Chapter 7

\(\cdot\) A fisherman reels in 12.0 \(\mathrm{m}\) of line while landing a fish, using a constant forward pull of 25.0 \(\mathrm{N}\) . How much work does the tension in the line do on the fish?

A tennis player hits a 58.0 g tennis ball so that it goes straight up and reaches a maximum height of 6.17 m. How much work does gravity do on the ball on the way up? On the way down?

\(\cdot\) A boat with a horizontal tow rope pulls a water skier. She skis off to the side, so the rope makes an angle of \(15.0^{\circ}\) with the forward direction of motion. If the tension in the rope is \(180 \mathrm{N},\) how much work does the rope do on the skier during a forward displacement of 300.0 \(\mathrm{m} ?\)

\(\cdot\) A constant horizontal pull of 8.50 \(\mathrm{N}\) drags a box along a hor- izontal floor through a distance of 17.4 \(\mathrm{m} .\) (a) How much work does the pull do on the box? (b) Suppose that the same pull is exerted at an angle above the horizontal. If this pull now does 65.0 \(\mathrm{J}\) of work on the box while pulling it through the same dis- tance, what angle does the force make with the horizontal?

\(\cdot\) You push your physics book 1.50 \(\mathrm{m}\) along a horizontal tabletop with a horizontal push of 2.40 \(\mathrm{N}\) while the opposing force of friction is 0.600 \(\mathrm{N} .\) How much work does each of the following forces do on the book? (a) your 2.40 \(\mathrm{N}\) push, (b) the friction force, (c) the normal force from the table, and (d) grav- ity? (e) What is the net work done on the book?

\(\cdot\) A 128.0 \(\mathrm{N}\) carton is pulled up a frictionless baggage ramp inclined at \(30.0^{\circ}\) above the horizontal by a rope exerting a 72.0 \(\mathrm{N}\) pull parallel to the ramp's surface. If the carton travels 5.20 \(\mathrm{m}\) along the surface of the ramp, calculate the work done on it by (a) the rope, (b) gravity, and (c) the normal force of the ramp. (d) What is the net work done on the carton? (e) Sup- pose that the rope is angled at \(50.0^{\circ}\) above the horizontal, instead of being parallel to the ramp's surface. How much work does the rope do on the carton in this case?

\(\cdot\) A factory worker moves a 30.0 \(\mathrm{kg}\) crate a distance of 4.5 \(\mathrm{m}\) along a level floor at constant velocity by pushing horizontally on it. The coefficient of kinetic friction between the crate and the floor is 0.25 . (a) What magnitude of force must the worker apply? (b) How much work is done on the crate by the worker's push? (c) How much work is done on the crate by friction? (d) How much work is done by the normal force? By gravity? (e) What is the net work done on the crate?

\(\cdot\) \(\cdot\) An 8.00 kg package in a mail-sorting room slides 2.00 \(\mathrm{m}\) down a chute that is inclined at \(53.0^{\circ}\) below the horizontal. The coefficient of kinetic friction between the package and the chute's surface is 0.40 . Calculate the work done on the package by (a) friction (b) gravity, and (c) the normal force. (d) What is the net work done on the package?

\(\cdot\) \(\cdot\) Two tugboats pull a disabled supertanker. Each tug exerts a constant force of \(1.80 \times 10^{6} \mathrm{N}\) , one \(14^{\circ}\) west of north and the other \(14^{\circ}\) east of north, as they pull the tanker 0.75 \(\mathrm{km}\) toward the north. What is the total work they do on the supertanker?

\(\cdot\) \(\cdot\) A tow truck pulls a car 5.00 \(\mathrm{km}\) along a horizontal road- way using a cable having a tension of 850 \(\mathrm{N}\) (a) How much work does the cable do on the car if it pulls horizontally? If it pulls at \(35.0^{\circ}\) above the horizontal? (b) How much work does the cable do on the tow truck in both cases of part (a)? (c) How much work does gravity do on the car in part (a)?

\(\cdot\) \(\cdot\) A boxed 10.0 kg computer monitor is dragged by friction 5.50 m up along the moving surface of a conveyor belt inclined at an angle of \(36.9^{\circ}\) above the horizontal. If the monitor's speed is a constant \(2.10 \mathrm{cm} / \mathrm{s},\) how much work is done on the monitor by (a) friction, (b) gravity, and (c) the normal force of the conveyor belt?

\(\cdot\) It takes 4.186 \(\mathrm{J}\) of energy to raise the temperature of 1.0 \(\mathrm{g}\) of water by \(1.0^{\circ} \mathrm{C}\) (a) How fast would a 2.0 \(\mathrm{g}\) cricket have to jump to have that much kinetic energy? (b) How fast would a 4.0 \(\mathrm{g}\) cricket have to jump to have the same amount of kinetic energy?

