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**Hugh D. Young, College Physics, 9th edition, Addison Wesley 2011**

**Chapter 9**

Multiple-Choice Problems

1. When a wheel turns through one complete rotation, the angle (in radians) that it has turned through is closest to

A. 57. B. 6. C. 360.

2. Two points are on a disk that rotates about an axis perpendicular to the plane of the disk at its center. Point B is 3 times as far from the axis as point A. If the linear speed of point B is V, then the linear speed of point A is

A. 9V. B. 3V. C. V. D. V/3 E. V/9

3. A bicycle wheel rotating at a rate of 12 rad/s begins to accelerate at a rate of 2 rad/s2. After 5 seconds the rate of rotation will be

A. 2 rad/s B. 10 rad/s C. 17 rad/s D. 22 rad/s

4. You are designing a flywheel to store large amounts of kinetic energy. Which one of the following uniform shapes will be the most effective for storing the greatest amount of kinetic energy if all the objects have the same mass and same angular velocity?

A. A solid sphere of diameter D rotating about a diameter.

B. A solid cylinder of diameter D rotating about an axis perpendicular to each face through its center.

C. A thin-walled hollow cylinder of diameter D rotating about an axis perpendicular to the plane of the circular face at its center.

D. A solid thin bar of length D rotating about an axis perpendicular to it at its center.

5. Four uniform objects having the same mass and diameter are released simultaneously from rest at the same distance above the bottom of a hill and roll down without slipping. The objects are a solid sphere, a solid cylinder, a hollow cylinder, and a thin-walled hollow cylinder, as illustrated in Table 9.2. Which of these objects will be the first one to reach the bottom of the hill?

A. The solid sphere

B. The solid cylinder

C. The hollow cylinder

D. The thin-walled hollow cylinder

6. Which of the objects in question 5 will be the last one to reach the bottom of the hill?

A. The solid sphere

B. The solid cylinder

C. The hollow cylinder

D. The thin-walled hollow cylinder

7. For the objects in question 5, which of the following statements are correct? (There may be more than one correct choice.)

A. All these objects have the same forward speed at the bottom of the hill.

B. All these objects have the same kinetic energy at the bottom of the hill.

C. All these objects have the same rotational kinetic energy at the bottom of the hill.

D. All these objects have the same angular velocity at the bottom of the hill.

8. Two uniform solid spheres of the same size, but different mass, are released from rest simultaneously at the same height on a hill and roll to the bottom without slipping. Which of the following statements about these spheres are true? (There may be more than one correct choice.)

A. Both spheres reach the bottom at the same time.

B. The heavier sphere reaches the bottom ahead of the lighter one.

C. Both spheres arrive at the bottom with the same forward speed.

D. Both spheres arrive at the bottom with the same total kinetic energy.

9. A disk starts from rest and has a constant angular acceleration. If it takes time T to make its first revolution, in time 2T (starting from rest) the disk will make

A. ?2 revolutions.

B. 2 revolutions.

C. 4 revolutions.

D. 8 revolutions.

10. Two unequal masses m and 2m are attached to a thin bar of negligible mass that rotates about an axis perpendicular to the bar. When m is a distance 2d from the axis and 2m is a distance d from the axis, the moment of inertia of this combination is I. If the masses are now interchanged, the moment of inertia will be

A. (2/3)I B. I. C. (3/2)I D. 2I. E. 4I.

11. A thin uniform bar has a moment of inertia I about an axis perpendicular to it through its center. If both the mass and length of this bar are doubled, its moment of inertia about the same axis will be

A. 2l. B. 4l. C. 8l. D. 16l.

12. Two small objects of equal weight are attached to the ends of a thin weightless bar that spins about an axis perpendicular to the bar at its center. Each mass is a distance d from the axis of rotation. The system spins at a rate so that its kinetic energy is K. If both masses are now moved closer in so that each is from the rotation axis, while the angular velocity does not change, the kinetic energy of the system will be

A. (1/4)K B. (1/2)K C. 2K. D. 4K.

13. A disk starts from rest and rotates with constant angular acceleration. If the angular velocity is at the end of the first two revolutions, then at the end of the first eight revolutions it will be

A. ?2? B. 2? C. 4? D. 16?

14. Two identical merry-go-rounds are rotating at the same speed. One is crowded with riding children; the other is nearly empty. If both merry-go-rounds cut off their motors at the same time

and coast to a stop, slowed only by friction (which you can assume is the same for both merry-go-rounds), which will take longer to stop?

