Answer: Automotive Power I. A truck engine transmits 28.0

You push your physics book 1.50 m along a horizontal tabletop with a horizontal push of 2.40 N while the opposing force of friction is 0.600 N. How much work does each of the following forces do on the book: (a) your 2.40-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 km along a horizontal roadway using a cable having a tension of 850 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 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?

Suppose the worker in Exercise 6.3 pushes downward at an angle of below the horizontal. (a) What magnitude of force must the worker apply to move the crate at constant velocity? (b) How much work is done on the crate by this force when the crate is pushed a distance of 4.5 m? (c) How much work is done on the crate by friction during this displacement? (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 m long leaning against a vertical wall. The ladder makes a 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 one west of north and the other east of north, as they pull the tanker 0.75 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 cm to the right and the 12.0-N block moves 75.0 cm downward. During this process, how much work is done (a) on the 12.0-N block by (i) gravity and (ii) the tension in the string? (b) On the 20.0-N block by (i) gravity 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 to the cart as it undergoes a displacement How much work does the force you apply do on the grocery cart? s 1-9.0 m2n ? 13.0 m2n

A 0.800-kg ball is tied to the end of a string 1.60 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 m down a chute that is inclined at 53.0 below the horizontal. The coefficient of kinetic friction between the package and the chutes 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 m up along the moving surface of a conveyor belt inclined at an angle of 36.9 above the horizontal. If the monitors speed is a constant 2.10 cm 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 to a 380-kg car as the car travels 48.0 m in a direction that is 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 kg and have been clocked running at up to 72 mph (32 m/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 m/s, and at point B it has slowed to 1.25 m/s. (a) How much work was done on the book between A and B? (b) If -0.750 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 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 kg (around 150,000 tons) and hit the ground at a speed of (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 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 km/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 m, how many joules of kinetic energy will it have when it reaches the ground? (c) Is it reasonable that a 30-kg child could run fast enough to have 100 J of kinetic energy?

In Fig. E6.7 assume that there is no friction force on the 20.0-N block that sits on the tabletop. The pulley is light and frictionless. (a) Calculate the tension T in the light string that connects the blocks. (b) For a displacement in which the 12.0-N block descends 1.20 m, calculate the total work done on (i) the 20.0-N block and (ii) the 12.0-N block. (c) For the displacement in part (b), calculate the total work done on the system of the two blocks. How does your answer compare to the work done on the 12.0-N block by gravity? (d) If the system is released from rest, what is the speed of the 12.0-N block when it has descended 1.20 m?

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 watermelons (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 workenergy theorem to solve each of these problems. You can use Newtons 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 m into the air. How fast was the boulder moving just as it left the volcano? (c) A skier moving at 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 above the horizontal, a toboggan has a speed of 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 m above the ground, it is traveling at upward. Use the workenergy theorem to find (a) the rocks 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 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 Use the workenergy 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, and a

A mass m slides down a smooth inclined plane from an initial vertical height h, making an angle 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 workenergy 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 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 frictionless hill, starting from rest 15.0 m above the bottom

A sled with mass 8.00 kg moves in a straight line on a frictionless horizontal surface. At one point in its path, its speed is after it has traveled 2.50 m beyond this point, its speed is Use the workenergy theorem to find the force acting on the sled, assuming that this force is constant and that it acts in the direction of the sleds motion

A soccer ball with mass 0.420 kg is initially moving with speed A soccer player kicks the ball, exerting a constant force of magnitude 40.0 N in the same direction as the balls motion. Over what distance must the players foot be in contact with the ball to increase the balls speed to

A 12-pack of Omni-Cola (mass 4.30 kg) is initially at rest on a horizontal floor. It is then pushed in a straight line for 1.20 m by a trained dog that exerts a horizontal force with magnitude 36.0 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 kg straight upward with an initial speed of (a) How much work has gravity done on the baseball when it reaches a height of 20.0 m above the bat? (b) Use the workenergy 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 m? Explain

A little red wagon with mass 7.00 kg moves in a straight line on a frictionless horizontal surface. It has an initial speed of and then is pushed 3.0 m in the direction of the initial velocity by a force with a magnitude of 10.0 N. (a) Use the workenergy theorem to calculate the wagons final speed. (b) Calculate the acceleration produced by the force. Use this acceleration in the kinematic relationships of Chapter 2 to calculate the wagons final speed. Compare this result to that calculated in part (a).

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.

Stopping Distance. A car is traveling on a level road with speed at the instant when the brakes lock, so that the tires slide rather than roll. (a) Use the workenergy theorem to calculate the minimum stopping distance of the car in terms of g, and the coefficient of kinetic friction 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 m s in a direction west of north. How much work must be done on the crate to change its velocity to 5.62 m s in a direction south of east?

Heart Repair. A surgeon is using material from a donated heart to repair a patients 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 cm when a 1.50-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 cm, what is the greatest force it will be able to exert there?

To stretch a spring 3.00 cm from its unstretched length, 12.0 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 cm from its unstretched length? (c) How much work must be done to compress this spring 4.00 cm from its unstretched length, and what force is needed to compress it this distance?

Three identical 6.40-kg masses are hung by three identical springs, as shown in Fig. E6.33. Each spring has a force constant of and was 12.0 cm long before any masses were attached to it. (a) Draw a free-body diagram of each mass. (b) How long is each spring when hanging as shown? (Hint: First isolate only the bottom mass. Then treat the bottom two masses as a system. Finally, treat all three masses as a system.)

A child applies a force parallel to the x-axis to a 10.0-kg sled moving on the frozen surface of a small pond. As the child controls the speed of the sled, the x-component of the force she applies varies with the x-coordinate of the sled as shown in Fig. E6.34. Calculate the work done by the force when the sled moves (a) from to (b) from x = 8.0 m to (c) from to 12.0 x = 12.0 m; x = 0 m.

Suppose the sled in Exercise 6.34 is initially at rest at Use the workenergy theorem to find the speed of the sled at (a) and (b) You can ignore friction between the sled and the surface of the pond

A 2.0-kg box and a 3.0-kg box on a perfectly smooth horizontal floor have a spring of force constant compressed between them. If the initial compression of the spring is 6.0 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 on a horizontal, frictionless surface runs into a light spring of force constant Use the workenergy 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 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 m farther, and what maximum force must you apply?

(a) In Example 6.7 (Section 6.3) it was calculated that with the air track turned off, the glider travels 8.6 cm before it stops instantaneously. How large would the coefficient of static friction have to be to keep the glider from springing back to the left? (b) If the coefficient of static friction between the glider and the track is what is the maximum initial speed that the glider can be given and still remain at rest after it stops instantaneously? With the air track turned off, the coefficient of kinetic friction is mk = 0.47.

A 4.00-kg block of ice is placed against a horizontal spring that has force constant and is compressed 0.025 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?

A force is applied to a 2.0-kg radio-controlled model car parallel to the x-axis as it moves along a straight track. The x-component of the force varies with the x-coordinate of the car as shown in Fig. E6.41. Calculate the work done by the force when the car moves from (a) to (b) to (c) to (d) to (e) to x = 7.0 m x = 2.0 m

Suppose the 2.0-kg model car in Exercise 6.41 is initially at rest at x = 0 and \(\vec{F}\) is the net force acting on it. Use the work–energy theorem to find the speed of the car at (a) x = 3.0 m; (b) x = 4.0 m; (c) x = 7.0 m.

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 N/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 kg are pushed against the other end, compressing the spring 0.375 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 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?

A small glider is placed against a compressed spring at the bottom of an air track that slopes upward at an angle of \(40.0^{\circ}\) above the horizontal. The glider has mass 0.0900 kg. The spring has k = 640 N/m and negligible mass. When the spring is released, the glider travels a maximum distance of 1.80 m along the air track before sliding back down. Before reaching this maximum distance, the glider loses contact with the spring. (a) What distance was the spring originally compressed? (b) When the glider has traveled along the air track 0.80 m from its initial position against the compressed spring, is it still in contact with the spring? What is the kinetic energy of the glider at this point?

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 and negligible mass. When the spring is released, the brick is propelled upward. If the brick has mass 1.80 kg and is to reach a maximum height of 3.6 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 is applied to a 6.00-kg box that is sitting on the horizontal, frictionless surface of a frozen lake. is the only horizontal force on the box. If the box is initially at rest at , what is its speed after it has traveled 14.0 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 N/s)t. What is the instantaneous power supplied by this force at t = 5.00 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?

Should You Walk or Run? It is 5.0 km from your home to the physics lab. As part of your physical fitness program, you could run that distance at 10 km h (which uses up energy at the rate of 700 W), or you could walk it leisurely at 3.0 km h (which uses energy at 290 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 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 m/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 N to maintain a speed of Find the power required per rider, assuming that each contributes equally. Express your answer in watts and in horsepower

When its 75-kW (100-hp) engine is generating full power, a small single-engine airplane with mass 700 kg gains altitude at a rate of 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 hp? (b) How many crates for an average power output of 100 W?

An elevator has mass 600 kg, not including passengers. The elevator is designed to ascend, at constant speed, a vertical distance of 20.0 m (five floors) in 16.0 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 kg.

A ski tow operates on a slope of length 300 m. The rope moves at and provides power for 50 riders at one time, with an average mass per rider of 70.0 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 km/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 g, and assume the wings move an average downward distance of 1.0 cm during each stroke. Assuming 100 downward strokes per second, estimate the average power output of the insect

A balky cow is leaving the barn as you try harder and harder to push her back in. In coordinates with the origin at the barn door, the cow walks from x = 0 to x = 6.9 m as you apply a force with x-component \(F_{x}=-[20.0 \mathrm{~N}+(3.0 \mathrm{~N} / \mathrm{m}) x]\). How much work does the force you apply do on the cow during this displacement?

