Solved: Youre an engineer for a company that makes bungee-jump cords, and youre asked to

Give two examples of situations in which you might think youre doing work but in which, in the technical sense, you do no work.

If the scalar product of two nonzero vectors is zero, what can you conclude about their relative directions?

Must you do work to whirl a ball around on the end of a string? Explain.

If you pick up a suitcase and put it down, how much total work have you done on the suitcase? Does your answer change if you pick up the suitcase and drop it?

You want to raise a piano a given height using a ramp. With a fixed, nonzero coefficient of friction, will you have to do more work if the ramp is steeper or more gradual? Explain.

Does the gravitational force of the Sun do work on a planet in a circular orbit? On a comet in an elliptical orbit? Explain.

A pendulum bob swings back and forth on the end of a string, describing a circular arc. Does the tension force in the string do any work?

Does your cars kinetic energy change if you drive at constant speed for 1 hour?

A watt-second is a unit of what quantity? Relate it to a more standard SI unit.

A truck is moving northward at 55 mi/h. Later, its moving eastward at the same speed. Has its kinetic energy changed? Has work been done on the truck? Has a force acted on the truck? Explain.

A news article reports that a new solar farm will produce 143 kilowatt-hours of electricity. Criticize this statement. What did the writer probably mean?

Is it possible for you to do work on an object without changing the objects kinetic energy? Explain.

How much work do you do as you exert a 75-N force to push a shopping cart through a 12-m-long supermarket aisle?

If the coefficient of kinetic friction is 0.21, how much work do you do when you slide a 50-kg box at constant speed across a 4.8-m-wide room?

A crane lifts a 650-kg beam vertically upward 23 m and then swings it eastward 18 m. How much work does the crane do? Neglect friction, and assume the beam moves with constant speed.

The worlds highest waterfall, the Cherun-Meru in Venezuela, has a total drop of 980 m. How much work does gravity do on a cubic meter of water dropping down the Cherun-Meru?

A meteorite plunges to Earth, embedding itself 75 cm in the ground. If it does 140 MJ of work in the process, what average force does the meteorite exert on the ground?

An elevator of mass m rises a vertical distance h with upward acceleration equal to one-tenth g. Find an expression for the work the elevator cable does on the elevator.

Show that the scalar product obeys the distributive law: A S # 1B S + C S 2 = A S # B S + A S # C S .

Find the work done by a force F S = 1.8 ni + 2.2 nj N as it acts on an object moving from the origin to the point 56 ni + 31 nj m.

To push a stalled car, you apply a 470-N force at 17 to the cars motion, doing 860 J of work in the process. How far do you push the car?

Find the total work done by the force shown in Fig. 6.15 as the object on which it acts moves (a) from x = 0 to x = 3 km and (b) from x = 3 km to x = 4 km. Distance (km) 10 10 0 Force (N) 2 3 4 20 30 40 50 Figure 6.15 Exercise 22

How much work does it take to stretch a spring with k = 200 N/m (a) 10 cm from equilibrium and (b) from 10 cm to 20 cm from equilibrium?

Uncompressed, the spring for an automobile suspension is 45 cm long. It needs to be fitted into a space 32 cm long. If the spring constant is 3.8 kN/m, how much work does a mechanic have to do to fit the spring?

You do 8.5 J of work to stretch a spring with k = 190 N/m, starting with the spring unstretched. How far does the spring stretch?

Spider silk is a remarkable elastic material. A particular strand has spring constant 70 mN/m, and it stretches 9.6 cm when a fly hits it. How much work did the flys impact do on the silk strand?

Whats the kinetic energy of a 2.4*105 @kg airplane cruising at 900 km/h?

A cyclotron accelerates protons from rest to 21 Mm/s. How much work does it do on each proton?

At what speed must a 950-kg subcompact car be moving to have the same kinetic energy as a 3.2*104 @kg truck going 20 km/h?