\(\cdot\) A bullet is fired into a large stationary absorber and comes to rest. Temperature measurements of the absorber show that the bullet lost 1960 \(\mathrm{J}\) of kinetic energy, and high-speed photos of the bullet show that it was moving at 965 \(\mathrm{m} / \mathrm{s}\) just as it struck the absorber. What is the mass of the bullet?

\(\cdot\) \(\cdot\) Animal energy. Adult cheetahs, the fastest of the great cats, have a mass of about 70 \(\mathrm{kg}\) and have been clocked at up to 72 mph \((32 \mathrm{m} / \mathrm{s})\) . (a) How many joules of kinetic energy does such a swift cheetah have? (b) By what factor would its kinetic energy change if its speed were doubled?

\(\cdot\) \(\cdot\) A racing dog is initially running at \(10.0 \mathrm{m} / \mathrm{s},\) but is slow- ing down. (a) How fast is the dog moving when its kinetic energy has been reduced by half? (b) By what fraction has its kinetic energy been reduced when its speed has been reduced by half?

\(\cdot\) \(\cdot\) If a running house cat has 10.0 \(\mathrm{J}\) of kinetic energy at speed \(v,\) (a) At what speed (in terms of \(v )\) will she have 20.0 \(\mathrm{J}\) of kinetic energy? (b) What would her kinetic energy be if she ran half as fast as the speed in part (a)?

\(\cdot\) A 0.145 kg baseball leaves a pitcher's hand at a speed of 32.0 \(\mathrm{m} / \mathrm{s} .\) If air drag is negligible, how much work has the pitcher done on the ball by throwing it?

\(\cdot\) \(\cdot\) \(\cdot\) A 1.50 \(\mathrm{kg}\) book is sliding along a rough horizontal sur- face. At point \(A\) it is moving at \(3.21 \mathrm{m} / \mathrm{s},\) and at point \(B\) it has slowed to 1.25 \(\mathrm{m} / \mathrm{s}\) (a) How much work was done on the book between \(A\) and \(B ?\) (b) If \(-0.750 \mathrm{J}\) of work is done on the book from \(B\) to \(C,\) how fast is it moving at point \(C ?\) (c) How fast would it be moving at \(C\) if \(+0.750 \mathrm{J}\) of work were done on it from \(B\) to \(C\) ?

\(\cdot\) \(\cdot\) Stopping distance of a car. The driver of an 1800 \(\mathrm{kg}\) car (including passengers) traveling at 23.0 \(\mathrm{m} / \mathrm{s}\) slams on the brakes, locking the wheels on the dry pavement. The coeffi- cient of kinetic friction between rubber and dry concrete is typically 0.700 . (a) Use the work-energy principle to calculate how far the car will travel before stopping. (b) How far would the car travel if it were going twice as fast? (c) What happened to the car's original kinetic energy?

\(\cdot\) Meteor crater. About \(50,000\) years ago, a meteor crashed into the earth near present-day Flagstaff, Arizona. Recent \((2005)\) measurements estimate that this meteor had a mass of about \(1.4 \times 10^{8} \mathrm{kg}\) (around \(150,000\) tons) and hit the ground at 12 \(\mathrm{km} / \mathrm{s}\) . (a) How much kinetic energy did this meteor deliver to the ground? (b) How does this energy compare to the energy produced in one day by a standard coal-fired power plant, which generates about 1 billion joules per second?

\(\cdot\) \(\cdot\) You throw a 20 \(\mathrm{N}\) rock into the air from ground level and observe that, when it is 15.0 \(\mathrm{m}\) high, it is traveling upward at 25.0 \(\mathrm{m} / \mathrm{s} .\) Use the work-energy principle to find (a) the rock's speed just as it left the ground and (b) the maximum height the rock will reach.

\(\bullet\) A 0.420 kg soccer ball is initially moving at 2.00 \(\mathrm{m} / \mathrm{s}\) . A soccer player kicks the ball, exerting a constant 40.0 \(\mathrm{N}\) force in the same direction as the ball's motion. Over what distance must her foot be in contact with the ball to increase the ball's speed to 6.00 \(\mathrm{m} / \mathrm{s} ?\)

\(\bullet\) A 61 kg skier on level snow coasts 184 m to a stop from a speed of 12.0 \(\mathrm{m} / \mathrm{s}\) . (a) Use the work-energy principle to find the coefficient of kinetic friction between the skis and the snow. (b) Suppose a 75 \(\mathrm{kg}\) skier with twice the starting speed coasted the same distance before stopping. Find the coefficient of kinetic friction between that skier's skis and the snow.