Rotating Bar. A thin, uniform 12.0-kg bar that is 2.00 m long rotates uniformly about a pivot at one end, making 5.00 complete revolutions every 3.00 seconds. What is the kinetic energy of this bar? (Hint: Different points in the bar have different speeds. Break the bar up into infinitesimal segments of mass dm and integrate to add up the kinetic energies of all these segments.)

A Near-Earth Asteroid. On April 13, 2029 (Friday the 13th!), the asteroid 99942 Apophis will pass within 18,600 mi of the earthabout the distance to the moon! It has a density of can be modeled as a sphere 320 m in diameter, and will be traveling at (a) If, due to a small disturbance in its orbit, the asteroid were to hit the earth, how much kinetic energy would it deliver? (b) The largest nuclear bomb ever tested by the United States was the Castle/Bravo bomb, having a yield of 15 megatons of TNT. (A megaton of TNT releases of energy.) How many Castle/Bravo bombs would be equivalent to the energy of Apophis?

A luggage handler pulls a 20.0-kg suitcase up a ramp inclined at above the horizontal by a force of magnitude 140 N that acts parallel to the ramp. The coefficient of kinetic friction between the ramp and the incline is If the suitcase travels 3.80 m along the ramp, calculate (a) the work done on the suitcase by the force (b) the work done on the suitcase by the gravitational force; (c) the work done on the suitcase by the normal force; (d) the work done on the suitcase by the friction force; (e) the total work done on the suitcase. (f) If the speed of the suitcase is zero at the bottom of the ramp, what is its speed after it has traveled 3.80 m along the ramp?

Chin-Ups. While doing a chin-up, a man lifts his body 0.40 m. (a) How much work must the man do per kilogram of body mass? (b) The muscles involved in doing a chin-up can generate about 70 J of work per kilogram of muscle mass. If the man can just barely do a 0.40-m chin-up, what percentage of his body's mass do these muscles constitute? (For comparison, the total percentage of muscle in a typical 70-kg man with 14% body fat is about 43%.) (c) Repeat part (b) for the man's young son, who has arms half as long as his fathers but whose muscles can also generate 70 J of work per kilogram of muscle mass. (d) Adults and children have about the same percentage of muscle in their bodies. Explain why children can commonly do chin-ups more easily than their fathers.

A 20.0-kg crate sits at rest at the bottom of a 15.0-m-long ramp that is inclined at \(34.0^{\circ}\) above the horizontal. A constant horizontal force of 290 N is applied to the crate to push it up the ramp. While the crate is moving, the ramp exerts a constant frictional force on it that has magnitude 65.0 N. (a) What is the total work done on the crate during its motion from the bottom to the top of the ramp? (b) How much time does it take the crate to travel to the top of the ramp?

Consider the blocks in Exercise 6.7 as they move 75.0 cm. Find the total work done on each one (a) if there is no friction between the table and the 20.0-N block, and (b) if and between the table and the 20.0-N block.

The space shuttle, with mass 86,400 kg, is in a circular orbit of radius around the earth. It takes 90.1 min for the shuttle to complete each orbit. On a repair mission, the shuttle is cautiously moving 1.00 m closer to a disabled satellite every 3.00 s. Calculate the shuttles kinetic energy (a) relative to the earth and (b) relative to the satellite.

A 5.00-kg package slides 1.50 m down a long ramp that is inclined at below the horizontal. The coefficient of kinetic friction between the package and the ramp is \(\mu_{\mathrm{k}}=0.310\). Calculate (a) the work done on the package by friction; (b) the work done on the package by gravity; (c) the work done on the package by the normal force; (d) the total work done on the package. (e) If the package has a speed of 2.20 m/s at the top of the ramp, what is its speed after sliding 1.50 m down the ramp?

Whiplash Injuries. When a car is hit from behind, its passengers undergo sudden forward acceleration, which can cause a severe neck injury known as whiplash. During normal acceleration, the neck muscles play a large role in accelerating the head so that the bones are not injured. But during a very sudden acceleration, the muscles do not react immediately because they are flexible, so most of the accelerating force is provided by the neck bones. Experimental tests have shown that these bones will fracture if they absorb more than 8.0 J of energy. (a) If a car waiting at a stoplight is rear-ended in a collision that lasts for 10.0 ms, what is the greatest speed this car and its driver can reach without breaking neck bones if the drivers head has a mass of 5.0 kg (which is about right for a 70-kg person)? Express your answer in m s and in mph. (b) What is the acceleration of the passengers during the collision in part (a), and how large a force is acting to accelerate their heads? Express the acceleration in m s2 and in gs

A net force along the x-axis that has x-component \(F_x=-12.0\mathrm{\ N}+ \left(0.300 \mathrm {\ \ N}/ \mathrm{m}^2 \right)x^2\) is applied to a 5.00-kg object that is initially at the origin and moving in the -x-direction with a speed of 6.00 m/s. What is the speed of the object when it reaches the point x = 5.00 m?

An object is attracted toward the origin with a force given by (Gravitational and electrical forces have this distance dependence.) (a) Calculate the work done by the force when the object moves in the x- direction from to If is the work done by positive or negative? (b) The only other force acting on the object is a force that you exert with your hand to move the object slowly from to How much work do you do? If is the work you do positive or negative? (c) Explain the similarities and differences between your answers to parts (a) and (b).

The gravitational pull of the earth on an object is inversely proportional to the square of the distance of the object from the center of the earth. At the earths surface this force is equal to the objects normal weight mg, where and at large distances, the force is zero. If a 20,000-kg asteroid falls to earth from a very great distance away, what will be its minimum speed as it strikes the earths surface, and how much kinetic energy will it impart to our planet? You can ignore the effects of the earths atmosphere.

A box is sliding with a speed of 4.50 m/s on a horizontal surface when, at point P, it encounters a rough section. On the rough section, the coefficient of friction is not constant, but starts at 0.100 at P and increases linearly with distance past P, reaching a value of 0.600 at 12.5 m past point P. (a) Use the work–energy theorem to find how far this box slides before stopping. (b) What is the coefficient of friction at the stopping point? (c) How far would the box have slid if the friction coefficient didn’t increase but instead had the constant value of 0.100?

Consider a spring that does not obey Hookes law very faithfully. One end of the spring is fixed. To keep the spring stretched or compressed an amount x, a force along the x-axis with x-component must be applied to the free end. Here and Note that when the spring is stretched and when it is compressed. (a) How much work must be done to stretch this spring by 0.050 m from its unstretched length? (b) How much work must be done to compress this spring by 0.050 m from its unstretched length? (c) Is it easier to stretch or compress this spring? Explain why in terms of the dependence of on x. (Many real springs behave qualitatively in the same way.)

A small block with a mass of 0.0900 kg is attached to a cord passing through a hole in a frictionless, horizontal surface (Fig. P6.75). The block is originally revolving at a distance of 0.40 m from the hole with a speed of The cord is then pulled from below, shortening the radius of the circle in which the block revolves to 0.10 m. At this new distance, the speed of the block is observed to be (a) What is the tension in the cord in the original situation when the block has speed (b) What is the tension in the cord in the final situation when the block has speed (c) How much work was done by the person who pulled on the cord?

Proton Bombardment. A proton with mass is propelled at an initial speed of directly toward a uranium nucleus 5.00 m away. The proton is repelled by the uranium nucleus with a force of magnitude where x is the separation between the two objects and Assume that the uranium nucleus remains at rest. (a) What is the speed of the proton when it is from the uranium nucleus? (b) As the proton approaches the uranium nucleus, the repulsive force slows down the proton until it comes momentarily to rest, after which the proton moves away from the uranium nucleus. How close to the uranium nucleus does the proton get? (c) What is the speed of the proton when it is again 5.00 m away from the uranium nucleus?

A block of ice with mass 4.00 kg is initially at rest on a frictionless, horizontal surface. A worker then applies a horizontal force \(\overrightarrow{\boldsymbol{F}}\) to it. As a result, the block moves along the x-axis such that its position as a function of time is given by \(x(t)=\alpha t^{2}+\beta t^{3}\), where \(\alpha=0.200 \mathrm{~m} / \mathrm{s}^{2} \text { and } \beta=0.0200 \mathrm{~m} / \mathrm{s}^{3}\). (a) Calculate the velocity of the object when t = 4.00 s. (b) Calculate the magnitude of when \(\overrightarrow{\boldsymbol{F}}\) when t = 4.00 s. (c) Calculate the work done by the force \(\overrightarrow{\boldsymbol{F}}\) during the first 4.00 s of the motion.

You and your bicycle have combined mass 80.0 kg. When you reach the base of a bridge, you are traveling along the road at (Fig. P6.78). At the top of the bridge, you have climbed a vertical distance of 5.20 m and have slowed to You can ignore work done by friction and any inefficiency in the bike or your legs. (a) What is the total work done on you and your bicycle when you go from the base to the top of the bridge? (b) How much work have you done with the force you apply to the pedals?

You are asked to design spring bumpers for the walls of a parking garage. A freely rolling 1200-kg car moving at 0.65 m/s is to compress the spring no more than 0.090 m before stopping. What should be the force constant of the spring? Assume that the spring has negligible mass.

The spring of a spring gun has force constant and negligible mass. The spring is compressed 6.00 cm, and a ball with mass 0.0300 kg is placed in the horizontal barrel against the compressed spring. The spring is then released, and the ball is propelled out the barrel of the gun. The barrel is 6.00 cm long, so the ball leaves the barrel at the same point that it loses contact with the spring. The gun is held so the barrel is horizontal. (a) Calculate the speed with which the ball leaves the barrel if you can ignore friction. (b) Calculate the speed of the ball as it leaves the barrel if a constant resisting force of 6.00 N acts on the ball as it moves along the barrel. (c) For the situation in part (b), at what position along the barrel does the ball have the greatest speed, and what is that speed? (In this case, the maximum speed does not occur at the end of the barrel.)

