Solution: Working Like a Horse. Your job is to lift 30-kg

The sign of many physical quantities depends on the choice of coordinates. For example, ay 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

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

A rope tied to a body is pulled, causing the body to accelerate. But according to Newtons 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 bodys kinetic energy change? Explain.

If it takes total work W to give an object a speed v and kinetic energy K, starting from rest, what will be the objects 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?

If there is a net nonzero force on a moving object, can the total work done on the object be zero? Explain, using an example.

In Example 5.5 (Section 5.1), how does the work done on the bucket by the tension in the cable compare with the work done on the cart by the tension in the cable?

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

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?

A force F S is in the x-direction and has a magnitude that depends on x. Sketch a possible graph of F versus x such that the force does zero work on an object that moves from x1 to x2, even though the force magnitude is not zero at all x in this range.

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.

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

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.

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?

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.

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.

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.)

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.

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?

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.

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.

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 straightline sections and for which the peak power is equal to twice the average power.

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

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?

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?

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?

Using a cable with a tension of 1350 N, a tow truck pulls a car 5.00 km along a horizontal roadway. (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)?

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 30 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 and leans against a vertical wall. The ladder makes a 30.0 angle with the wall. (a) How much work does gravity do on the painter? (b) Does the answer to part (a) depend on whether the painter climbs at constant speed or accelerates up the ladder?

Two tugboats pull a disabled supertanker. Each tug exerts a constant force of 1.80 * 106 N, one 14 west of north and the other 14 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 tugboats pull a disabled supertanker. Each tug exerts a constant force of 1.80 * 106 N, one 14 west of north and the other 14 east of north, as they pull the tanker 0.75 km toward the north. What is the total work they do on the supertanker? (i) gravity, (ii) the tension in the string, (iii) friction, and (iv) the normal force? (c) Find the total work done on each block.

A loaded grocery cart is rolling across a parking lot in a strong wind. You apply a constant force \(\overrightarrow{\boldsymbol{F}}=(30 \mathrm{~N}) \hat{\imath}-(40 \mathrm{~N}) \hat{\jmath}\) 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?

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.

A 12.0-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 128.0-N carton is pulled up a frictionless baggage ramp inclined at 30.0 above the horizontal by a rope exerting a 72.0-N pull parallel to the ramps surface. If the carton travels 5.20 m along the surface of the ramp, calculate the work done on it by (a) the rope, (b) gravity, and (c) the normal force of the ramp. (d) What is the net work done on the carton? (e) Suppose that the rope is angled at 50.0 above the horizontal, instead of being parallel to the ramps surface. How much work does the rope do on the carton in this case?

A boxed 10.0-kg computer monitor is dragged by friction 5.50 m upward along 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?

A large crate sits on the floor of a warehouse. Paul and Bob apply constant horizontal forces to the crate. The force applied by Paul has magnitude 48.0 N and direction 61.0 south of west. How much work does Pauls force do during a displacement of the crate that is 12.0 m in the direction 22.0 east of north?

You apply a constant force \(\vec F = (-68.0 \ \mathrm N)\hat {i}+ (36.0 \ \mathrm N)\hat {j}\) to a 380-kg car as the car travels 48.0 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?

A large crate sits on the floor of a warehouse. Paul and Bob apply constant horizontal forces to the crate. The force applied by Paul has magnitude 48.0 N and direction 61.0 south of west. How much work does Pauls force do during a displacement of the crate that is 12.0 m in the direction 22.0 east of north?

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?

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 132 m/s2. (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?

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?

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 * 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 million tons of TNT, and 1.0 ton of TNT releases 4.184 * 109 J of energy.)

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 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?

Use the workenergy theorem to solve each of these problems. You can use Newtons laws to check your answers. (a) A skier moving at 5.00 m>s encounters a long, rough horizontal patch of snow having a coefficient of kinetic friction of 0.220 with her skis. How far does she travel on this patch before stopping? (b) Suppose the rough patch in part (a) was only 2.90 m long. How fast would the skier be moving when she reached the end of the patch? (c) At the base of a frictionless 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?