A. The crowded merry-go-round

B. The empty merry-go-round

C. The same time for both.

15. A solid sphere and a hollow sphere, both uniform and having the same mass and radius, are released from rest at the same height on a ramp. If they roll down the ramp without slipping, which one will have the greater forward velocity at the bottom of the ramp?

A. The solid sphere

B. The hollow sphere

C. Both will have the same forward velocity.

16. A uniform ball rolls without slipping toward a hill with a forward speed V. In which case will this ball go farther up the hill?

A. There is enough friction on the hill to prevent slipping of the ball.

B. The hill is frictionless.

C. The ball will reach the same height in both cases, since it has the same initial kinetic energy in each case.

**Problems**

1. A flexible straight wire 75.0 cm long is bent into the arc of a circle of radius 2.50 m. What angle (in radians and degrees) will this arc subtend at the center of the circle?

2. (a) What angle in radians is subtended by an arc 1.50 m in length on the circumference of a circle of radius 2.50 m? What is this angle in degrees? (b) An arc 14.0 cm in length on the circumference of a circle subtends an angle of 128°. What is the radius of the circle? (c) The angle between two radii of a

circle with radius 1.50 m is 0.700 rad. What length of arc is intercepted on the circumference of the circle by the two radii?

3. (a) Calculate the angular velocity (in rad/s) of the second, minute, and hour hands on a wall clock. (b) What is the period of each of these hands?

4. The once-popular LP (long-play) records were 12 in. in diameter and turned at a constant rpm. Find (a) the angular speed of the LP in rad/s and (b) its period in seconds.

5. If a wheel 212 cm in diameter takes 2.25 s for each revolution, find its (a) period and (b) angular speed in rad/s.

6. Find the angular velocity, in rad/s, of (a) the earth due to its daily spin on its axis, (b) the earth due to its yearly motion around the sun, and (c) our moon due to its monthly motion around the earth. Consult Appendix E as needed.

7. A laser beam aimed from the earth is swept across the face of the moon. (a) If the beam is rotated at an angular velocity of 1.50 ? 10–3 rad/s, at what speed does the laser light move across the moon’s surface? (b) If the diameter of the laser spot on the moon is 6.00 km, what is the angle of divergence of the laser beam?

8. Communications satellites. Communications satellites are placed in orbits so that they always remain above the same point of the earth’s surface. (a) What must be the period of such a satellite? (b) What is its angular velocity in rad/s.

9. An airplane propeller is rotating at 1900 rpm. (a) Compute the propeller’s angular velocity in rad/s (b) How many seconds does it take for the propeller to turn through 35°? (c) If the propeller were turning at 18 rad/s, at how many rpm would it be turning? (d) What is the period (in seconds) of this propeller?

10. A wall clock on Planet X has two hands that are aligned at midnight and turn in the same direction at uniform rates, one at 0.0425 rad/s and the other at 0.0163 rad/s. At how many seconds after midnight are these hands (a) first aligned and (b) next aligned?

11. A turntable that spins at a constant 78.0 rpm takes 3.50 s to reach this angular speed after it is turned on. Find (a) its angular acceleration (in rad/s2) assuming it to be constant, and (b) the number of degrees it turns through while speeding up.

12. When the power is turned off on a turntable spinning at 78.0 rpm, you find that it takes 10.5 revolutions for it to stop while slowing down at a uniform rate. (a) What is the angular acceleration (in of this turntable? (b) How long does it take to stop after the power is turned off?

13. DVDs. The angular speed of digital video discs (DVDs) varies with whether the inner or outer part of the disc is being read. (CDs function in the same way.) Over a 133 min playing time, the angular speed varies from 570 rpm to 1600 rpm. Assuming it to be constant, what is the angular acceleration (in rad/s2) of such a DVD?

14. A circular saw blade 0.200 m in diameter starts from rest. In 6.00 s, it reaches an angular velocity of 140 rad/s with constant angular acceleration. Find the angular acceleration and the angle through which the blade has turned in this time.

