Chapter 6

You push your physics book 1.50 \(\mathrm{m}\) along a horizontal table top 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 tabletop, and (d) gravity? (e) What is the net work done on the book?

A tow truck pulls a car 5.00 \(\mathrm{km}\) along a horizontal roadway 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)?

A factory worker pushes a 30.0 -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 this force? (c) How much work is done on the crate by friction? (d) How much work is done on the crate by the normal force? By gravity? (e) What is the total work done on the crate?

A 75.0 -kg painter climbs a ladder that is 2.75 \(\mathrm{m}\) long leaning against a vertical wall. The ladder makes a \(30.0^{\circ}\) angle with the wall. (a) How much work does gravity do on the painter? (b) Does the answer to part (a) depend on whether the painter climbs at constant speed or accelerates up the ladder?

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?

Two blocks are connected by a very light string passing over a massless and frictionless pulley (Fig. E6.7). Traveling at constant speed, the 20.0 -N block moves 75.0 \(\mathrm{cm}\) to the right and the 12.0 -N block moves 75.0 \(\mathrm{cm}\) downward. During this process, how much work is done (a) on the \(12.0-\mathrm{N}\) block by (i) gravity and (ii) the tension in the string? (b) On the 20.0 -N block by (i) gravity, (ii) the tension in the string, (iii) friction, and (iv) the normal force? (c) Find the total work done on each block.

A loaded grocery cart is rolling across a parking lot in a strong wind. You apply a constant force \(\vec{F}=(30 \mathrm{N}) \hat{\imath}-(40 \mathrm{N}) \hat{\mathrm{J}}\) to the cart as it undergoes a displacement \(\vec{s}=(-9.0 \mathrm{m}) \hat{\boldsymbol{\imath}}-(3.0 \mathrm{m}) \hat{\boldsymbol{J}}\) . How much work does the force you apply do on the grocery cart?

\(\mathrm{A} 0.800\) -kg ball is tied to the end of a string 1.60 \(\mathrm{m}\) long and swung in a vertical circle. (a) During one complete circle, starting anywhere, calculate the total work done on the ball by (i) the tension in the string and (ii) gravity. (b) Repeat part (a) for motion along the semicircle from the lowest to the highest point on the path.

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?

A boxed 10.0 -kg computer monitor is dragged by friction 5.50 \(\mathrm{m}\) 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?

You apply a constant force \(\vec{\boldsymbol{F}}=(-68.0 \mathrm{N}) \hat{\boldsymbol{\imath}}+(36.0 \mathrm{N}) \hat{\boldsymbol{j}}\) to a 380 -kg car as the car travels 48.0 \(\mathrm{m}\) in a direction that is \(240.0^{\circ}\) . counterclockwise from the \(+x\) -axis. How much work does the force you apply do on the car?

Animal Energy. BIO Adult cheetahs, the fastest of the great cats, have a mass of about 70 \(\mathrm{kg}\) and have been clocked running at up to 72 \(\mathrm{mph}(32 \mathrm{m} / \mathrm{s})\) . (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?

A 1.50 -kg book is sliding along a rough horizontal surface. 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 \(C\) ?

Meteor Crater. About \(50,000\) years ago, a meteor crashed into the earth near present-day Flagstaff, Arizona. Measurements from 2005 estimate that this meteor had a mass of about \(1.4 \times 10^{8}\) kg (around \(150,000\) tons) and hit the ground at a speed of 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 released by a \(1.0-\) megaton nuclear bomb? (A megaton bomb releases the same amount of energy as a million tons of TNT, and 1.0 ton of TNT releases \(4.184 \times 10^{9}\) J of energy.)