\(\bullet\) \(\bullet\) A block of ice with mass 2.00 kg slides 0.750 m down an inclined plane that slopes downward at an angle of \(36.9^{\circ}\) below the horizontal. If the block of ice starts from rest, what is its final speed? You can ignore friction.

\(\bullet\) To stretch a certain spring by 2.5 \(\mathrm{cm}\) from its equilibrium position requires 8.0 \(\mathrm{J}\) of work. (a) What is the force constant of this spring? (b) What was the maximum force required to stretch it by that distance?

\(\bullet\) \(\bullet\) A spring is 17.0 \(\mathrm{cm}\) long when it is lying on a table. One end is then attached to a hook and the other end is pulled by a force that increases to 25.0 \(\mathrm{N}\) , causing the spring to stretch to a length of 19.2 \(\mathrm{cm} .\) (a) What is the force constant of this spring? (b) How much work was required to stretch the spring from 17.0 \(\mathrm{cm}\) to 19.2 \(\mathrm{cm}\) (c) How long will the spring be if the 25 \(\mathrm{N}\) force is replaced by a 50 \(\mathrm{N}\) force?

\(\bullet\) \(\bullet\) A spring of force constant 300.0 \(\mathrm{N} / \mathrm{m}\) and unstretched length 0.240 \(\mathrm{m}\) is stretched by two forces, pulling in opposite directions at opposite ends of the spring, that increase to 15.0 \(\mathrm{N} .\) How long will the spring now be, and how much work was required to stretch it that distance?

\(\bullet\) \(\bullet\) An unstretched spring has a force constant of 1200 \(\mathrm{N} / \mathrm{m}\) . How large a force and how much work are required to stretch the spring: (a) by 1.0 \(\mathrm{m}\) from its unstretched length, and (b) by 1.0 \(\mathrm{m}\) beyond the length reached in part (a)?

\(\bullet\) How high can we jump? The maximum height a typical human can jump from a crouched start is about 60 \(\mathrm{cm} .\) By how much does the gravitational potential energy increase for a 72 \(\mathrm{kg}\) person in such a jump? Where does this energy come from?

\(\bullet\) A 72.0 -kg swimmer jumps into the old swimming hole from a diving board 3.25 m above the water. Use energy conserva- tion to find his speed just he hits the water (a) if he just holds his nose and drops in, (b) if he bravely jumps straight up (but just beyond the board!) at \(2.50 \mathrm{m} / \mathrm{s},\) and \((\mathrm{c})\) if he manages to jump downward at 2.50 \(\mathrm{m} / \mathrm{s}\) .

\(\bullet\) \(\bullet\) A 2.50 -kg mass is pushed against a horizontal spring of force constant 25.0 \(\mathrm{N} / \mathrm{cm}\) on a frictionless air table. The spring is attached to the tabletop, and the mass is not attached to the spring in any way. When the spring has been compressed enough to store 11.5 \(\mathrm{J}\) of potential energy in it, the mass is sud- denly released from rest. (a) Find the greatest speed the mass reaches. When does this occur? (b) What is the greatest accel- eration of the mass, and when does it occur?

\(\bullet\) A force of magnitude 800.0 \(\mathrm{N}\) stretches a certain spring by 0.200 \(\mathrm{m}\) from its equilibrium position. (a) What is the force constant of this spring? (b) How much elastic potential energy is stored in the spring when it is: (i) stretched 0.300 \(\mathrm{m}\) from its equilibrium position and (ii) compressed by 0.300 \(\mathrm{m}\) from its equilibrium position? (c) How much work was done in stretch- ing the spring by the original 0.200 \(\mathrm{m} ?\)

\(\bullet\) Tendons. Tendons are strong elastic fibers that attach mus- cles to bones. To a reasonable approximation, they obey Hooke's law. In laboratory tests on a particular tendon, it was found that, when a 250 g object was hung from it, the tendon stretched 1.23 \(\mathrm{cm}\) . (a) Find the force constant of this tendon in \(\mathrm{N} / \mathrm{m}\) . (b) Because of its thickness, the maximum tension this tendon can support without rupturing is 138 \(\mathrm{N}\) . By how much can the tendon stretch without rupturing, and how much energy is stored in it at that point?