15. A wheel turns with a constant angular acceleration of 0.640 rad/s2. (a) How much time does it take to reach an angular velocity of 8.00 rad/s starting from rest? (b) Through how many revolutions does the wheel turn in this interval?

16. An electric fan is turned off, and its angular velocity decreases uniformly from 500.0 rev/min to 200.0 rev/min in 4.00 s. (a) Find the angular acceleration in rev/s2 and the number of revolutions made by the motor in the 4.00 s interval. (b) How many more seconds are required for the fan to come to rest if the angular acceleration remains constant at the value calculated in part (a)?

17. A flywheel in a motor is spinning at 500.0 rpm when a power failure suddenly occurs. The flywheel has mass 40.0 kg and diameter 75.0 cm. The power is off for 30.0 s, and during this time the flywheel slows down uniformly due to friction in its axle bearings. During the time the power is off, the flywheel makes 200.0 complete revolutions. (a) At what rate is the flywheel spinning when the power comes back on? (b) How long after the beginning of the power failure would it have taken the flywheel to stop if the power had not come back on, and how many revolutions would the wheel have made during this time?

18. A flywheel having constant angular acceleration requires 4.00 s to rotate through 162 rad. Its angular velocity at the end of this time is 108 rad/s. Find (a) the angular velocity at the beginning of the 4.00 s interval; (b) the angular acceleration of the flywheel.

19. Emilie’s potter’s wheel rotates with a constant 2.25 rad/s2 angular acceleration. After 4.00 s, the wheel has rotated through an angle of 60.0 rad. What was the angular velocity of the wheel at the beginning of the 4.00 s interval?

20. Derive Eq. 9.12 by combining Eqs. 9.7 and 9.11 to eliminate t.

21. A car is traveling at a speed of 63 mi/h on a freeway. If its tires have diameter 24.0 in and are rolling without sliding or slipping, what is their angular velocity?

22. (a) A cylinder 0.150 m in diameter rotates in a lathe at 620 rpm. What is the tangential speed of the surface of the cylinder? (b) The proper tangential speed for machining cast iron is about 0.600 m/s. At how many revolutions per minute should a piece of stock 0.0800 m in diameter be rotated in a lathe to produce this tangential speed?

23. A wheel rotates with a constant angular velocity of 6.00 rad/s. (a) Compute the radial acceleration of a point 0.500 m from the axis, using the relation arad = ?2r. (b) Find the tangential speed of the point, and compute its radial acceleration from the relation arad = v2/r.

24. Ultracentrifuge. Find the required angular speed (in rpm) of an ultracentrifuge for the radial acceleration of a point 2.50 cm from the axis to equal 400,000g.

25. Exercise! An exercise bike that you pedal in place has a bicycle chain connecting the wheel you pedal to the large wheel in front, as shown in Figure 9.24. For the wheel diameters shown, how many rpm must you produce to turn the large wheel at 75 rpm?

26. A flywheel with a radius of 0.300 m starts from rest and accelerates with a constant angular acceleration of 0.600 m/s2. Compute the magnitude of the tangential acceleration, the radial acceleration, and the resultant acceleration of a point on its rim (a) at the start, (b) after it has turned through 60.0°, and (c) after it has turned through 120.0°.

27. Electric drill. According to the shop manual, when drilling a 12.7-mm-diameter hole in wood, plastic, or aluminum, a drill should have a speed of 1250 rev/min. For a 12.7-mm-diameter drill bit turning at a constant 1250 rev/min find (a) the maximum linear speed of any part of the bit and (b) the maximum radial acceleration of any part of the bit.

28. Dental hygiene. Electric toothbrushes can be effective in removing dental plaque. One model consists of a head 1.1 cm in diameter that rotates back and forth through a 70.0° angle 7600 times per minute. The rim of the head contains a thin row of bristles. (a) What is the average angular speed in each direction of the rotating head, in rad/s? (b) What is the average linear speed in each direction of the bristles against the teeth? (c) Using your own observations, what is the approximate speed of the bristles against your teeth when you brush by hand with an ordinary toothbrush?