A 60-kg skateboarder comes over the top of a hill at 5.0 m/s and reaches 10 m/s at the bottom. Find the total work done on the skateboarder between the top and bottom of the hill.

After a tornado, a 0.50-g drinking straw was found embedded 4.5 cm in a tree. Subsequent measurements showed that the tree exerted a stopping force of 70 N on the straw. What was the straws speed?

From what height would you have to drop a car for its impact to be equivalent to a 20-mi/h collision?

A typical human diet is 2000 calories per day, where the calorie describing food energy is actually 1 kilocalorie. Express 2000 kcal/day in watts.

A horse plows a 200-m-long furrow in 5.0 min, exerting a 750-N force. Find its power output, measured in watts and in horsepower.

A typical car battery stores about 1 kW# h of energy. Whats its power output if it drains completely in (a) 1 minute, (b) 1 hour, and (c) 1 day?

A sprinter completes a 100-m dash in 10.6 s, doing 22.4 kJ of work. Whats her average power output?

How much work can a 3.5-hp lawnmower engine do in 1 h?

A 75-kg long-jumper takes 3.1 s to reach a prejump speed of 10 m/s. Whats his power output?

Estimate your power output as you do deep knee bends at the rate of one per second.

In midday sunshine, solar energy strikes Earth at the rate of about 1 kW/m2 . How long would it take a perfectly efficient solar collector of 15@m2 area to collect 40 kW# h of energy? (Note: This is roughly the energy content of a gallon of gasoline.)

It takes about 20 kJ to melt an ice cube. A typical microwave oven produces 900 W of microwave power. How long will it take a typical microwave to melt the ice cube?

Which consumes more energy, a 1.2-kW hair dryer used for 10 min or a 7-W night-light left on for 24 h?

You slide a box of books at constant speed up a 30 ramp, applying a force of 200 N directed up the slope. The coefficient of sliding friction is 0.18. (a) How much work have you done when the box has risen 1 m vertically? (b) Whats the mass of the box?

Two people push a stalled car at its front doors, each applying a 280-N force at 25 to the forward direction, as shown in Fig. 6.16. How much work does each person do in pushing the car 5.6 m? 25 25 Figure 6.16 Problem 44

Youre at the gym, doing arm raises. With each rep, you lift a 20-N weight 55 cm. (a) How many raises must you do before youve expended 200 kcal of work (see Exercise 33)? (b) If your workout takes 1.0 min, whats your average power output?

A locomotive does 7.9*1011 J of work in pulling a 3.4*106 @kg train 180 km. Find the average force in the coupling between the locomotive and the rest of the train.

You pull a box 23 m horizontally, using the rope shown in Fig. 6.17. If the rope tension is 120 N, and if the rope does 2500 J of work on the box, what angle u does the rope make with the horizontal? u = ? Figure 6.17 Problem 47

(a) Find the scalar products ni # ni , nj # nj , and kn # kn . (b) Find ni # nj, nj # kn , and kn # ni . (c) Use the distributive law to multiply out the scalar product of two arbitrary vectors A S = Ax ni + Ay nj + Az kn and B S = Bx ni + By nj + Bz kn, and use the results of (a) and (b) to verify Equation 6.4.

(a) Find the scalar product of the vectors ani + bnj and bni - anj, where a and b are arbitrary constants. (b) Whats the angle between the two vectors?

Looking to cut costs, the airline you work for asks you to investigate the efficiency of the tractors that push aircraft away from the gates. One model is supposed to do no more than 10 MJ of work in pushing a 747 aircraft 25 m. If the tractor exerts a 0.42-MN force, does it meet its specifications?

How much work does a force F S = 67ni + 23nj + 55k n N do as it acts on a body moving in a straight line from r ! 1 = 16ni + 31njm to r ! 2 = 21ni + 10nj + 14n k m?