A 2.50-kg textbook is forced against a horizontal spring of negligible mass and force constant compressing the spring a distance of 0.250 m. When released, the textbook slides on a horizontal tabletop with coefficient of kinetic friction m 0.30. k = Use the workenergy theorem to find how far the textbook moves from its initial position before coming to rest

Pushing a Cat. Your cat Ms. (mass 7.00 kg) is trying to make it to the top of a frictionless ramp 2.00 m long and inclined upward at above the horizontal. Since the poor cat cant get any traction on the ramp, you push her up the entire length of the ramp by exerting a constant 100-N force parallel to the ramp. If Ms. takes a running start so that she is moving at at the bottom of the ramp, what is her speed when she reaches the top of the incline? Use the workenergy theorem.

Crash Barrier. A student proposes a design for an automobile crash barrier in which a 1700-kg sport utility vehicle moving at crashes into a spring of negligible mass that slows it to a stop. So that the passengers are not injured, the acceleration of the vehicle as it slows can be no greater than 5.00g. (a) Find the required spring constant k, and find the distance the spring will compress in slowing the vehicle to a stop. In your calculation, disregard any deformation or crumpling of the vehicle and the friction between the vehicle and the ground. (b) What disadvantages are there to this design?

A physics professor is pushed up a ramp inclined upward at \(30.0^{\circ}\) above the horizontal as he sits in his desk chair that slides on frictionless rollers. The combined mass of the professor and chair is 85.0 kg. He is pushed 2.50 m along the incline by a group of students who together exert a constant horizontal force of 600 N. The professor’s speed at the bottom of the ramp is 2.00 m/s. Use the work-energy theorem to find his speed at the top of the ramp.

A 5.00-kg block is moving at \(v_0=6.00\mathrm{\ m}/\mathrm{s}\) along a frictionless, horizontal surface toward a spring with force constant k = 500 N/m that is attached to a wall (Fig. P6.85). The spring has negligible mass. (a) Find the maximum distance the spring will be compressed. (b) If the spring is to compress by no more than 0.150 m, what should be the maximum value of \(v_{0}\)?

Consider the system shown in Fig. P6.86. The rope and pulley have negligible mass, and the pulley is frictionless. The coefficient of kinetic friction between the 8.00-kg block and the tabletop is The blocks are released from rest. Use energy methods to calculate the speed of the 6.00-kg block after it has descended 1.50 m

Consider the system shown in Fig. P6.86. The rope and pulley have negligible mass, and the pulley is frictionless. Initially the 6.00-kg block is moving downward and the 8.00-kg block is moving to the right, both with a speed of The blocks come to rest after moving 2.00 m. Use the workenergy theorem to calculate the coefficient of kinetic friction between the 8.00-kg block and the tabletop.

CALC Bow and Arrow. Figure P6.88 shows how the force exerted by the string of a compound bow on an arrow varies as a function of how far back the arrow is pulled (the draw length). Assume that the same force is exerted on the arrow as it moves forward after being released. Full draw for this bow is at a draw length of 75.0 cm. If the bow shoots a 0.0250-kg arrow from full draw, what is the speed of the arrow as it leaves the bow?

On an essentially frictionless, horizontal ice rink, a skater moving at encounters a rough patch that reduces her speed to 1.65 m s due to a friction force that is 25% of her weight. Use the workenergy theorem to find the length of this rough patch.

Rescue. Your friend (mass 65.0 kg) is standing on the ice in the middle of a frozen pond. There is very little friction between her feet and the ice, so she is unable to walk. Fortunately, a light rope is tied around her waist and you stand on the bank holding the other end. You pull on the rope for 3.00 s and accelerate your friend from rest to a speed of while you remain at rest. What is the average power supplied by the force you applied?

A pump is required to lift 800 kg of water (about 210 gallons) per minute from a well 14.0 m deep and eject it with a speed of (a) How much work is done per minute in lifting the water? (b) How much work is done in giving the water the kinetic energy it has when ejected? (c) What must be the power output of the pump?

BIO All birds, independent of their size, must maintain a power output of 1025 watts per kilogram of body mass in order to fly by flapping their wings. (a) The Andean giant hummingbird (Patagona gigas) has mass 70 g and flaps its wings 10 times per second while hovering. Estimate the amount of work done by such a hummingbird in each wingbeat. (b) A 70-kg athlete can maintain a power output of 1.4 kW for no more than a few seconds; the steady power output of a typical athlete is only 500 W or so. Is it possible for a human-powered aircraft to fly for extended periods by flapping its wings? Explain.

A physics student spends part of her day walking between classes or for recreation, during which time she expends energy at an average rate of 280 W. The remainder of the day she is sitting in class, studying, or resting; during these activities, she expends energy at an average rate of 100 W. If she expends a total of \(1.1\times10^7\mathrm{\ J}\) of energy in a 24-hour day, how much of the day did she spend walking?

The Grand Coulee Dam is 1270 m long and 170 m high. The electrical power output from generators at its base is approximately 2000 MW. How many cubic meters of water must flow from the top of the dam per second to produce this amount of power if 92% of the work done on the water by gravity is converted to electrical energy? (Each cubic meter of water has a mass of 1000 kg.)

Power of the Human Heart. The human heart is a powerful and extremely reliable pump. Each day it takes in and discharges about 7500 L of blood. Assume that the work done by the heart is equal to the work required to lift this amount of blood a height equal to that of the average American woman (1.63 m). The density (mass per unit volume) of blood is \(1.05\times10^3\mathrm{\ kg}/\mathrm{m}^3\). (a) How much work does the heart do in a day? (b) What is the heart's power output in watts?

Six diesel units in series can provide 13.4 MW of power to the lead car of a freight train. The diesel units have total mass \(1.10\times 10^6\mathrm{\ kg}}\). The average car in the train has mass \(8.2\times 10^4\mathrm{\ kg}\) and requires a horizontal pull of 2.8 kN to move at a constant 27 m/s on level tracks. (a) How many cars can be in the train under these conditions? (b) This would leave no power for accelerating or climbing hills. Show that the extra force needed to accelerate the train is about the same for a \(0.10-\mathrm{m} / \mathrm{s}^{2}\) acceleration or a 1.0% slope (slope angle \(\alpha=\arctan 0.010\)). (c) With the 1.0% slope, show that an extra 2.9 MW of power is needed to maintain the speed of the diesel units. (d) With 2.9 MW less power available, how many cars can the six diesel units pull up a 1.0% slope at a constant 27 m/s?

It takes a force of 53 kN on the lead car of a 16-car passenger train with mass to pull it at a constant on level tracks. (a) What power must the locomotive provide to the lead car? (b) How much more power to the lead car than calculated in part (a) would be needed to give the train an acceleration of at the instant that the train has a speed of on level tracks? (c) How much more power to the lead car than that calculated in part (a) would be needed to move the train up a 1.5% grade (slope angle ) at a constant

An object has several forces acting on it. One of these forces is \(\vec{\boldsymbol{F}}=a x y \hat{\imath}\), a force in the x-direction whose magnitude depends on the position of the object, with \(\alpha=2.50\mathrm{\ N}/\mathrm{m}^2\). Calculate the work done on the object by this force for the following displacements of the object: (a) The object starts at the point x = 0, y = 3.00 m and moves parallel to the x-axis to the point x = 2.00 m, y = 3.00 m. (b) The object starts at the point x = 2.00 m, y = 0 and moves in the y-direction to the point x = 2.00 m, y = 3.00 m. (c) The object starts at the origin and moves on the line y = 1.5x to the point x = 2.00 m, y = 3.00 m.

Cycling. For a touring bicyclist the drag coefficient is 1.00, the frontal area A is and the coefficient of rolling friction is 0.0045. The rider has mass 50.0 kg, and her bike has mass 12.0 kg. (a) To maintain a speed of on a level road, what must the riders power output to the rear wheel be? (b) For racing, the same rider uses a different bike with coefficient of rolling friction 0.0030 and mass 9.00 kg. She also crouches down, reducing her drag coeffi- cient to 0.88 and reducing her frontal area to What must her power output to the rear wheel be then to maintain a speed of (c) For the situation in part (b), what power output is required to maintain a speed of Note the great drop in power requirement when the speed is only halved. (For more on aerodynamic speed limitations for a wide variety of human- powered vehicles, see The Aerodynamics of Human-Powered Land Vehicles, Scientific American, December 1983.)

Automotive Power I. A truck engine transmits 28.0 kW (37.5 hp) to the driving wheels when the truck is traveling at a constant velocity of magnitude on a 60.0 km>h 137.3 mi>h2 level road. (a) What is the resisting force acting on the truck? (b) Assume that 65% of the resisting force is due to rolling friction and the remainder is due to air resistance. If the force of rolling friction is independent of speed, and the force of air resistance is proportional to the square of the speed, what power will drive the truck at At Give your answers in kilowatts and in horsepower.