29. The spin cycles of a washing machine have two angular speeds, 423 rev/min and 640 rev/min. The internal diameter of the drum is 0.470 m. (a) What is the ratio of the maximum radial force on the laundry for the higher angular speed to that for the lower speed? (b) What is the ratio of the maximum tangential speed of the laundry for the higher angular speed to that for the lower speed? (c) Find the laundry’s maximum tangential speed and the maximum radial acceleration, in terms of g.

30. A twirler’s baton is made of a slender metal cylinder of mass M and length L. Each end has a rubber cap of mass m, and you can accurately treat each cap as a particle in this problem. Find the total moment of inertia of the baton about the usual twirling axis (perpendicular to the baton through its center).

31. A thin uniform bar has two small balls glued to its ends. The bar is 2.00 m long and has mass 4.00 kg, while the balls each have mass 0.500 kg and can be treated as point masses. Find the moment of inertia of this combination about each of the following axes: (a) an axis perpendicular to the bar through its center; (b) an axis perpendicular to the bar through one of the balls; (c) an axis parallel to the bar through both balls.

32. Use the formulas of Table 9.2 to find the moment of inertia about each of the following axes for a rod that is 0.300 cm in diameter and 1.50 m long, with a mass of 0.0420 kg: (a) about an axis perpendicular to the rod and passing through its center; (b) about an axis perpendicular to the rod and passing through one end; and (c) about an axis along the length of the rod.

33. Four small 0.200 kg spheres, each of which you can regard as a point mass, are arranged in a square 0.400 m on a side and connected by light rods. Find the moment of inertia of the system about an axis (a) through the center of the square, perpendicular to its plane at point O; (b) along the line AB; and (c) along the line CD.

34. Compound objects. Moment of inertia is a scalar. Therefore, if several objects are connected together, the moment of inertia of this compound object is simply the scalar (algebraic) sum of the moments of inertia of each of the component objects. Use this principle to answer each of the following questions about the moment of inertia of compound objects: (a) A thin uniform 2.50 kg bar 1.50 m long has a small 1.25 kg mass glued to each end. What is the moment of inertia of this object about an axis perpendicular to the bar through its center? (b) What is the moment of inertia of the object in part (a) about an axis perpendicular to the bar at one end? (c) A 725 g metal wire is bent into the shape of a hoop 60.0 cm in diameter. Six wire spokes, each of mass 112 g, are added from the center of the hoop to the rim. What is the moment of inertia of this object about an axis perpendicular to it through its center?

35. Energy from the Moon? Suppose that sometime in the future we decide to tap the moon’s rotational energy for use on earth. In additional to the astronomical data in Appendix E, you may need to know that the moon spins on its axis once every 27.3 days. Assume that the moon is uniform throughout. (a) How much total energy could we get from the moon’s rotation? (b) The world presently uses about of energy per year. If in the future the world uses five times as much energy yearly, for how many years would the moon’s rotation provide us energy? In light of your answer, does this seem like a cost-effective energy source in which to invest?

36. A wagon wheel is constructed as shown in Figure 9.27. The radius of the wheel is 0.300 m, and the rim has a mass of 1.40 kg. Each of the wheel’s eight spokes, which come out from the center and are 0.300 m long, has a mass of 0.280 kg. What is the moment of inertia of the wheel about an axis through its center and perpendicular to the plane of the wheel?

37. You need to design an industrial turntable that is 60.0 cm in diameter and has a kinetic energy of 0.250 J when turning at 45.0 rpm (a) What must be the moment of inertia of the turntable about the rotation axis? (b) If your workshop makes this turntable in the shape of a uniform solid disk, what must be its mass?

38. A grinding wheel in the shape of a solid disk is 0.200 m in diameter and has a mass of 3.00 kg. The wheel is rotating at 2200 rpm about an axis through its center. (a) What is its kinetic energy? (b) How far would it have to drop in free fall to acquire the same amount of kinetic energy?

39. The flywheel of a gasoline engine is required to give up 500 J of kinetic energy while its angular velocity decreases from 650 rev/min to 520 rev/min. What moment of inertia is required?

40. An airplane propeller is 2.08 m in length (from tip to tip) with mass 117 kg and is rotating at 2400 rpm about an axis through its center. You can model the propeller as a slender rod. (a) What is its rotational kinetic energy? (b) Suppose that, due to weight constraints, you had to reduce the propeller’s mass to 75.0% of its original mass, but you still needed to keep the same size and kinetic energy. What would its angular speed have to be, in rpm?