A force F S acts in the x-direction, its magnitude given by F = ax 2 , where x is in meters and a = 5.0 N/m2 . Find the work done by this force as it acts on a particle moving from x = 0 to x = 6.0 m.

A certain amount of work is required to stretch spring A a certain distance. Twice as much work is required to stretch spring B half that distance. Compare the spring constants of the two.

A force with magnitude F = a1x acts in the x-direction, where a = 9.5 N/m1/2. Calculate the work this force does as it acts on an object moving from (a) x = 0 to x = 3.0 m; (b) 3.0 m to 6.0 m; and (c) 6.0 m to 9.0 m.

The force exerted by a rubber band is given approximately by F = F0 c L0 - x L0 - L 2 0 1L0 + x22 d where L0 is the unstretched length, x is the stretch, and F0 is a constant. Find an expression for the work needed to stretch the rubber band a distance x.

You put your little sister (mass m) on a swing whose chains have length L and pull slowly back until the swing makes an angle f with the vertical. Show that the work you do is mgL11 - cosf2.

Two unknown elementary particles pass through a detection chamber. If they have the same kinetic energy and their mass ratio is 4: 1, whats the ratio of their speeds?

A tractor tows a plane from its airport gate, doing 8.7 MJ of work. The link from the plane to the tractor makes a 22 angle with the planes motion, and the tension in the link is 0.41 MN. How far does the tractor move the plane?

E. coli bacteria swim by means of flagella that rotate about 100 times per second. A typical E. coli bacterium swims at 22 m/s, its flagella exerting a force of 0.57 pN to overcome the resistance due to its liquid environment. (a) Whats the bacteriums power output? (b) How much work would it do in traversing the 25-mm width of a microscope slide?

On February 15, 2013, an asteroid moving at 19 km/s entered Earths atmosphere over Chelyabinsk, Russia, and exploded at an altitude of more than 20 km. This was the largest object known to have entered the atmosphere in over a century. The asteroids kinetic energy just before entering the atmosphere was estimated as the energy equivalent of 500 kilotons of the explosive TNT. (Kilotons [kt] and megatons [Mt] are energy units used to describe the explosive yields of nuclear weapons, and youll find the energy equivalent of 1 Mt in Appendix C).What was the approximate mass of the Chelyabinsk asteroid?

An elevator ascends from the ground floor to the 10th floor, a height of 41 m, in 35 s. If the mass of the elevator and passengers is 840 kg, whats the power necessary to lift the elevator? (Your answer is greater than the actual power needed because elevators are counterweighted, thus reducing the work the motor needs to do.)

Youre asked to assess the reliability of a nuclear power plant, as measured by the capacity factorthe ratio of the energy it actually produces to what it could produce if it operated all the time. The plant is rated at 840 MW of electrical power output, and in a full year it produces 6.8*109 kW# h of electrical energy. Whats its capacity factor?

A force pointing in the x-direction is given by F = F01x/x022 , where F0 and x0 are constants and x is position. Find an expression for the work done by this force as it acts on an object moving from x = 0 to x = x0.

A force pointing in the x-direction is given by F = ax 3/2, where a = 0.75 N/m3/2. Find the work done by this force as it acts on an object moving from x = 0 to x = 14 m.

Two vectors have equal magnitude, and their scalar product is one-third the square of their magnitude. Find the angle between them.

At what rate can a half-horsepower well pump deliver water to a tank 60 m above the water level in the well? Give your answer in kg/s and gal/min.

The rate at which the United States imports oil, expressed in terms of the energy content of the imported oil, is about 600 GW. Using the Energy Content of Fuels table in Appendix C, convert this figure to gallons per day.

By measuring oxygen uptake, sports physiologists have found that long-distance runners power output is given approximately by P = m1bv - c2, where m and v are the runners mass and speed, and b and c are constants given by b = 4.27 J/kg # m and c = 1.83 W/kg. Determine the work done by a 54-kg runner who runs a 10-km race at 5.2 m/s.