Some Typical Kinetic Energies. (a) In the Bohr model of the atom, the ground- state electron in hydrogen has an orbital speed of 2190 \(\mathrm{km} / \mathrm{s} .\) What is its kinetic energy? (Consult Appendix F.) (b) If you drop a 1.0-kg weight (about 2 lb) from a height of 1.0 \(\mathrm{m}\) , how many joules of kinetic energy will it have when it reaches the ground? (c) Is it reasonable that a \(30-\mathrm{kg}\) child could run fast enough to have 100 \(\mathrm{J}\) of kinetic energy?

A 4.80 -kg watermelon is dropped from rest from the roof of a 25.0 -m-tall building and feels no appreciable air resistance. (a) Calculate the work done by gravity on the watermelon during its displacement from the roof to the ground. (b) Just before it strikes the ground, what is the watermelon's (i) kinetic energy and (ii) speed? (c) Which of the answers in parts (a) and (b) would be different if there were appreciable air resistance?

Use the work-energy theorem to solve each of these problems. You can use Newton's laws to check your answers. Neglect air resistance in all cases. (a) A branch falls from the top of a 95.0 -m-tall redwood tree, starting from rest. How fast is it moving when it reaches the ground? (b) A volcano ejects a boulder directly upward 525 \(\mathrm{m}\) into the air. How fast was the boulder moving just as it left the volcano? (c) A skier moving at 5.00 \(\mathrm{m} / \mathrm{s}\) encounters a long, rough horizontal patch of snow having coefficient of kinetic friction 0.220 with her skis. How far does she travel on this patch before stopping? (d) Suppose the rough patch in part (c) was only 2.90 m long? How fast would the skier be moving when she reached the end of the patch? (e) At the base of a frictionless icy hill that rises at \(25.0^{\circ}\) above the horizontal, a toboggan has a speed of 12.0 \(\mathrm{m} / \mathrm{s}\) toward the hill. How high vertically above the base will it go before stopping?

You throw a 20 -N rock vertically into the air from ground level. You observe that when it is 15.0 \(\mathrm{m}\) above the ground, it is trav- eling at 25.0 \(\mathrm{m} / \mathrm{s}\) upward. Use the work-energy theorem to find (a) the rock's speed just as it left the ground and (b) its maximum height.

You are a member of an Alpine Rescue Team. You must project a box of supplies up an incline of constant slope angle \(\alpha\) so that it reaches a stranded skier who is a vertical distance \(h\) above the bottom of the incline. The incline is slippery, but there is some friction present, with kinetic friction coefficient \(\mu_{\mathrm{k}}.\) Use the work- energy theorem to calculate the minimum speed you must give the box at the bottom of the incline so that it will reach the skier. Express your answer in terms of \(g, h, \mu_{\mathrm{k}},\) and \(\alpha\).

A mass \(m\) slides down a smooth inclined plane from an initial vertical height \(h,\) making an angle \(\alpha\) with the horizontal. (a) The work done by a force is the sum of the work done by the components of the force. Consider the components of gravity parallel and perpendicular to the surface of the plane. Calculate the work done on the mass by each of the components, and use these results to show that the work done by gravity is exactly the same as if the mass had fallen straight down through the air from a height \(h\) . (b) Use the work-energy theorem to prove that the speed of the mass at the bottom of the incline is the same as if it had been dropped from height \(h,\) independent of the angle \(\alpha\) of the incline. Explain how this speed can be independent of the slope angle. (c) Use the results of part (b) to find the speed of a rock that slides down an icy friction- less hill, starting from rest 15.0 m above the bottom.

A sled with mass 8.00 \(\mathrm{kg}\) moves in a straight line on a frictionless horizontal surface. At one point in its path, its speed is \(4.00 \mathrm{m} / \mathrm{s} ;\) after it has traveled 2.50 \(\mathrm{m}\) beyond this point, its speed is 6.00 \(\mathrm{m} / \mathrm{s}\) . Use the work-energy theorem to find the force acting on the sled, assuming that this force is constant and that it acts in the direction of the sled's motion.