Automotive Power II. (a) If 8.00 hp are required to drive a 1800-kg automobile at on a level road, what is the total retarding force due to friction, air resistance, and so on? (b) What power is necessary to drive the car at up a 10.0% grade (a hill rising 10.0 m vertically in 100.0 m horizontally)? (c) What power is necessary to drive the car at What power is necessary to drive the car at down a 1.00% grade? (d) Down what percent grade would the ca

On a winter day in Maine, a warehouse worker is shoving boxes up a rough plank inclined at an angle \(\alpha\) above the horizontal. The plank is partially covered with ice, with more ice near the bottom of the plank than near the top, so that the coefficient of friction increases with the distance x along the plank: \(\mu = A x\) where A is a positive constant and the bottom of the plank is at x = 0. (For this plank the coefficients of kinetic and static friction are equal: \(\boldsymbol{\mu}_{\mathrm{k}}=\boldsymbol{\mu}_{\mathrm{s}}=\boldsymbol{\mu}\).) The worker shoves a box up the plank so that it leaves the bottom of the plank moving at speed \(v_{0}\). Show that when the box first comes to rest, it will remain at rest if \(v_{0}^{2} \geq \frac{3 g \sin ^{2} \alpha}{A \cos \alpha}\) Text Transcription: alpha mu = Ax mu_k = mu_s = mu v_0 v_0^2 geq 3g sin ^2 alpha/A cos alpha

CALC A Spring with Mass. We usually ignore the kinetic energy of the moving coils of a spring, but lets try to get a reasonable approximation to this. Consider a spring of mass M, equilibrium length and spring constant k. The work done to stretch or compress the spring by a distance L is where Consider a spring, as described above, that has one end fixed and the other end moving with speed Assume that the speed of points along the length of the spring varies linearly with distance l from the fixed end. Assume also that the mass M of the spring is distributed uniformly along the length of the spring. (a) Calculate the kinetic energy of the spring in terms of M and (Hint: Divide the spring into pieces of length dl; find the speed of each piece in terms of l, and L; find the mass of each piece in terms of dl, M, and L; and integrate from 0 to L. The result is not since not all of the spring moves with the same speed.) In a spring gun, a spring of mass 0.243 kg and force constant is compressed 2.50 cm from its unstretched length. When the trigger is pulled, the spring pushes horizontally on a 0.053-kg ball. The work done by friction is negligible. Calculate the balls speed when the spring reaches its uncompressed length (b) ignoring the mass of the spring and (c) including, using the results of part (a), the mass of the spring. (d) In part (c), what is the final kinetic energy of the ball and of the spring?

CALC An airplane in flight is subject to an air resistance force proportional to the square of its speed But there is an additional resistive force because the airplane has wings. Air flowing over the wings is pushed down and slightly forward, so from Newtons third law the air exerts a force on the wings and airplane that is up and slightly backward (Fig. P6.104). The upward force is the lift force that keeps the airplane aloft, and the backward force is called induced drag. At flying speeds, induced drag is inversely proportional to so that the total air resistance force can be expressed by where and are positive constants that depend on the shape and size of the airplane and the density of the air. For a Cessna 150, a small single-engine airplane, and In steady flight, the engine must provide a forward force that exactly balances the air resistance force. (a) Calculate the speed at which this airplane will have the maximum range (that is, travel the greatest distance) for a given quantity of fuel. (b) Calculate the speed (in ) for which the airplane will have the maximum endurance (that is, remain in the air the longest time).

Problem 1DQ The sign of many physical quantities depends on the choice of coordinates. For example, a y for free-fall motion can be negative or positive, depending on whether we choose upward or downward as positive. Is the same true of work? In other words, can we make positive work negative by a different choice of coordinates? Explain.

Problem 1E You push your physics book 1.50 m along a horizontal table-top with a horizontal push of 2.40 N while the opposing force of friction is 0.600 N. How much work does each of the following forces do on the book: (a) your 2.40-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?

Problem 2DQ An elevator is hoisted by its cables at constant speed. Is the total work done on the elevator positive, negative, or zero? Explain.

Problem 2E A tow truck pulls a car 5.00 km along a horizontal roadway using a cable having a tension of 850 N. (a) How much work does the cable do on the car if it pulls horizontally? If it pulls at 35.0° 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)?

Problem 3DQ A rope tied to a body is pulled, causing the body to accelerate. But according to Newton’s third law, the body pulls back on the rope with a force of equal magnitude and opposite direction. Is the total work done then zero? If so, how can the body’s kinetic energy change? Explain.

Problem 3E A factory worker pushes a 30.0-kg crate a distance of 4.5 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?

Problem 4DQ If it takes total work W to give an object a speed v and kinetic energy K , starting from rest, what will be the object’s speed (in terms of v) and kinetic energy (in terms of K) if we do twice as much work on it, again starting from rest?

Problem 4E Suppose the worker in Exercise 6.3 pushes downward at an angle of 30o below the horizontal. (a) What magnitude of force must the worker apply to move the crate at constant velocity? (b) How much work is done on the crate by this force when the crate is pushed a distance of 4.5 m? (c) How much work is done on the crate by friction during this displacement? (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? 6.3 . A factory worker pushes a 30.0-kg crate a distance of 4.5 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?

If there is a net nonzero force on a moving object, is it possible for the total work done on the object to be zero? Explain, with an example that illustrates your answer.

Problem 5E A 75.0-kg painter climbs a ladder that is 2.75 m long and leans against a vertical wall. The ladder makes a 30.0o 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?

In Example 5.5 (Section 5.1), how does the work done on the bucket by the tension in the cable compare to the work done on the cart by the tension in the cable? Equation Transcription: Text Transcription: w_1 sin 15^° w_1 cos 15^°

Problem 6E Two tugboats pull a disabled supertanker. Each tug exerts a constant force of 1.80 X 106 N, one 14o west of north and the other 14o east of north, as they pull the tanker 0.75 km toward the north. What is the total work they do on the supertanker?

In the conical pendulum in Example 5.20 (Section 5.4), which of the forces do work on the bob while it is swinging?

Two blocks are connected by a very light string passing over a massless and frictionless pulley (Fig. E6.7). Traveling at constant speed, the \(20.0-\mathrm{N}\) block moves \(75.0 \mathrm{~cm}\) to the right and the \(\text { 12.0-N }\) block moves \(75.0 \mathrm{~cm}\) downward. During this process, how much work is done (a) on the \(\text { 12.0-N }\) block by (i) gravity and (ii) the tension in the string? (b) On the \(20.0-\mathrm{N}\) 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. Equation Transcription: Text Transcription: 20.0-N 75.0 cm 12.0-N 75.0 cm 12.0-N 20.0-N

For the cases shown in Fig. Q6.8, the object is released from rest at the top and feels no friction or air resistance. In which (if any) cases will the mass have (i) the greatest speed at the bottom and (ii) the most work done on it by the time it reaches the bottom? Equation Transcription: Text Transcription: 2m

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

A force \(\vec{F}\) is in the x-direction and has a magnitude that depends on x. Sketch a possible graph of ???? versus x such that the force does zero work on an object that moves from \(\mathrm{X}_{1} \text { to } \mathrm{X}_{2}\), even though the force magnitude is not zero at all x in this range. Equation Transcription: to Text Transcription: Vec F X_1 to x_2

Problem 9E A 0.800-kg ball is tied to the end of a string 1.60 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.

Problem 10DQ Does a car’s kinetic energy change more when the car speeds up from 10 to 15 m/s or from 15 to 20 m/s? Explain.

Problem 10E An 8.00-kg package in a mail-sorting room slides 2.00 m down a chute that is inclined at 53.0° 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)

Problem 11DQ A falling brick has a mass of 1.5 kg and is moving straight downward with a speed of 5.0 m/s. A 1.5-kg physics book is sliding across the floor with a speed of 5.0 m/s. A 1.5-kg melon is traveling with a horizontal velocity component 3.0 m/s to the right and a vertical component 4.0 m/s upward. Do all of these objects have the same velocity? Do all of them have the same kinetic energy? For both questions, give your reasoning.

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 cm/s, how much work is done on the monitor by (a) friction, (b) gravity, and (c) the normal force of the conveyor belt?

Problem 12DQ Can the total work done on an object during a displacement be negative? Explain. If the total work is negative, can its magnitude be larger than the initial kinetic energy of the object? Explain.

You apply a constant force \(\vec{F}=(-68.0 \mathrm{~N}) \hat{i}+(36.0 \mathrm{~N}) \hat{\jmath}\) to a \(380-\mathrm{kg}\) 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? Equation Transcription: ° Text Transcription: \vec{F}=(-68.0 \{~N}) \hat{i}+(36.0 \{~N}) \hat\j} 380-kg 48.0 m 240.0^°

Problem 13DQ A net force acts on an object and accelerates it from rest to a speed v1. In doing so, the force does an amount of work W1. By what factor must the work done on the object be increased to produce three times the final speed, with the object again starting from rest?

Problem 13E BIO Animal Energy. Adult cheetahs, the fastest of the great cats, have a mass of about 70 kg and have been clocked to run at up to 72 mi/h (32 m/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 truck speeding down the highway has a lot of kinetic energy relative to a stopped state trooper, but no kinetic energy relative to the truck driver. In these two frames of reference, is the same amount of work required to stop the truck? Explain.

Problem 14E A 1.50-kg book is sliding along a rough horizontal surface. At point A it is moving at 3.21 m/s, and at point B it has slowed to 1.25 m/s. (a) How much work was done on the book between A and B? (b) If - 0.750 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 J of work was done on it from B to C?

Problem 15DQ You are holding a briefcase by the handle, with your arm straight down by your side. Does the force your hand exerts do work on the briefcase when (a) you walk at a constant speed down a horizontal hallway and (b) you ride an escalator from the first to second floor of a building? In both cases justify your answer.

Problem 15E 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 X 108 kg (around 150,000 tons) and hit the ground at a speed of 12 km/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 mil-lion tons of TNT, and 1.0 ton of TNT releases 4.184 X 109 J of energy.)

Problem 16DQ When a book slides along a tabletop, the force of friction does negative work on it. Can friction ever do positive work? Explain. (Hint: Think of a box in the back of an accelerating truck.)