41. Storing energy in flywheels. It has been suggested that we should use our power plants to generate energy in the off-hours (such as late at night) and store it for use during the day. One idea put forward is to store the energy in large flywheels. Suppose we want to build such a flywheel in the shape of a hollow cylinder of inner radius 0.500 m and outer radius 1.50 m, using concrete of density 2.20 ? 103 kg/m3 (a) If, for stability, such a heavy flywheel is limited to 1.75 second for each revolution and has negligible friction at its axle, what must be its length to store 2.5 MJ of energy in its rotational motion? (b) Suppose that by strengthening the frame you could safely double the flywheel’s rate of spin. What length of flywheel would you need in that case?

42. A light string is wrapped around the outer rim of a solid uniform cylinder of diameter 75.0 cm that can rotate without friction about an axle through its center. A 3.00 kg stone is tied to the free end of the string, as shown in Figure 9.28. When the system is released from rest, you determine that the stone reaches a speed of 3.50 m/s after having fallen 2.50 m. What is the mass of the cylinder?

43. A solid uniform 3.25 kg cylinder, 65.0 cm in diameter and 12.4 cm long, is connected to a 1.50 kg weight over two massless frictionless pulleys as shown in Figure 9.29. The cylinder is free to rotate about an axle through its center perpendicular to its circular faces, and the system is released from rest. (a) How far must the 1.50 kg weight fall before it reaches a speed of 2.50 m/s? (b) How fast is the cylinder turning at this instant?

44. A compound disk of outside diameter 140.0 cm is made up of a uniform solid disk of radius 50.0 cm and area density 3.00 g/cm2 surrounded by a concentric ring of inner radius 50.0 cm, outer radius 70.0 cm, and area density 2.00 g/cm2. Find the moment of inertia of this object about an axis perpendicular to the plane of the object and passing through its center.

45. Gymnastics. We can roughly model a gymnastic tumbler as a uniform solid cylinder of mass 75 kg and diameter 1.0 m. If this tumbler rolls forward at 0.50 rev/s (a) how much total kinetic energy does he have and (b) what percent of his total kinetic energy is rotational?

46. A bicycle racer is going downhill at 11.0 m/s when, to his horror, one of his 2.25 kg wheels comes off when he is 75.0 m above the foot of the hill. We can model the wheel as a thin-walled cylinder 85.0 cm in diameter and neglect the small mass of the spokes. (a) How fast is the wheel moving when it reaches the foot of the hill if it rolled without slipping all the way down? (b) How much total kinetic energy does the wheel have when it reaches the bottom of the hill?

47. A 2.20-kg hoop 1.20 m in diameter is rolling to the right without slipping on a horizontal floor at a steady (a) How fast is its center moving? (b) What is the total kinetic energy of the hoop?

48. A solid uniform sphere and a uniform spherical shell, both having the same mass and radius, roll without slipping down a hill that rises at an angle ? above the horizontal. Both spheres start from rest at the same vertical height h. (a) How fast is each sphere moving when it reaches the bottom of the hill? (b) Which sphere will reach the bottom first, the hollow one or the solid one?

49. A size-5 soccer ball of diameter 22.6 cm and mass 426 g rolls up a hill without slipping, reaching a maximum height of 5.00 m above the base of the hill. We can model this ball as a thin-walled hollow sphere. (a) At what rate was it rotating at the base of the hill? (b) How much rotational kinetic energy

did it then have?

50. A solid uniform marble and a block of ice, each with the same mass, start from rest at the same height H above the bottom of a hill and move down it. The marble rolls without slipping, but the ice slides without friction. (a) Find the speed of each of these objects when it reaches the bottom of the hill. (b) Which object is moving faster at the bottom, the ice or the marble? (c) Which object has more kinetic energy at the bottom, the ice or the marble?

51. What fraction of the total kinetic energy is rotational for the following objects rolling without slipping on a horizontal surface? (a) a uniform solid cylinder; (b) a uniform sphere; (c) a thin-walled, hollow sphere; (d) a hollow cylinder with outer radius R and inner radius R/2

52. A string is wrapped several times around the rim of a small hoop with a radius of 0.0800 m and a mass of 0.180 kg. If the free end of the string is held in place and the hoop is released from rest (see Figure 9.30), calculate the angular speed of the rotating hoop after it has descended 0.750 m.