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

A 12 -pack of Omni-Cola (mass 4.30 \(\mathrm{kg}\) ) is initially at rest on a horizontal floor. It is then pushed in a straight line for 1.20 \(\mathrm{m}\) by a trained dog that exerts a horizontal force with magnitude 36.0 \(\mathrm{N}\) . Use the work-energy theorem to find the final speed of the 12-pack if (a) there is no friction between the 12 -pack and the floor, and (b) the coefficient of kinetic friction between the 12 -pack and the floor is \(0.30 .\)

A batter hits a baseball with mass 0.145 \(\mathrm{kg}\) straight upward with an initial speed of 25.0 \(\mathrm{m} / \mathrm{s}\) . (a) How much work has gravity done on the baseball when it reaches a height of 20.0 \(\mathrm{m}\) above the bat? (b) Use the work-energy theorem to calculate the speed of the baseball at a height of 20.0 m above the bat. You can ignore air resistance. (c) Does the answer to part (b) depend on whether the baseball is moving upward or downward at a height of 20.0 \(\mathrm{m} ?\) Explain.

A little red wagon with mass 7.00 \(\mathrm{kg}\) moves in a straight line on a frictionless horizontal surface. It has an initial speed of 4.00 \(\mathrm{m} / \mathrm{s}\) and then is pushed 3.0 \(\mathrm{m}\) in the direction of the initial velocity by a force with a magnitude of 10.0 \(\mathrm{N}\) . (a) Use the work-energy theorem to calculare the wagon's final speed. (b) Cal- culate the acceleration produced by the force. Use this acceleration in the kinematic relationships of Chapter 2 to calculate the wagon's final speed. Compare this result to that calculated in part (a).

A block of ice with mass 2.00 \(\mathrm{kg}\) slides 0.750 \(\mathrm{m}\) 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.

Stopping Distance. Acar is traveling on a level road with speed \(v_{0}\) at the instant when the brakes lock, so that the tires slide rather than roll. (a) Use the work-energy theorem to calculate the minimum stopping distance of the car in terms of \(v_{0}, g,\) and the coefficient of kinetic friction \(\mu_{\mathrm{k}}\) between the tires and the road. b) By what factor would the minimum stopping distance change if (i) the coefficient of kinetic friction were doubled, or (ii) the initial speed were doubled, or (iii) both the coefficient of kinetic friction and the initial speed were doubled?

A 30.0 -kg crate is initially moving with a velocity that has magnitude 3.90 \(\mathrm{m} / \mathrm{s}\) in a direction \(37.0^{\circ}\) west of north. How much work must be done on the crate to change its velocity to 5.62 \(\mathrm{m} / \mathrm{s}\) in a direction \(63.0^{\circ}\) south of east?

B10 Heart Repair. A surgeon is using material from a donated heart to repair a patient's damaged aorta and needs to know the elastic characteristics of this aortal material. Tests performed on a 16.0 -cm strip of the donated aorta reveal that it stretches 3.75 \(\mathrm{cm}\) when a \(1.50-\mathrm{N}\) pull is exerted on it. (a) What is the force constant of this strip of aortal material? (b) If the maximum distance it will be able to stretch when it replaces the aorta in the damaged heart is 1.14 \(\mathrm{cm}\) , what is the greatest force it will be able to exert there?

\(\bullet\) \(\bullet\) To stretch a spring 3.00 \(\mathrm{cm}\) from its unstretched length, 12.0 \(\mathrm{J}\) of work must be done. (a) What is the force constant of this spring? (b) What magnitude force is needed to stretch the spring 3.00 \(\mathrm{cm}\) from its unstretched length? (c) How much work must be done to compress this spring 4.00 \(\mathrm{cm}\) from its unstretched length, and what force is needed to compress it this distance?

A 2.0 -kg box and a 3.0 -kg box on a perfectly smooth horizontal floor have a spring of force constant 250 \(\mathrm{N} / \mathrm{m}\) compressed between them. If the initial compression of the spring is \(6.0 \mathrm{cm},\) find the acceleration of each box the instant after they are released. Be sure to include free-body diagrams of each box as part of your solution.