Problem 16E Some Typical Kinetic Energies. (a) In the Bohr model of the atom, the ground-state electron in hydrogen has an orbital speed of 2190 km/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 m, how many joules of kinetic energy will it have when it reaches the ground? (c) Is it reasonable that a 30-kg child could run fast enough to have 100 J of kinetic energy?

Problem 17DQ Time yourself while running up a flight of steps, and compute the average rate at which you do work against the force of gravity. Express your answer in watts and in horsepower.

In Fig. E6.7 assume that there is no friction force on the 20.0-N block that sits on the tabletop. The pulley is light and frictionless. (a) Calculate the tension ???? in the light string that connects the blocks. (b) For a displacement in which the 12.0-N block descends 1.20 m, calculate the total work done on (i) the 20.0-N block and (ii) the 12.0-N block. (c) For the displacement in part (b), calculate the total work done on the system of the two blocks. How does your answer compare to the work done on the 12.0-N block by gravity? (d) If the system is released from rest, what is the speed of the 12.0-N block when it has descended 1.20 m? Equation Transcription: Text Transcription: 20.0-N 12.0 N 1.20 m 20.0-N 12.0-N 12.0-N 12.0-N 1.20 m

Problem 18DQ Fractured Physics. Many terms from physics are badly misused in everyday language. In both cases, explain the errors involved. (a) A strong person is called powerful. What is wrong with this use of power? (b) When a worker carries a bag of concrete along a level construction site, people say he did a lot of work. Did he?

Problem 18E A 4.80-kg watermelon is dropped from rest from the roof of an 18.0-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?

An advertisement for a portable electrical generating unit claims that the unit’s diesel engine produces 28,000 hp to drive an electrical generator that produces 30 MW of electrical power. Is this possible? Explain.

Problem 19E 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 m into the air. How fast was the boulder moving just as it left the volcano? (c) A skier moving at 5.00 m/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 friclionless icy hill that rises at 25.0° above the horizontal, a toboggan has a speed of 12.0 m/s toward the hill. How high vertically above the base will it go before stopping?

A car speeds up while the engine delivers constant power. Is the acceleration greater at the beginning of this process or at the end? Explain.

You throw a 20-N rock vertically into the air from ground level. You observe that when it is 15.0 m above the ground, it is traveling at 25.0 m/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.

Problem 21DQ Consider a graph of instantaneous power versus time, with the vertical P-axis starting at P = 0. What is the physical significance of the area under the P-versus-t curve between vertical lines at t1 and t2? How could you find the average power from the graph? Draw a P-versus-t curve that consists of two straight-line sections and for which the peak power is equal to twice the average power.

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 nonzero net force acts on an object. Is it possible for any of the following quantities to be constant: (a) the particle’s speed; (b) the particle’s velocity; (c) the particle’s kinetic energy?

Problem 22E A mass m slides down a smooth inclined plane from an initial vertical height h, making an angle ? 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 the mass had been dropped from height h, independent of the angle ? 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 frictionless hill, starting from rest 15.0 m above the bottom.

Problem 23DQ When a certain force is applied to an ideal spring, the spring stretches a distance x from its unstretched length and does work W. If instead twice the force is applied, what distance (in terms of x) does the spring stretch from its unstretched length, and how much work (in terms of W) is required to stretch it this distance?

Problem 23E A sled with mass 8.00 kg moves in a straight line on a frictionless horizontal surface. At one point in its path, its speed is 4.00 m/s; after it has traveled 2.50 m beyond this point, its speed is 6.00 m/s. Use the work-energy theorem to find the force acting on the sled, assuming that this force is constant anti that it acts in the direction of the sled’s motion.

Problem 24DQ If work W is required to stretch a spring a distance x from its unstretched length, what work (in terms of W) is required to stretch the spring an additional distance x?

A soccer ball with mass 0.420 kg is initially moving with speed 2.00 m/s. A soccer player kicks the ball, exerting a constant force of magnitude 40.0 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 m/s?

Problem 25E A 12-pack of Omni-Cola (mass 4.30 kg) is initially at rest on a horizontal floor. It is then pushed in a straight line for 1.20 m by a trained dog that exerts a horizontal force with magnitude 36.0 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.

Problem 26E A batter hits a baseball with mass 0.145 kg straight upward with an initial speed of 25.0 m/s. (a) How much work has gravity done on the baseball when it reaches a height of 20.0 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 m? Explain.

A little red wagon with mass 7.00 kg moves in a straight line on a frictionless horizontal surface. It has an initial speed of 4.00 m/s and then is pushed 3.0 m in the direction of the initial velocity by a force with a magnitude of 10.0 N. (a) Use the work–energy theorem to calculate the wagon’s final speed. (b) Calculate 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 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.

Problem 29E Stopping Distance. A car is traveling on a level road with speed v0 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 v0, g, and the coefficient of kinetic friction 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?

Problem 30E A 30.0-kg crate is initially moving with a velocity that has magnitude 3.90 m/s in a direction 37.0o west of north. How much work must be done on the crate to change its velocity to 5.62 m/s in a direction 63.0o south of east?

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 cm when a 1.50-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 cm, what is the greatest force it will be able to exert there?

Problem 32E To stretch a spring 3.00 cm from its unstretched length, 12.0 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 cm from its unstretched length? (c) How much work must be done to compress this spring 4.00 cm from its unstretched length, and what force is needed to compress it this distance?

Three identical \(6.40-\mathrm{kg}\) masses are hung by three identical springs, as shown in Fig. E6.33. Each spring has a force constant of \(7.80 \mathrm{kN} / \mathrm{m}\) and was \(12.0 \mathrm{~cm}\) long before any masses were attached to it. (a) Draw a free-body diagram of each mass. (b) How long is each spring when hanging as shown? (Hint: First isolate only the bottom mass. Then treat the bottom two masses as a system. Finally, treat all three masses as a system.) Equation Transcription: Text Transcription: 6.40-kg 7.80 kN/m 12.0 cm

A child applies a force \(\vec{F}\) parallel to the x-axis to a 10.0-kg sled moving on the frozen surface of a small pond. As the child controls the speed of the sled, the x-component of the force she applies varies with the x-coordinate of the sled as shown in Fig. E6.34. Calculate the work done by the force\(\vec{F}\) when the sled moves (a) from \(x=0 \text { to } x=8.0 \mathrm{~m}\); (b) from \(x=8.0 \mathrm{~m} \text { to } x=12.0 \mathrm{~m}\); (c) from \(x=0 \text { to } 12.0 \mathrm{~m}\). Equation Transcription: Text Transcription: Vec F Vec F x=0 to x=8.0 m x=8.0 m to x=12.0 m x=0 to 12.0 m

Suppose the sled in Exercise 6.34 is initially at rest at = 0. Use the work–energy theorem to find the speed of the sled at (a) \(x=8.0 \mathrm{~m}\) and (b) \(x=12.0 \mathrm{~m}\). You can ignore friction between the sled and the surface of the pond. Equation Transcription: Text Transcription: x=8.0 m x=12.0 m

A 2.0-kg box and a 3.0-kg box on a perfectly smooth horizontal floor have a spring of force constant 250 N/m compressed between them. If the initial compression of the spring is 6.0 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.

Problem 37E A 6.0-kg box moving at 3.0 m/s on a horizontal, friction-less surface runs into a light spring of force constant 75 N/cm. Use the work–energy theorem to find the maximum compression of the spring.

Problem 38E 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 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 m farther, and what maximum force must you apply?

(a) In Example 6.7 (Section 6.3) it was calculated that with the air track turned off, the glider travels 8.6 cm before it stops instantaneously. How large would the coefficient of static friction \(\mu_{s}\)have to be to keep the glider from springing back to the left? (b) If the coefficient of static friction between the glider and the track is what is the maximum initial speed that the glider can be given and still remain at rest after it stops \(\mu_{\mathrm{s}}=0.60\), what is the maximum initial speed \(v_{1}\)that the glider can be given and still remain at rest after it stops instantaneously? With the air track turned off, the coefficient of kinetic friction is \(\mu_{\mathrm{k}}=0.47\). Equation Transcription: Text Transcription: \mu_{s} \mu_{\{s}}=0.60 V_{1} \mu_{\{k}}=0.47

Problem 40E A 4.00-kg block of ice is placed against a horizontal spring that has force constant k = 200 N/m and is compressed 0.025 m. The spring is released and accelerates the block along a horizontal surface. 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?

A force \(\overrightarrow{\boldsymbol{F}}\) is applied to a 2.0-kg radio-controlled model car parallel to the x-axis as it moves along a straight track. The x-component of the force varies with the x-coordinate of the car as shown in Fig. E6.41. Calculate the work done by the force \(\overrightarrow{\boldsymbol{F}}\) when the car moves from (a) x = 0 to x = 3.0 m; (b) x = 3.0 m to x = 4.0 m; (c) x = 4.0 m to x = 7.0 m; (d) x = 0 to x = 7.0 m; (e) x = 7.0 m to x = 2.0 m.

Suppose the 2.0-kg model car in Exercise 6.41 is initially at rest at x = 0 and \(\vec{F}\) is the net force acting on it. Use the work–energy theorem to find the speed of the car at (a) \(x=3.0 \mathrm{~m}\); (b)\(x=4.0 \mathrm{~m}\);; (c) \(x=7.0 \mathrm{~m}\);. 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 ???? = 40.0 N/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 kg are pushed against the other end, compressing the spring 0.375 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 m? Equation Transcription: Text Transcription: x=0 Vec F x=3.0 m x=4.0 m x=7.0m

Problem 43E 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 N/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 kg are pushed against the other end, compressing the spring 0.375 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 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?

Problem 46E 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 N/m and negligible mass. When the spring is released, the brick is propelled upward. If the brick has mass 1.80 kg and is to reach a maximum height of 3.6 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?)