\(\bullet\) In designing a machine part, you need a spring that is 8.50 \(\mathrm{cm}\) long when no forces act on it and that will store 15.0 \(\mathrm{J}\) of energy when it is compressed by 1.20 \(\mathrm{cm}\) from its equilibrium position. (a) What should be the force constant of this spring? (b) Can the spring store 850 \(\mathrm{J}\) by compression?

\(\bullet\) \(\bullet\) Food calories. The food calorie, equal to \(4186 \mathrm{J},\) is a measure of how much energy is released when food is metabo- lized by the body. A certain brand of fruit-and-cereal bar con- tains 140 food calories per bar. (a) If a 65 \(\mathrm{kg}\) hiker eats one of these bars, how high a mountain must he climb to "work off" the calories, assuming that all the food energy goes only into increasing gravitational potential energy? (b) If, as is typical, only 20\(\%\) of the food calories go into mechanical energy, what would be the answer to part (a)? (Note: In this and all other problems, we are assuming that 100\(\%\) of the food calories that are eaten are absorbed and used by the body. This actually not true. A person's "metabolic efficiency" is the percentage of calories eaten that are actually used; the rest are eliminated by the body. Metabolic efficiency varies considerably from person to person.)

\(\bullet\) \(\bullet\) A good workout. You overindulged on a delicious dessert, so you plan to work off the extra calories at the gym. To accomplish this, you decide to do a series of arm raises hold- ing a 5.0 kg weight in one hand. The distance from your elbow to the weight is \(35 \mathrm{cm},\) and in each arm raise you start with your arm horizontal and pivot it until it is vertical. Assume that the weight of your arm is small enough compared with the weight you are lifting that you can ignore it. As is typical, your muscles are 20\(\%\) efficient in converting the food energy they use up into mechanical energy, with the rest going into heat. If your dessert contained 350 food calories, how many arm raises must you do to work off these calories? Is it realistic to do them all in one session?

\(\bullet\) \(\bullet\) An exercise program. \(A 75\) kg person is put on an exer- cise program by a physical therapist, the goal being to burn up 500 food calories in each daily session. Recall that human muscles are about 20\(\%\) efficient in converting the energy they use up into mechanical energy. The exercise program consists of a set of consecutive high jumps, each one 50 \(\mathrm{cm}\) into the air (which is pretty good for a human) and lasting \(2.0 \mathrm{s},\) on the average. How many jumps should the person do per session, and how much time should be set aside for each session? Do you think that this is a physically reasonable exercise session?

\(\bullet\) Tall Pacific Coast redwood trees (Sequoia sempervirens) can reach heights of about 100 \(\mathrm{m} .\) If air drag is negligibly small, how fast is a sequoia cone moving when it reaches the ground if it dropped from the top of a 100 \(\mathrm{m}\) tree?

\(\bullet\) The total height of Yosemite Falls is 2425 ft. (a) How many more joules of gravitational potential energy are there for each kilogram of water at the top of this waterfall com- pared with each kilogram of water at the foot of the falls? (b) Find the kinetic energy and speed of each kilogram of water as it reaches the base of the waterfall, assuming that there are no losses due to friction with the air or rocks and that the mass of water had negligible vertical speed at the top. How fast (in \(\mathrm{m} / \mathrm{s}\) and mph) would a 70 \(\mathrm{kg}\) person have to run to have that much kinetic energy? (c) How high would Yosemite Falls have to be so that each kilogram of water at the base had twice the kinetic energy you found in part (b); twice the speed you found in part (b)?

\(\bullet\) The speed of hailstones. Although the altitude may vary considerably, hailstones sometimes originate around 500 \(\mathrm{m}\) (about 1500 \(\mathrm{ft} )\) above the ground. (a) Neglecting air drag, how fast will these hailstones be moving when they reach the ground, assuming that they started from rest? Express your answer in \(\mathrm{m} / \mathrm{s}\) and in mph. (b) From your own experience, are hailstones actually falling that fast when they reach the ground? Why not? What has happened to most of the initial potential energy?

\(\bullet\) \(\bullet\) Pebbles of weight \(w\) are launched from the edge of a verti- cal cliff of height \(h\) at speed \(v_{0} .\) How fast (in terms of the quan- tities just given) will these pebbles be moving when they reach the ground if they are launched (a) straight up, (b) straight down, (c) horizontally away from the cliff, and (d) at an angle \(\theta\) above the horizontal? (e) How would the answers to the pre- vious parts change if the pebbles weighed twice as much?