Youre writing performance specifications for a new car model. The 1750-kg car delivers energy to its drive wheels at the rate of 35 kW. Neglecting air resistance, what do you list for the greatest speed at which it can climb a 4.5 slope?

A 1400-kg car ascends a mountain road at a steady 60 km/h, against a 450-N force of air resistance. If the engine supplies energy to the drive wheels at the rate of 38 kW, whats the slope angle of the road?

You do 2.2 kJ of work pushing a 78-kg trunk at constant speed 3.1 m along a ramp inclined upward at 22. Whats the frictional coefficient between trunk and ramp?

(a) Find the work done in lifting 1 L of blood (mass 1 kg) from the foot to the head of a 1.7-m-tall person. (b) If blood circulates through the body at the rate of 5.0 L/min, estimate the hearts power output. (Your answer underestimates the power by a factor of about 5 because it neglects fluid friction and other factors.)

(a) What power is needed to push a 95-kg crate at 0.62 m/s along a horizontal floor where the coefficient of friction is 0.78? (b) How much work is done in pushing the crate 11 m?

You mix flour into bread dough, exerting a 45-N force on the spoon, which you move at 0.29 m/s. (a) What power do you supply? (b) How much work do you do if you stir for 1.0 min?

A machine does work at a rate given by P = ct2, where c = 18 W/s2 and t is time. Find the work done between t = 10 s and t = 20 s.

A typical bumblebee has mass 0.25 mg. It beats its wings 100 times per second, and the wings undergo an average displacement of about 1.5 mm. When the bee is hovering over a flower, the average force between wings and air must support the bees weight. Estimate the average power the bee expends in hovering.

Youre trying to decide whether to buy an energy-efficient 225-W refrigerator for $1150 or a standard 425-W model for $850. The standard model will run 20% of the time, but better insulation means the energy-efficient model will run 11% of the time. If electricity costs 9.5./kW# h, how long would you have to own the energy-efficient model to make up the difference in cost? Neglect interest you might earn on your money.

Your friend does five reps with a barbell, on each rep lifting 45 kg 0.50 m. She claims the work done is enough to burn off a chocolate bar with energy content 230 kcal (see Exercise 33). Is that true? If not, how many lifts would it take?

A machine delivers power at a decreasing rate P = P0t 2 0 /1t + t022 , where P0 and t0 are constants. The machine starts at t = 0 and runs forever. Show that it nevertheless does only a finite amount of work, equal to P0t0.

A locomotive accelerates a freight train of total mass M from rest, applying constant power P. Determine the speed and position of the train as functions of time, assuming all the power goes to increasing the trains kinetic energy.

A force given by F = b/1x acts in the x-direction, where b is a constant with the units N# m1/2. Show that even though the force becomes arbitrarily large as x approaches zero, the work done in moving from x1 to x2 remains finite even as x1 approaches zero. Find an expression for that work in the limit x1 S 0.

Youre assisting a cardiologist in planning a stress test for a 75-kg patient. The test involves rapid walking on an inclined treadmill, and the patient is to reach a peak power output of 350 W. If the patients maximum walking speed is 8.0 km/h, what should be the treadmills inclination angle?

Youre an engineer for a company that makes bungee-jump cords, and youre asked to develop a formula for the work involved in stretching cords to double their length. Your cords have forcedistance relations described by F = -1kx + bx 2 + cx 3 + dx 4 2, where k, b, c, and d are constants. (a) Given a cord with unstretched length L0, whats your formula? (b) Evaluate the work done in doubling the stretch of a 10-m cord with k = 420 N/m, b = -86 N/m2 , c = 12 N/m3 , and d = -0.50 N/m4 .

You push an object of mass m slowly, partway up a loop-the-loop track of radius R, starting from the bottom, and ending at a height h 6 R above the bottom. The coefficient of friction between the object and the track is a constant m. Show that the work you do against friction is mmg22hR - h2 .