Problem 45E A small glider is placed against a compressed spring at the bottom of an air track that slopes upward at an angle of 40.0o above the horizontal. The glider has mass 0.0900 kg. The spring has k = 640 N/m and negligible mass. When the spring is released, the glider travels a maximum distance of 1.80 m along the air track before sliding back down. Before reaching this maximum distance, the glider loses contact with the spring. (a) What distance was the spring originally compressed? (b) When the glider has traveled along the air track 0.80 m from its initial position against the compressed spring, is it still in contact with the spring? What is the kinetic energy of the glider at this point?

Problem 47E CALC A force in the + x-direction with magnitude F(x) = 18.0 N – (0.530 N/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 m?

Problem 48E A crate on a motorized cart starts from rest and moves with a constant eastward acceleration of a = 2.80 m/s2. A worker assists the cart by pushing on the crate with a force that is east-ward and has magnitude that depends on time according to F(t) = (5.40 N/s)t. What is the instantaneous power supplied by this force at t = 5.00 s?

Problem 49E 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?

Problem 50E BIO Should You Walk or Run? It is 5.0 km from your home to the physics lab. As part of your physical fitness program, you could run that distance at 10 km/h (which uses up energy at the rate of 700 W), or you could walk it leisurely at 3.0 km/h (which uses energy at 290 W). Which choice would burn up more energy, and how much energy (in joules) would it burn? Why does the more intense exercise burn 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 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?

Problem 52E A 20.0-kg rock is sliding on a rough, horizontal surface at 8.00 m/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 N to maintain a speed of 9.00 m/s. Find the power required per rider, assuming that each contributes equally. Express your answer in watts and in horsepower.

Problem 54E When its 75-kW (100-hp) engine is generating full power, a small single-engine airplane with mass 700 kg gains altitude at a rate of 2.5 m/s (150 m/min, or 500 ft/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.)

Problem 55E 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. How many crates would you have to load onto the truck in 1 minute (a) for the average power output you use to lift the crates to equal 0.50 hp; (b) for an average power output of 100 W?

Problem 56E An elevator has mass 600 kg, not including passengers. The elevator is designed to ascend, at constant speed, a vertical distance of 20.0 m (five floors) in 16.0 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 kg.

Problem 57E A ski tow operates on a 15.0o slope of length 300 m. The rope moves at 12.0 km/h and provides power for 50 riders at one time, with an average mass per rider of 70.0 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 km/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?

Problem 59E BIO While hovering, a typical flying insect applies an average force equal to twice its weight during each downward stroke. Take the mass of the insect to be 10 g, and assume the wings move an average downward distance of 1.0 cm during each stroke. Assuming 100 downward strokes per second, estimate the average power output of the insect.

Problem 60P CALC A balky cow is leaving the barn as you try harder and harder to push her back in. In coordinates with the origin at the barn door, the cow walks from x = 0 to x = 6.9 m as you apply a force with x-component Fx = -[20.0 N + (3.0 N/m)x]. How much work does the force you apply do on the cow during this displacement?

Problem 61P Rotating Bar. A thin, uniform 12.0-kg bar that is 2.0 m long rotates uniformly about a pivot at one end, making 5.0 complete revolutions every 3.00 seconds. What is the kinetic energy of this bar? (Hint: Different points in the bar have different speeds Break the bar up into infinitesimal segments of mass dm and integrate to add up the kinetic energies of all these segments.)

A Near-Earth Asteroid. On April 13, 2029 (Friday the 13th!), the asteroid 99942 Apophis will pass within 18,600 mi of the earth—about \(\frac{1}{13}\) the distance to the moon! It has a density of \(2600 \mathrm{~kg} / \mathrm{m}^{3}\), can be modeled as a sphere 320 m in diameter, and will be traveling at 12.6 km/s. (a) If, due to a small disturbance in its orbit, the asteroid were to hit the earth, how much kinetic energy would it deliver? (b) The largest nuclear bomb ever tested by the United States was the “Castle/Bravo” bomb, having a yield of 15 megatons of TNT. (A megaton of TNT releases \(4.184 \times 10^{15}\)J of energy.) How many Castle/Bravo bombs would be equivalent to the energy of Apophis? Equation Transcription: Text Transcription: \frac{1}{13} 4.184 times 10^15

A luggage handler pulls a 20.0-kg suitcase up a ramp inclined at \(25.0^{\circ}\) above the horizontal by a force \(\vec{F}\) of magnitude 140 N that acts parallel to the ramp. The coefficient of kinetic friction between the ramp and the incline is \(\mu_{\mathrm{k}}=0.300\). If the suitcase travels 3.80 m along the ramp, calculate (a) the work done on the suitcase by the force \(\vec{F}\) (b) the work done on the suitcase by the gravitational force; (c) the work done on the suitcase by the normal force; (d) the work done on the suitcase by the friction force; (e) the total work done on the suitcase. (f) If the speed of the suitcase is zero at the bottom of the ramp, what is its speed after it has traveled 3.80 m along the ramp? Equation Transcription: 25.0° Text Transcription: 25.0^° Vec F \mu_{\{k}}=0.300 Vec F

Problem 64P BIO Chin-ups. While doing a chin-up, a man lifts his body 0.40 m. (a) How much work must the man do per kilogram of body mass? (b) The muscles involved in doing a chin-up can generate about 70 J of work per kilogram of muscle mass. If the man can just barely do a 0.40-m chin-up, what percentage of his body’s mass do these muscles constitute? (For comparison, the total percentage of muscle in a typical 70-kg man with 14% body fat is about 43%.) (c) Repeat part (b) for the man’s young son, who has arms half as long as his father’s but whose muscles can also generate 70 J of work per kilogram of muscle mass. (d) Adults and children have about the same percentage of muscle in their bodies. Explain why children can commonly do chin-ups more easily than their fathers.

A 20.0-kg crate sits at rest at the bottom of a 15.0-m-long ramp that is inclined at \(34.0^{\circ}\) above the horizontal. A constant horizontal force of 290 N is applied to the crate to push it up the ramp. While the crate is moving, the ramp exerts a constant frictional force on it that has magnitude 65.0 N. (a) What is the total work done on the crate during its motion from the bottom to the top of the ramp? (b) How much time does it take the crate to travel to the top of the ramp?

Consider the blocks in Exercise 6.7 as they move 75.0 cm. Find the total work done on each one (a) if there is no friction between the table and the 20.0-N block, and (b) if \(\mu_{s}=0.500\) and \(\mu_{\mathrm{k}}=0.325\) between the table and the 20.0-N block. Equation Transcription: Text Transcription: \(\mu_{s}=0.500\) mu_{k}}=0.325

Problem 67P The space shuttle, with mass 86.400 kg. is in a circular-orbit of radius 6.66 × 106 m around the earth. It takes 90.1 min for the shuttle to complete each orbit. On a repair mission, the shuttle is cautiously moving 1.00 m closer to a disabled satellite every 3.00 s. Calculate the shuttle’s kinetic energy (a) relative to the earth and (b) relative to the satellite.

Problem 68P A 5.00-kg package slides 1.50 m down a long amp that is inclined at 24.0° below the horizontal. The coefficient of kinetic friction between the package and the ramp is ?k = 0.310 Calculate (a) the work done on the package by friction; (b) the work done on the package by gravity; (c) the work done on the package by the normal force; (d) the total work done on the package. (e) If the package has a speed of 2.20 m/s at the top of the ramp, what is its speed after sliding 1.50 m down the ramp?

Problem 69P CP BIO Whiplash Injuries. When a car is hit from behind, its passengers undergo sudden forward acceleration, which can cause a severe neck injury known as whiplash. During normal acceleration, the neck muscles play a large role in accelerating the head so that the bones are not injured. But during a very sudden acceleration, the muscles do not react immediately because they are flexible; most of the accelerating force is provided by the neck bones. Experiments have shown that these bones will fracture if they absorb more than 8.0 J of energy. (a) If a car waiting at a stoplight is rear-ended in a collision that lasts for 10.0 ms, what is the greatest speed this car and its driver can reach without breaking neck bones if the driver’s head has a mass of 5.0 kg (which is about right for a 70-kg person)? Express your answer in m/s and in mi/h. (b) What is the acceleration of the passengers during the collision in part (a), and how large a force is acting to accelerate their heads? Express the acceleration in m/s2 and in g’s.

Problem 70P CALC A net force along the x -axis that has x- component Fx = -12.0 N + (0.300 N/m2)x2 is applied to a 5.00-kg object that is initially at the origin and moving in the -x-direction with a speed of 6.00 m/s. What is the speed of the object when it reaches the point x = 5.00 m?

Problem 71P An object is attracted toward the origin with a force given by Fx = ?k/x2. (Gravitational and electrical forces have this distance dependence.) (a) Calculate the work done by the force Fx when the object moves in the x-direction from x1 to x2. If x2 > x1, is the work done by Fx positive or negative? (b) The only other force acting on the object is a force that you exert with your hand to move the object slowly from x1 to x2. How much work do you do? If x2 > x1, is the work you do positive or negative? (c) Explain the similarities and differences between your answers to parts (a) and (b).

Problem 72P The gravitational pull of the earth on an object is inversely proportional to the square of the distance of the object from the center of the earth. At the earth’s surface this force is equal to the object’s normal weight mg, where g = 9.8 m/s2, and at large distances, the force is zero. If a 20,000-kg asteroid falls to earth from a very great distance away, what will be its minimum ’speed as it strikes the earth’s surface, and how much kinetic energy will it impart to our planet? You can ignore the effects of the earth’s atmosphere.