\(\bullet\) Volcanoes on Io. Io, a satellite of Jupiter, is the most volcanically active moon or planet in the solar system. It has volcanoes that send plumes of matter over 500 \(\mathrm{km}\) high (see the accompanying fig- ure). Due to the satellite's small mass, the acceleration due to gravity on Io is only \(1.81 \mathrm{m} / \mathrm{s}^{2},\) and \(\mathrm{Io}\) has no appreciable atmosphere. As- sume that there is no varia- tion in gravity over the distance traveled. (a) What must be the speed of material just as it leaves the volcano to reach an altitude of 500 \(\mathrm{km} ?\) (b) If the gravitational potential energy is zero at the surface, what is the potential energy for a 25 kg fragment at its maximum height on Io? How much would this gravitational potential energy be if it were at the same height above earth?

\(\bullet\) \(\bullet\) Human energy vs. insect energy. For its size, the com- mon flea is one of the most accomplished jumpers in the animal world. A \(2.0-\mathrm{mm}\) -long, 0.50 \(\mathrm{mg}\) critter can reach a height of 20 \(\mathrm{cm}\) in a single leap. (a) Neglecting air drag, what is the take- off speed of such a flea? (b) Calculate the kinetic energy of this flea at takeoff and its kinetic energy per kilogram of mass. (c) If a 65 kg, 2.0 -m-tall human could jump to the same height com- pared with his length as the flea jumps compared with its length, how high could he jump, and what takeoff speed would he need? (d) In fact, most humans can jump no more than 60 \(\mathrm{cm}\) from a crouched start. What is the kinetic energy per kilogram of mass at takeoff for such a 65 kg person? (e) Where does the flea store the energy that allows it to make such a sudden leap?

\(\bullet\) \(\bullet\) \(\mathrm{A} 25 \mathrm{kg}\) child plays on a swing having support ropes that are 2.20 \(\mathrm{m}\) long. A friend pulls her back until the ropes are \(42^{\circ}\) from the vertical and releases her from rest. (a) What is the potential energy for the child just as she is released, compared with the potential energy at the bottom of the swing? (b) How fast will she be moving at the bottom of the swing? (c) How much work does the tension in the ropes do as the child swings from the initial position to the bottom?

\(\bullet\) Tarzan and Jane. Tarzan, in one tree, sights Jane in another tree. He grabs the end of a vine with length 20 \(\mathrm{m}\) that makes an angle of \(45^{\circ}\) with the vertical, steps off his tree limb, and swings down and then up to Jane's open arms. When he arrives, his vine makes an angle of \(30^{\circ}\) with the vertical. Deter- mine whether he gives her a tender embrace or knocks her off her limb by calculating Tarzan's speed just before he reaches Jane. You can ignore air resistance and the mass of the vine.

\(\bullet\) \(\bullet\) A slingshot obeying Hooke's law is used to launch pebbles vertically into the air. You observe that if you pull a pebble back 20.0 \(\mathrm{cm}\) against the elastic band, the pebble goes 6.0 \(\mathrm{m}\) high. (a) Assuming that air drag is negligible, how high will the peb- ble go if you pull it back 40.0 \(\mathrm{cm}\) instead? (b) How far must you pull it back so it will reach 12.0 \(\mathrm{m}\) (c) If you pull a pebble that is twice as heavy back \(20.0 \mathrm{cm},\) how high will it go?

\(\bullet\) \(\bullet\) When a piece of wood is pressed against a spring and com- presses the spring by \(5.0 \mathrm{cm},\) the wood gains a maximum kinetic energy \(K\) when it is released. How much kinetic energy (in terms of \(K\) would the piece of wood gain if the spring were compressed 10.0 \(\mathrm{cm}\) instead?

\(\bullet\) \(\bullet\) A 12.0 N package of whole wheat flour is suddenly placed on the pan of a scale such as you find in grocery stores. The pan is supported from below by a vertical spring of force con- stant 325 \(\mathrm{N} / \mathrm{m}\) . If the pan has negligible weight, find the maxi- mum distance the spring will be compressed if no energy is dissipated by friction.

\(\bullet\) \(\bullet\) A spring of negligible mass has force constant \(k=\) 1600 \(\mathrm{N} / \mathrm{m}\) (a) How far must the spring be compressed for 3.20 \(\mathrm{J}\) of potential energy to be stored in it? (b) You place the spring vertically with one end on the floor. You then drop a \(1.20-\) -kg book onto it from a height of 0.80 m above the top of the spring. Find the maximum distance the spring will be compressed.

\(\bullet\) A fun-loving 11.4 \(\mathrm{kg}\) otter slides up a hill and then back down to the same place. If she starts up at 5.75 \(\mathrm{m} / \mathrm{s}\) and returns at 3.75 \(\mathrm{m} / \mathrm{s}\) , how much mechanical energy did she lose on the hill, and what happened to that energy?