A 6.0-kg box moving at 3.0 \(\mathrm{m} / \mathrm{s}\) on a horizontal, frictionless surface runs into a light spring of force constant 75 \(\mathrm{N} / \mathrm{cm}\) . Use the work-energy theorem to find the maximum compression of the spring.

Leg Presses. As part of your daily workout, you lie on your back and push with your feet against a platform attached to two stiff springs arranged side by side so that they are parallel to each other. When you push the platform, you compress the springs. You do 80.0 J of work when you compress the springs 0.200 \(\mathrm{m}\) from their uncompressed length.(a) What magnitude of force must you apply to hold the platform in this position? (b) How much additional work must you do to move the platform 0.200 \(\mathrm{m}\) farther, and what maximum force must you apply?

A 4.00-kg block of ice is placed against a horizontal spring that has force constant \(k=200 \mathrm{N} / \mathrm{m}\) and is compressed 0.025 \(\mathrm{m}\) . The spring is released and accelerates the block along a horizontal surface. You can ignore friction and the mass of the spring. (a) Calculate the work done on the block by the spring during the motion of the block from its initial position to where the spring has returned to its uncompressed length. (b) What is the speed of the block after it leaves the spring?

At a waterpark, sleds with riders are sent along a slippery, horizontal surface by the release of a large compressed spring. The spring with force constant \(k=40.0 \mathrm{N} / \mathrm{cm}\) and negligible mass rests on the frictionless horizontal surface. One end is in contact with a stationary wall. A sled and rider with total mass 70.0 \(\mathrm{kg}\) are pushed against the other end, compressing the spring 0.375 \(\mathrm{m}\) . The sled is then released with zero initial velocity. What is the sled's speed when the spring (a) returns to its uncompressed length and (b) is still compressed 0.200 \(\mathrm{m} ?\)

Half of a Spring. (a) Suppose you cut a massless ideal spring in half. If the full spring had a force constant \(k\) , what is the force constant of each half, in terms of \(k ?\) (Hint: Think of the original spring as two equal halves, each producing the same force as the entire spring. Do you see why the forces must be equal? (b) If you cut the spring into three equal segments instead, what is the force constant of each one, in terms of \(k ?\)

An ingenious bricklayer builds a device for shooting bricks up to the top of the wall where he is working. He places a brick on a vertical compressed spring with force constant \(k=450 \mathrm{N} / \mathrm{m}\) and negligible mass. When the spring is released, the brick is propelled upward. If the brick has mass 1.80 \(\mathrm{kg}\) and is to reach a maximum height of 3.6 \(\mathrm{m}\) above its initial position on the compressed spring, what distance must the bricklayer compress the spring initially? (The brick loses contact with the spring when the spring returns to its uncompressed length. Why?

CALC A force in the \(+x\) -direction with magnitude \(F(x)=18.0 \mathrm{N}-(0.530 \mathrm{N} / \mathrm{m}) x\) is applied to a 6.00 -kg box that is sitting on the horizontal, frictionless surface of a frozen lake. \(F(x)\) is the only horizontal force on the box. If the box is initially at rest at \(x=0,\) what is its speed after it has traveled 14.0 \(\mathrm{m}\) ?

A crate on a motorized cart starts from rest and moves with a constant eastward acceleration of \(a=2.80 \mathrm{m} / \mathrm{s}^{2}\) . A worker assists the cart by pushing on the crate with a force that is eastward and has magnitude that depends on time according to \(F(t)=\) \((5.40 \mathrm{N} / \mathrm{s}) t .\) What is the instantaneous power supplied by this force at \(t=5.00 \mathrm{s} ?\)

How many joules of energy does a \(100-\) watt light bulb use per hour? How fast would a 70 -kg person have to run to have that amount of kinetic energy?