You are a member of an Alpine Rescue Team. You must project a box of supplies up an incline of constant slope angle a 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 mk. 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, mk, and a

You throw a 3.00-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

. A sled with mass 12.00 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 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 mass m slides down a smooth inclined plane from an initial vertical height h, making an angle a 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 the mass had been dropped from height h, independent of the angle a 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 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 workenergy 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 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 balls motion. Over what distance must the players foot be in contact with the ball to increase the balls speed to 6.00 m>s?

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 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 1.35 m down an inclined plane that slopes downward at an angle of 36.9 below the horizontal. If the block of ice starts from rest, what is its final speed? Ignore friction.

. 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 workenergy theorem to calculate the minimum stopping distance of the car in terms of v0, g, and the coefficient of kinetic friction mk 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 \(37.0^{\circ}\) west of north. How much work must be done on the crate to change its velocity to 5.62 m/s in a direction \(63.0^{\circ}\) 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 8.50-kg masses are hung by three identical springs (Fig. E6.35). Each spring has a force constant of 7.80 kN/m 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 \(\vec{F}\) 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. E6.36. Calculate the work done by \(\vec{F}\) when the sled moves (a) from x = 0 to x = 8.0 m; (b) from x = 8.0 m to x = 12.0 m; (c) from x = 0 to 12.0 m.

Suppose the sled in Exercise 6.36 is initially at rest at x = 0. Use the workenergy theorem to find the speed of the sled at (a) x = 8.0 m and (b) x = 12.0 m. Ignore friction between the sled and the surface of the pond.

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

A 6.0-kg box moving at 3.0 m/s on a horizontal, frictionless 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.

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 ms 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 ms = 0.60, what is the maximum initial speed v1 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 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 F S 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 (Fig. E6.43). Calculate the work done by the force F S 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 = 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 6.43 is initially at rest at x = 0 and F S 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 sleds 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 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 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?)

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?

A crate on a motorized cart starts from rest and moves with a constant eastward acceleration of a = 2.80 m>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 F1t2 = 15.40 N>s2t. 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 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?

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.

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 1150 m>min, or 500 ft>min2. 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. 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?

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 k

Effect on Blood of Walking. While a person is walking, his arms swing through approximately a 45° angle in \(\frac{1}{2} s\). As a reasonable approximation, assume that the arm moves with constant speed during each swing. A typical arm is 70.0 cm long, measured from the shoulder joint. (a) What is the acceleration of a 1.0-g drop of blood in the fingertips at the bottom of the swing? (b) Draw a free-body diagram of the drop of blood in part (a). (c) Find the force that the blood vessel must exert on the drop of blood in part (a). Which way does this force point? (d) What force would the blood vessel exert if the arm were not swinging?

You are applying a constant horizontal force F S = 1-8.00 N2nd + 13.00 N2ne to a crate that is sliding on a factory floor. At the instant that the velocity of the crate is v S = 13.20 m>s2nd + 12.20 m>s2ne, what is the instantaneous power supplied by this force?

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.

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?

A luggage handler pulls a 20.0-kg suitcase up a ramp inclined at 32.0° above the horizontal by a force \(\overrightarrow{\boldsymbol{F}}\) of magnitude 160 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 \(\overrightarrow{\boldsymbol{F}}\); (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?

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 bodys 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 mans 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

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) \(\mu_{\mathrm{s}}=0.500 \text { and } \mu_{\mathrm{k}}=0.325\) between the table and the 20.0-N block.

A 5.00-kg package slides 2.80 m down a long ramp that is inclined at \(24.0^\circ\) below the horizontal. The coefficient of kinetic friction between the package and the ramp is \(\mu k = 0.310\). 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 it has slid 2.80 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; 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 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 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>s 2 and in gs.

A net force along the x-axis that has x-component Fx = -12.0 N + 10.300 N>m2 2x2 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?