Problem 73P CALC Varying Coefficient of Friction. A box is sliding with a speed of 4.50 m/s on a horizontal surface when, at point P, it encounters a rough section. The coefficient of friction there is not constant; it starts at 0.100 at P and increases linearly with distance past P, reaching a value of 0.600 at 12.5 m past point P. (a) Use the work–energy theorem to find how far this box slides before stopping. (b) What is the coefficient of friction at the stopping point? (c) How far would the box have slid if the friction coefficient didn’t increase but instead had the constant value of 0.100?

Problem 74P CALC Consider a spring that does not obey Hooke’s law very faithfully. One end of the spring is fixed. To keep the spring stretched or compressed an amount x, a force along the x-axis with x-component Fx = kx - bx2 + cx3 must be applied to the free end. Here k = 100 N/m, b = 700 N/m2, and c = 12,000 N/m3. Note that x > 0 when the spring is stretched and x < 0 when it is compressed. (a) How much work must be done to stretch this spring by 0.050 m from its unstretched length? (b) How much work must be done to compress this spring by 0.050 m from its unstretched length? (c) Is it easier to stretch or compress this spring? Explain why in terms of the dependence of Fx on x. (Many real springs behave qualitatively in the same way.)

A small block with a mass of 0.0900 kg is attached to a cord passing through a hole in a frictionless, horizontal surface (Fig. P6.75). The block is originally revolving at a distance of 0.40 m from the hole with a speed of 0.70 m/s. The cord is then pulled from below, shortening the radius of the circle in which the block revolves to 0.10 m. At this new distance, the speed of the block is observed to be 2.80 m/s. (a) What is the tension in the cord in the original situation when the block has speed \(v = 0.70 \mathrm{\ m}/\mathrm{s}\)? (b) What is the tension in the cord in the final situation when the block has speed \(v = 2.80 \mathrm{\ m}/\mathrm{s}\)? (c) How much work was done by the person who pulled on the cord? Text Transcription: v = 0.70 m/s v = 2.80 m/s

Problem 76P CALC Proton Bombardment. A proton with mass 1.67 X 10-27 kg is propelled at an initial speed of 3.00 X 105 m/s directly toward a uranium nucleus 5.00 m away. The proton is repelled by the uranium nucleus with a force of magnitude F = ?/x2, where x is the separation between the two objects and ? = 2.12 X 10-26 N.m2. Assume that the uranium nucleus remains at rest. (a) What is the speed of the proton when it is 8.00 X 10-10 m from the uranium nucleus? (b) As the proton approaches the uranium nucleus, the repulsive force slows down the proton until it comes momentarily to rest, after which the pro-ton moves away from the uranium nucleus. How close to the uranium nucleus does the proton get? (c) What is the speed of the proton when it is again 5.00 m away from the uranium nucleus?

A block of ice with mass 4.00 kg is initially at rest on a frictionless, horizontal surface. A worker then applies a horizontal force \(\vec{F}\) to it. As a result, the block moves along the x-axis such that its position as a function of time is given by \(x(t)=a t^{2}+\beta t^{3}\), where \(a=0.200 \mathrm{~m} / \mathrm{s}^{2}\) and (a) \(\beta=0.0200 \mathrm{~m} / \mathrm{s}^{3}\).Calculate the velocity of the object when t=4.00 s. (b) Calculate the magnitude of \(\vec{F}\) when (c) Calculate the work done by the force \(\vec{F}\) during the first 4.00 s of the motion.

You and your bicycle have combined mass 80.0 kg. When you reach the base of a bridge, you are traveling along the road at 5.00 m/s (Fig. P6.78). At the top of the bridge, you have climbed a vertical distance of 5.20 m and have slowed 1.50 m/s. to You can ignore work done by friction and any inefficiency in the bike or your legs. (a) What is the total work done on you and your bicycle when you go from the base to the top of the bridge? (b) How much work have you done with the force you apply to the pedals? Equation Transcription: Text Transcription: m=80.0 kg 5.20 m

Problem 79P You are asked to design spring bumpers for the walls of a parking garage. A freely rolling 1200-kg car moving at 0.65 m/s is to compress the spring no more than 0.090 m before stopping. What should be the force constant of the spring? Assume that the spring has negligible mass.

Problem 80P The spring of a spring gun has force constant k = 400 N/m and negligible mass. The spring is compressed 6.00 cm, and a ball with mass 0.0300 kg is placed in the horizontal barrel against the compressed spring. The spring is then released, and the ball is propelled out the barrel of the gun. The barrel is 6.00 cm long, so the ball leaves the barrel at the same point that it loses contact with the spring. The gun is held so that the barrel is horizontal. (a) Calculate the speed with which the ball leaves the barrel if you can ignore friction. (b) Calculate the speed of the ball as it leaves the barrel if a constant resisting force of 6.00 N acts on the ball as it moves along the barrel. (c) For the situation in part (b), at what position along the barrel does the ball have the greatest speed, and what is that speed? (In this case, the maximum speed does not occur at the end of the barrel.)

Problem 81P A 2.50-kg textbook is forced against a horizontal spring of negligible mass and force constant 250 N/m, compressing the spring a distance of 0.250 m. When released, the textbook slides on a horizontal tabletop with coefficient of kinetic friction µk = 0.30. Use the work–energy theorem to find how far the text-book moves from its initial position before it comes to rest.

Problem 82P Pushing a Cat. Your cat “Ms.” (mass 7.00 kg) is trying to make it to the top of a friclionless ramp 2.00 m long and inclined upward at 30.0° above the horizontal. Since the poor cat can’t get any traction on the ramp, you push her up the entire length of the ramp by exerting a constant 100-N force parallel to the ramp. If Ms. takes a running start so that she is moving at 2.40 m/s at the bottom of the ramp, what is her speed when she reaches the top of the incline? Use the work-energy theorem.

Problem 83P Crash Barrier. A student proposes a design for an automobile crash barrier in which a 1700-kg sport utility vehicle moving at 20.0 m/s crashes into a spring of negligible mass that slows it to a stop. So that the passengers are not injured, the acceleration of the vehicle as it slows can be no greater than 5.00g. (a) Find the required spring constant k, and find the distance the spring will compress in slowing the vehicle to a stop. In your calculation, disregard any deformation or crumpling of the vehicle and the friction between the vehicle and the ground. (b) What disadvantages are there to this design?

Problem 84P A physics professor is pushed up a ramp inclined up-ward at 30.0o above the horizontal as she sits in her desk chair, which slides on frictionless rollers. The combined mass of the professor and chair is 85.0 kg. She is pushed 2.50 m along the incline by a group of students who together exert a constant horizontal force of 600 N. The professor’s speed at the bottom of the ramp is 2.00 m/s. Use the work–energy theorem to find her speed at the top of the ramp.

A 5.00-kg block is moving at \(v_{0}=6.00 \mathrm{~m} / \mathrm{s}\) along a frictionless, horizontal surface toward a spring with force constant \(k=500 \mathrm{~N} / \mathrm{m}\) that is attached to a wall (Fig. P6.85). The spring has negligible mass. (a) Find the maximum distance the spring will be compressed. (b) If the spring is to compress by no more than 0.150 m, what should be the maximum value of \(v_{0}\)? Equation Transcription: Text Transcription: v_0=6.00 m/s k=500 N/m v_0

Consider the system shown in Fig. P6.86. The rope and pulley have negligible mass, and the pulley is frictionless. The coefficient of kinetic friction between the 8.00-kg block and the tabletop is \(\mu_{k}=0.250\). The blocks are released from rest. Use energy methods to calculate the speed of the 6.00-kg block after it has descended 1.50 m. Equation Transcription: Text Transcription: \mu_{k}=0.250

Consider the system shown in Fig. P6.81. The rope and pulley have negligible mass, and the pulley is frictionless. Initially the \(6.00-\mathrm{kg}\) block is moving downward and the \(8.00-\mathrm{kg}\) block is moving to the right, both with a speed of \(0.900 \mathrm{~m} / \mathrm{s}\). The blocks come to rest after moving \(2.00 \mathrm{~m}\). Use the work-energy theorem to calculate the coefficient of kinetic friction between the \(8.00-\mathrm{kg}\) block and the tabletop.

Bow and Arrow. Figure P6.88 shows how the force exerted by the string of a compound bow on an arrow varies as a function of how far back the arrow is pulled (the draw length). Assume that the same force is exerted on the arrow as it moves forward after being released. Full draw for this bow is at a draw length of \(75.0 \mathrm{~cm}\). If the bow shoots a \(0.0250-\mathrm{kg}\) arrow from full draw, what is the speed of the arrow as it leaves the bow? Equation Transcription: Text Transcription: 75.0 cm 0.0250-kg

Problem 89P On an essentially frictionless, horizontal ice rink, a skater moving at 3.0 m/s encounters a rough patch that reduces her speed to 1.65 m/s due to a friction force that is 25% of her weight. Use the work–energy theorem to find the length of this rough patch.

Problem 90P Rescue. Your friend (mass 65.0 kg) is standing on the ice in the middle of a frozen pond There is very little friction between her feet and the ice, so she is unable to walk. Fortunately, a light rope is tied around her waist and you stand on the bank holding the other end. You pull on the rope for 3.00 s and accelerate your friend from rest to a speed of 6.00 m/s while you remain at rest. What is the average power supplied by the force you applied?

Problem 91P A pump is required to lift 800 kg of water (about 210 gallons) per minute from a well 14.0 m deep and eject it with a speed of 18.0 m/s. (a) How much work is done per minute in lifting the water? (b) How much work is done in giving the water the kinetic energy it has when ejected? (c) What must be the power output of the pump?