BIO Should You Walk or Run? It is 5.0 \(\mathrm{km}\) from your home to the physics lab. As part of your physical fitness program, you could run that distance at 10 \(\mathrm{km} / \mathrm{h}\) (which uses up energy at the rate of 700 \(\mathrm{W}\) ), or you could walk it leisurely at 3.0 \(\mathrm{km} / \mathrm{h}\) (which uses energy at 290 \(\mathrm{W}\) W). Which choice would burn up more energy, and how much energy (in joules) would it burn? Why is it that the more intense exercise actually burns up less energy than the less intense exercise?

Magnetar. On December \(27,2004,\) astronomers observed the greatest flash of light ever recorded from outside the solar system. It came from the highly magnetic neutron star SGR \(1806-20\) (a magnetar). During 0.20 \(\mathrm{s}\) , this star released as much energy as our sun does in \(250,000\) years. If \(P\) is the average power output of our sun, what was the average power output (in terms of \(P )\) of this magnetar?

A 20.0 -kg rock is sliding on a rough, horizontal surface at 8.00 \(\mathrm{m} / \mathrm{s}\) and eventually stops due to friction. The coefficient of kinetic friction between the rock and the surface is \(0.200 .\) What average power is produced by friction as the rock stops?

A tandem (two-person) bicycle team must overcome a force of 165 \(\mathrm{N}\) to maintain a speed of 9.00 \(\mathrm{m} / \mathrm{s} .\) Find the power required per rider, assuming that each contributes equally. Express your answer in watts and in horsepower.

When its \(75-\mathrm{kW}(100 \mathrm{-hp})\) engine is generating full power, a small single-engine airplane with mass 700 \(\mathrm{kg}\) gains altitude at a rate of 2.5 \(\mathrm{m} / \mathrm{s}(150 \mathrm{m} / \mathrm{min}\) , or 500 \(\mathrm{ft} / \mathrm{min}\) ). What fraction of the engine power is being used to make the airplane climb? (The remainder is used to overcome the effects of air resistance and of inefficiencies in the propeller and engine.)

Working Like a Horse. Your job is to lift 30 -kg crates a vertical distance of 0.90 m from the ground onto the bed of a truck. (a) How many crates would you have to load onto the truck in 1 minute for the average power output you use to lift the crates to equal 0.50 \(\mathrm{hp} ?\) (b) How many crates for an average power output of 100 \(\mathrm{W} ?\)

An elevator has mass \(600 \mathrm{kg},\) not including passengers. The elevator is designed to ascend, at constant speed, a vertical distance of 20.0 \(\mathrm{m}\) (five floors) in 16.0 \(\mathrm{s}\) , and it is driven by a motor that can provide up to 40 hp to the elevator. What is the maximum number of passengers that can ride in the elevator? Assume that an average passenger has mass 65.0 \(\mathrm{kg}\) .

A ski tow operates on a \(15.0^{\circ}\) slope of length 300 \(\mathrm{m} .\) The rope moves at 12.0 \(\mathrm{km} / \mathrm{h}\) and provides power for 50 riders at one time, with an average mass per rider of 70.0 \(\mathrm{kg} .\) Estimate the power required to operate the tow.

The aircraft carrier John \(F\) . Kennedy has mass \(7.4 \times 10^{7} \mathrm{kg}.\) When its engines are developing their full power of \(280,000\) hp, the John \(F .\) Kennedy travels at its top speed of 35 knots \((65 \mathrm{km} / \mathrm{h}) .\) If 70\(\%\) of the power output of the engines is applied to pushing the ship through the water, what is the magnitude of the force of water resistance that opposes the carrier's motion at this speed?

A typical flying insect applies an average force equal to twice its weight during each downward stroke while hovering. Take the mass of the insect to be \(10 \mathrm{g},\) and assume the wings move an average downward distance of 1.0 \(\mathrm{cm}\) during each stroke. Assuming 100 downward strokes per second, estimate the average power output of the insect.