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?

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 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 7 0 when the spring is stretched and x 6 0 when it is compressed. How much work must be done (a) to stretch this spring by 0.050 m from its unstretched length? (b) 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 0.0600 kg is attached to a cord passing through a hole in a frictionless, horizontal surface (Fig. P6.71). 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 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?

Proton Bombardment. A proton with mass 1.67 * 10-27 kg is propelled at an initial speed of 3.00 * 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 = a>x2, where x is the separation between the two objects and a = 2.12 * 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 * 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 protonmoves 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?

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.

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. P6.74). At the top of the bridge, you have climbed a vertical distance of 5.20 m and slowed to 1.50 m>s. 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?

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 mk = 0.30. Use the workenergy theorem to find how far the textbook moves from its initial position before it comes to rest

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.)

One end of a horizontal spring with force constant 130.0 N/m is attached to a vertical wall. A 4.00-kg block sitting on the floor is placed against the spring. The coefficient of kinetic friction between the block and the floor is \(\mu_{\mathrm{k}}=0.400\). You apply a constant force \(\overrightarrow{\boldsymbol{F}}\) to the block. \(\overrightarrow{\boldsymbol{F}}\) has magnitude F = 82.0 N and is directed toward the wall. At the instant that the spring is compressed 80.0 cm, what are (a) the speed of the block, and (b) the magnitude and direction of the block’s acceleration?

One end of a horizontal spring with force constant 76.0 N>m is attached to a vertical post. A 2.00-kg block of frictionless ice is attached to the other end and rests on the floor. The spring is initially neither stretched nor compressed. A constant horizontal force of 54.0 N is then applied to the block, in the direction away from the post. (a) What is the speed of the block when the spring is stretched 0.400 m? (b) At that instant, what are the magnitude and direction of the acceleration of the block?

A 5.00-kg block is moving at v0 = 6.00 m>s along a frictionless, horizontal surface toward a spring with force constant k = 500 N>m that is attached to a wall (Fig. P6.79). 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 v0?

A physics professor is pushed up a ramp inclined upward at 30.0 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 professors speed at the bottom of the ramp is 2.00 m>s. Use the workenergy theorem to find her speed at the top of the ramp

Consider the system shown in Fig. P6.81. 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 0.900 m/s. The blocks come to rest after moving 2.00 m. Use the work–energy theorem to calculate the coefficient of kinetic friction between the 8.00-kg block and the tabletop.

Consider the system shown in Fig. P6.81. 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 \(\mu_{\mathrm{k}}=0.250\). 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.

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 workenergy theorem to find the length of this rough patch.

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 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?

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.)

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 * 107 J of energy in a 24-hour day, how much of the day did she spend walking?

An object has several forces acting on it. One of these forces is F S = axynd, a force in the x-direction whose magnitude depends on the position of the object, with a = 2.50 N>m2 . Calculate the work done on the object by this force for the following displacements of the object: (a) The object starts at the point 1x = 0, y = 3.00 m2 and moves parallel to the x-axis to the point 1x = 2.00 m, y = 3.00 m2. (b) The object starts at the point 1x = 2.00 m, y = 02 and moves in the y-direction to the point 1x = 2.00 m, y = 3.00 m2. (c) The object starts at the origin and moves on the line y = 1.5x to the point 1x = 2.00 m, y = 3.00 m2.

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 * 103 kg>m3 . (a) How much work does the heart do in a day? (b) What is the hearts power output in watts?

