy car rolling on a carpeted floor ¢. a baseball being thrown into the air 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?Power RATE OF ENERGY TRANSFER ‘The rate at which work is done is called power, 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 Win a time interval Av, then the average power delivered to the object over this time interval is written as follows: POWER ie At power=work time interval 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 w_ od ar ar ‘The distance moved per unit time is just the speed of the object. ceptual Challenge 1. Mountain Reads Many mountain roads are built so that they zigzag up the mountain rather than go straight up toward the peak. Discuss the advan- tages of such a design from the viewpoint of energy conservation and power. 2. Light Bulbs A light bulb is described as having 60 watts. What's wrong with this statement? SECTION OBJECTIVES = Relate the concepts of ‘energy, time, and power. = Calculate power in two different ways. = Explain the effect of machines on work and power. power a quantity that measures the rate at which work Is done or energy Is transformed eden Integrating Chemistry Visit go,hnw.com for the activity “Chemical Reactions. L gif Keyword HEWRKxPOWER (ALTERNATIVE FORM) P= power = force x speed ‘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, Figure 13 “The power of each of these bulbs while a bright bulb can use up to 500 W. Decorative lights use about 0.7 W tslsyouthe rte twhich nergy is each for indoor light and 7.0 W each for outdoor lights, esprboorchiiseneslisher sail In Sample Problem F, the three motors would lift the curtain at different range fom 07 W co 200 W. 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, PT en Power PROBLEM A 193 kg curtain needs to be raised 7.5 m, at constant speed, in as close to 5.0 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? SOLUTION Given: 193kg At Unknown: ‘Use the definition of power, Substitute the equation for work. W_Fa_mad At At At (193 kg) (9.81 m/s*)(7.5 m) 3.08 P=28x 10? W=2.8kW ‘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 180 Chapter SPower 1. A1.0x 10" kg elevator carries a maximum load of 800.0 kg. A constant frictional force of 4.0x 10° 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. Acar with a mass of 1.50 x 10° kg starts from rest and accelerates to a speed of 18.0 mis 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. Arain cloud contains 2.66 x 10” 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 x 10” J of work? 5. A150 10° 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? x=2.| 1. 450.0 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. C1 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? ical Thinking How are energy, time, and power related? Work and Energy 184