Problem 92P BIO All birds, independent of their size, must maintain a power output of 10–25 watts per kilogram of body mass in order to fly by flapping their wings. (a) The Andean giant hummingbird (Patagona gigas) has mass 70 g and flaps its wings 10 times per second while hovering. Estimate the amount of work done by such a hummingbird in each wingbeat. (b) A 70-kg athlete can maintain a power output of 1.4 kW for no more than a few seconds; the steady power output of a typical athlete is only 500 W or so. Is it possible for a human-powered aircraft to fly for extended periods by flapping its wings? Explain.

Problem 93P A physics student spends part of her day walking between classes or for recreation, during which time she expends energy at an average rate of 280 W. The remainder of the day she is sitting in class, studying, or resting; during these activities, she expends energy at an average rate of 100 W. If she expends a total of 1.1 X 107 J of energy in a 24-hour day, how much of the day did she spend walking?

Problem 94P The Grand Coulee Dam is 1270 m long and 170 m high. The electrical power output from generators at its base is approximately 2000 MW. How many cubic meters of water must flow from the top of the dam per second to produce this amount of power if 92% of the work done on the water by gravity is con-verted to electrical energy? (Each cubic meter of water has a mass of 1000 kg.)

Problem 95P BIO Power of the Human Heart. The human heart is a powerful and extremely reliable pump. Each day it takes in and discharges about 7500 L of blood. Assume that the work done by the heart is equal to the work required to lift this amount of blood a height equal to that of the average American woman (1.63 m). The density (mass per unit volume) of blood is 1.05 X 103 kg/m3. (a) How much work does the heart do in a day? (b) What is the heart’s power output in watts?

Problem 96P Six diesel units in series can provide 13.4 MW of power to the lead car of a freight train. The diesel units have total mass 1.10 × 106 kg. The average car in the train has mass 8.2 × 104 kg and requires a horizontal pull of 2.8 kN to move at a constant 27 m/s on level tracks. (a) How many cars can be in the train under these conditions? (b) This would leave no power for accelerating or climbing hills. Show that the extra force needed to accelerate the train is about the same for a 0.10-m/s2 acceleration or a 1.0% slope (slope angle ? = arctan 0.010). (c) With the 1.0% slope, show that an extra 2.9 MW of power is needed to maintain the 27-m/s speed of the diesel units. (d) With 2.9 MW less power available how many cars can the six diesel units pull up a 1.0% slope a constant 27 m/s?

Problem 97P It takes a force of 53 kN on the lead car of a 16-car passenger train with mass 9.1 × 103 kg to pull it at a constant 45 m/s (101 mi/h) on level tracks. (a) What power must the locomotive provide to the lead car? (b) How much more power to the lead car than calculated in part (a) would be needed to give the train an acceleration of 1.5 m/s2, at the instant that the train has a speed of 45 m/s on level tracks? (c) How much more power to the lead car than that calculated in part (a) would be needed to move the train up a 1.5% grade (slope angle ? = arctan 0.015) at a constant 45 m/s?

An object has several forces acting on it. One of these forces is \(\vec{F}\)a force in the x-direction whose magnitude depends on the position of the object, with \(\alpha=2.50 \mathrm{~N} / \mathrm{m}^{2}\)Calculate the work done on the object by this force for the following displacements of the object: (a) The object starts at the point \(x=0, y=3.00 \mathrm{~m}\) and moves parallel to the x-axis to the point \(x=2.00 \mathrm{~m}, \quad y=3.00 \mathrm{~m}\). (b) The object starts at the point \(x=2.00 \mathrm{~m}, y=0\) and moves in the y-direction to the point (c) The object starts at the origin and moves on the line \(y=1.5 x\) to the point \(x=2.00 \mathrm{~m}, y=3.00 \mathrm{~m}\). Equation Transcription: Text Transcription: Vec F \alpha=2.50{~N} {m}^{2} x=0, y=3.00 m x=2.00 m, y=3.00m x=2.00 m, y=0 y=1.5x x=2.00 m, y=3.00 m

Cycling. For a touring bicyclist the drag coefficient \(C\left(f_{\text {air }}=\frac{1}{2} C A \rho v^{2}\right)\) is 1.00, the frontal area A is 0.463 \(m^{2}\) and the coefficient of rolling friction is 0.0045. The rider has mass 50.0 kg, and her bike has mass 12.0 kg. (a) To maintain a speed of 12.0 m/s (about 27 mi/h) on a level road, what must the rider’s power output to the rear wheel be? (b) For racing, the same rider uses a different bike with coefficient of rolling friction 0.0030 and mass 9.00 kg. She also crouches down, reducing her drag coefficient to 0.88 and reducing her frontal area to 0.366\(m^{2}\). What must her power output to the rear wheel be then to maintain a speed of 12.0 m/s? (c) For the situation in part (b), what power output is required to maintain a speed of 6.0 m/s? Note the great drop in power requirement when the speed is only halved. (For more on aerodynamic speed limitations for a wide variety of human-powered vehicles, see “The Aerodynamics of Human-Powered Land Vehicles,” Scientific American, December 1983.) Equation Transcription: Text Transcription: C\left(f_{\text {air }}=\frac{1}{2} C A \rho v^{2}\right) M_2 m_2

Problem 100P Automotive Power I. A truck engine transmits 28.0 kW (37.5 hp) to the driving wheels when the truck is traveling at a constant velocity of magnitude 60.0 km/h (37.3 mi/h) on a level road. (a) What is the resisting force acting on the truck? (b) Assume that 65% of the resisting force is due to rolling friction and the remainder is due to air resistance. If the force of rolling friction is independent of speed, and the force of air resistance is proportional to the square of the speed, what power will drive the truck at 30.0 km/h? At 120.0 km/h? Give your answers in kilo watts and in horsepower.

Automotive Power II. (a) If 8.00 hp are required to drive a 1800-kg automobile at 60.0 km/h on a level road, what is the total retarding force due to friction, air resistance, and so on? (b) What power is necessary to drive the car at 60.0 km/h up a 10.0% grade (a hill rising 10.0 m vertically in 100.0 m horizontally)? (c) What power is necessary to drive the car at 60.0 km/h down a 1.00% grade? (d) Down what percent grade would the car coast at 60.0 km/h?

On a winter day in Maine, a warehouse worker is shoving boxes up a rough plank inclined at an angle above the horizontal. The plank is partially covered with ice, with more ice near the bottom of the plank than near the top, so that the coefficient of friction increases with the distance x along the plank: \(\mu=A x\), where A is a positive constant and the bottom of the plank is at . (For this plank the coefficients of kinetic and static friction are equal:\(\mu_{k}=\mu_{s}=\mu\) The worker shoves a box up the plank so that it leaves the bottom of the plank moving at speed \(v_{0}\). Show that when the box first comes to rest, it will remain at rest if \(v_{0}^{2} \geq \frac{3 g \sin ^{2} \alpha}{A \cos \alpha}\) Equation Transcription: Text Transcription: \mu=A x x=0 \mu_{k}=\mu_{s}=\mu V_o v_{0}^{2} \geq \frac{3 g \sin ^{2} \alpha}{A \cos \alpha}

A Spring with Mass. We usually ignore the kinetic energy of the moving coils of a spring, but let’s try to get a reasonable approximation to this. Consider a spring of mass ????, equilibrium length \(L_{0}\), and spring constant k. The work done to stretch or compress the spring by a distance is \(\frac{1}{2} k X^{2}\)where \(X=L-L_{0}\). Consider a spring, as described above, that has one end fixed and the other end moving with speed . Assume that the speed of points along the length of the spring varies linearly with distance l from the fixed end. Assume also that the mass ???? of the spring is distributed uniformly along the length of the spring. (a) Calculate the kinetic energy of the spring in terms of M and (Hint: Divide the spring into pieces of length dl; find the speed of each piece in terms of l, ????, and ; find the mass of each piece in terms of dl, ????, and ; and integrate from 0 to . The result is \(\frac{1}{2} M v^{2}\), since not all of the spring moves with the same speed.) In a spring gun, a spring of mass 0.243 kg and force constant 3200 N/m is compressed 2.50 cm from its unstretched length. When the trigger is pulled, the spring pushes horizontally on a 0.053-kg ball. The work done by friction is negligible. Calculate the ball’s speed when the spring reaches its uncompressed length (b) ignoring the mass of the spring and (c) including, using the results of part (a), the mass of the spring. (d) In part (c), what is the final kinetic energy of the ball and of the spring? Equation Transcription: Text Transcription: Lo 12kX2 X=L-L0 12Mv2

An airplane in flight is subject to an air resistance force proportional to the square of its speed \(v\). But there is an additional resistive force because the airplane has wings. Air flowing over the wings is pushed down and slightly forward, so from Newton’s third law the air exerts a force on the wings and airplane that is up and slightly backward (Fig. P6.104). The upward force is the lift force that keeps the airplane aloft, and the backward force is called induced drag. At flying speeds, induced drag is inversely proportional to \(v^{2}\), so that the total air resistance force can be expressed by \(F_{\text {air }}=\alpha v^{2}+\beta / v^{2}\), where \(\alpha\) and \(\beta\) are positive constants that depend on the shape and size of the airplane and the density of the air. For a Cessna 150, a small single-engine airplane, \(\alpha=0.30 \mathrm{~N} \cdot \mathrm{s}^{2} / \mathrm{m}^{2}\) and \(\beta=3.5 \times 10^{5} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{s}^{2}\). In steady flight, the engine must provide a forward force that exactly balances the air resistance force. (a) Calculate the speed (in km/h) at which this airplane will have the maximum range (that is, travel the greatest distance) for a given quantity of fuel. (b) Calculate the speed (in km/h) for which the airplane will have the maximum endurance (that is, remain in the air the longest time). Equation Transcription: Text Transcription: v v2 Fair=alpha v2+beta/v alpha beta alpha=0.30 N cdot s^2/m^2 beta=3.5105 N cdot m^2/s^2 km/h km/h