A particle moves from the origin to the point x = 3 m, y = 6 m along the curve y = ax 2 - bx, where a = 2 m-1 and b = 4. Its subject to a force cxyni + dnj, where c = 10 N/m2 and d = 15 N. Calculate the work done by the force.

Repeat Problem 85 for the following cases: (a) the particle moves first along the x-axis from the origin to the point (3 m, 0) and then parallel to the y-axis until it reaches (3 m, 6 m); (b) it moves first along the y-axis from the origin to the point (0, 6 m) and then parallel to the x-axis until it reaches (3 m, 6 m).

The worlds fastest elevator, in Taiwans Taipei 101 skyscraper (Fig. 6.18), ascends at the rate of 1010 m/min. Counterweights balance the weight of the elevator car, so the motor doesnt have to lift the cars weight. If the motor produces 330 kW of power, whats the maximum number of 67-kg people the elevator can accommodate? (Your answer somewhat overestimates the actual rated load of 24 people.) Controller Governor Traction machine Aerodynamic capsule Airtight soundproong car (atmospheric pressure control) Safety Buffer Figure 6.18 Problem 87

An experimental measurement of the force required to stretch a slingshot is given in the table below. Plot the forcedistance curve for this slingshot and use graphical integration to determine the work done in stretching the slingshot the full 40-cm distance. Stretch (cm) Force (N) 0 0 5.00 0.885 10.0 1.89 15.0 3.05 20.0 4.48 25.0 6.44 30.0 8.22 35.0 9.95 40.0 12.7

Youre an expert witness in a medical malpractice lawsuit. A hospital patients leg slipped off a stretcher and his heel hit the floor. The defense attorney for the hospital claims the leg, with mass 8 kg, hit the floor with a force equal to the weight of the legabout 80 Nand any damage was due to a prior injury. You argue that the leg and heel dropped freely for 0.7 m, then hit the floor and stopped in 2 cm. What do you tell the jury about the force on the heel?

The energy in a batted baseball comes from the power delivered while the bat is in contact with the ball. The most powerful hitters can supply some 10 horsepower during the brief contact time, propelling the ball to over 100 miles per hour. Figure 6.19 shows data taken from a particular hit, giving the power the bat delivers to the ball as a function of time. Which of the following is greatest at the peak of the curve? a. the balls kinetic energy b. the balls speed c. the rate at which the bat supplies energy to the ball d. the total work the bat has done on the ball

The energy in a batted baseball comes from the power delivered while the bat is in contact with the ball. The most powerful hitters can supply some 10 horsepower during the brief contact time, propelling the ball to over 100 miles per hour. Figure 6.19 shows data taken from a particular hit, giving the power the bat delivers to the ball as a function of time. The ball has its maximum speed at about a. 85 ms. b. 145 ms. c. 185 ms. d. whenever the force is greatest.

The energy in a batted baseball comes from the power delivered while the bat is in contact with the ball. The most powerful hitters can supply some 10 horsepower during the brief contact time, propelling the ball to over 100 miles per hour. Figure 6.19 shows data taken from a particular hit, giving the power the bat delivers to the ball as a function of time. As a result of being hit, the balls kinetic energy increases by about a. 550 J. b. 1.3 kJ. c. 7.0 kJ. d. You cant tell because you dont know its speed coming from the pitcher.

The energy in a batted baseball comes from the power delivered while the bat is in contact with the ball. The most powerful hitters can supply some 10 horsepower during the brief contact time, propelling the ball to over 100 miles per hour. Figure 6.19 shows data taken from a particular hit, giving the power the bat delivers to the ball as a function of time. The force on the ball is greatest approximately a. at 185 ms. b. at the peak in Fig. 6.19. c. before the peak in Fig. 6.19. d. after the peak in Fig. 6.19 but before 185 ms.