Figure P6.90 shows the results of measuring the force F exerted on both ends of a rubber band to stretch it a distance x from its unstretched position. (Source: www.sciencebuddies.org) The data points are well fit by the equation F = 33.55x0.4871, where F is in newtons and x is in meters. (a) Does this rubber band obey Hookes law over the range of x shown in the graph? Explain. (b) The stiffness of a spring that obeys Hookes law is measured by the value of its force constant k, where k = F>x. This can be written as k = dF>dx to emphasize the quantities that are changing. Define keff = dF>dx and calculate keff as a function of x for this rubber band. For a spring that obeys Hookes law, keff is constant, independent of x. Does the stiffness of this band, as measured by keff, increase or decrease as x is increased, within the range of the data? (c) How much work must be done to stretch the rubber band from x = 0 to x = 0.0400 m? From x = 0.0400 m to x = 0.0800 m? (d) One end of the rubber band is attached to a stationary vertical rod, and the band is stretched horizontally 0.0800 m from its unstretched length. A 0.300-kg object on a horizontal, frictionless surface is attached to the free end of the rubber band and released from rest. What is the speed of the object after it has traveled 0.0400 m?

In a physics lab experiment, one end of a horizontal spring that obeys Hookes law is attached to a wall. The spring is compressed 0.400 m, and a block with mass 0.300 kg is attached to it. The spring is then released, and the block moves along a horizontal surface. Electronic sensors measure the speed v of the block after it has traveled a distance d from its initial position against the compressed spring. The measured values are listed in the table. (a) The data show that the speed v of the block increases and then decreases as the spring returns to its unstretched length. Explain why this happens, in terms of the work done on the block by the forces that act on it. (b) Use the workenergy theorem to derive an expression for v2 in terms of d. (c) Use a computer graphing program (for example, Excel or Matlab) to graph the data as v2 (vertical axis) versus d (horizontal axis). The equation that you derived in part (b) should show that v2 is a quadratic function of d, so, in your graph, fit the data by a second-order polynomial (quadratic) and have the graphing program display the equation for this trendline. Use that equation to find the blocks maximum speed v and the value of d at which this speed occurs. (d) By comparing the equation from the graphing program to the formula you derived in part (b), calculate the force constant k for the spring and the coefficient of kinetic friction for the friction force that the surface exerts on the block.

For a physics lab experiment, four classmates run up the stairs from the basement to the top floor of their physics buildinga vertical distance of 16.0 m. The classmates and their masses are: Tatiana, 50.2 kg; Bill, 68.2 kg; Ricardo, 81.8 kg; and Melanie, 59.1 kg. The time it takes each of them is shown in Fig. P6.92. (a) Considering only the work done against gravity, which person had the largest average power output? The smallest? (b) Chang is very fit and has mass 62.3 kg. If his average power output is 1.00 hp, how many seconds does it take him to run up the stairs?

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 L0, and force constant k. The work done to stretch or compress the spring by a distance L is 1 2 kX2 , where X = L - L0. Consider a spring, as described above, that has one end fixed and the other end moving with speed v. 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 v. (Hint: Divide the spring into pieces of length dl; find the speed of each piece in terms of l, v, 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 1 2Mv2 , 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 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?

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 Newtons third law the air exerts a force on the wings and airplane that is up and slightly backward (Fig. P6.94). 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 v2 , so the total air resistance force can be expressed by Fair = av2 + b>v2 , where a and b 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, a = 0.30 N # s 2>m2 and b = 3.5 * 105 N # m2>s 2 . In steady flight, the engine must provide a forward force that exactly balances the air resistance force. (a) Calculate the speed 1in km>h2 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).

Based on the given data, how does the energy used in biking 1 km compare with that used in walking 1 km? Biking takes (a) \(\frac{1}{3}\) of the energy of walking the same distance; (b) the same energy as walking the same distance; (c) 3 times the energy of walking the same distance; (d) 9 times the energy of walking the same distance.

A 70-kg person walks at a steady pace of 5.0 km>h on a treadmill at a 5.0% grade. (That is, the vertical distance covered is 5.0% of the horizontal distance covered.) If we assume the metabolic power required is equal to that required for walking on a flat surface plus the rate of doing work for the vertical climb, how much power is required? (a) 300 W; (b) 315 W; (c) 350 W; (d) 370 W

How many times greater is the kinetic energy of the person when biking than when walking? Ignore the mass of the bike. (a) 1.7; (b) 3; (c) 6; (d) 9.