53. An apparatus for launching a small boat consists of a 150.0 kg cart that rides down a set of tracks on four solid steel wheels, each with radius 20.0 cm and mass 45.0 kg. The tracks slope at an angle of 7.50° to the horizontal, and the boat’s mass is 750.0 kg. If the boat is released from rest a distance of 16.0 m from the water (measured along the slope), how fast will it be moving when it reaches the water? Assume the wheels roll without slipping, and that there is no energy loss due to friction.

54. A uniform marble rolls down a symmetric bowl, starting from rest at the top of the left side. The top of each side is a distance h above the bottom of the bowl. The left half of the bowl is rough enough to cause the marble to roll without slipping, but the right half has no friction because it is coated with oil. (a) How far up the smooth side will the marble go, measured vertically from the bottom? (b) How high would the marble go if both sides were as rough as the left side? (c) How do you account for the fact that the marble goes higher with friction on the right side than without friction?

55. A 7300 N elevator is to be given an acceleration of 0.150g by connecting it to a cable of negligible weight wrapped around a turning cylindrical shaft. If the shaft’s diameter can be no larger than 16.0 cm due to space limitations, what must be its minimum angular acceleration to provide the required acceleration of the elevator?

56. A 392-N wheel comes off a moving truck and rolls without slipping along a highway. At the bottom of a hill it is rotating at 25.0 rad/s. The radius of the wheel is 0.600 m, and its moment of inertia about its rotation axis is 0.800MR2. Friction does work on the wheel as it rolls up the hill to a stop, a height h above the bottom of the hill; this work has absolute value 3500 J. Calculate h.

57. Odometer. The odometer (mileage gauge) of a car tells you the number of miles you have driven, but it doesn’t count the miles directly. Instead, it counts the number of revolutions of your car’s wheels and converts this quantity to mileage, assuming a standard size tire and that your tires do not slip on the pavement. (a) A typical midsize car has tires 24 inches in diameter. How many revolutions of the wheels must the odometer count in order to show a mileage of 0.10 mile? (b) What will the odometer read when the tires have made 5,000 revolutions? (c) Suppose you put oversize 28-inch-diameter tires on your car. How many miles will you really have driven when your odometer reads 500 miles?

58. Speedometer. Your car’s speedometer works in much the same way as its odometer (see the previous problem), except that it converts the angular speed of the wheels to a linear speed of the car, assuming standard-size tires and no slipping on the pavement. (a) If your car has standard 24-inch-diameter tires, how fast are your wheels turning when you are driving at a freeway speed of 55 mph? (b) How fast are you going when your wheels are turning at 500 rpm? (c) If you put on undersize 20-inch-diameter tires, what will the speedometer read when you are actually traveling at 50mph?

59. When a toy car is rapidly scooted across the floor, it stores energy in a flywheel. The car has mass 0.180 kg, and its flywheel has moment of inertia 4.00 ? 10–5 kg m2. The car is 15.0 cm long. An advertisement claims that the car can travel at a scale speed of up to 700 km/h. The scale speed is the speed of the toy car multiplied by the ratio of the length of an actual car to the length of the toy. Assume a length of 3.0 m for a real car. (a) For a scale speed of 700 km/h, what is the actual translational speed of the car? (b) If all the kinetic energy that is initially in the flywheel is converted to the translational kinetic energy of the toy, how much energy is originally stored in the flywheel? (c) What initial angular velocity of the flywheel was needed to store the amount of energy calculated in part (b)?

60. A passenger bus in Zurich, Switzerland, derived its motive power from the energy stored in a large flywheel. Whenever the bus was stopped at a station, the wheel was brought up to speed with the use of an electric motor that could then be attached to the electric power lines. The flywheel was a solid cylinder with a mass of 1000 kg and a diameter of 1.80 m; its top angular speed was 3000 rev/min. At this angular speed, what was the kinetic energy of the flywheel?

61. Kinetic energy of the earth. Consult Appendix E as necessary. Assuming that the earth is of uniform density inside, find the kinetic energy of the earth (a) due to its yearly motion around the sun and (b) due to its daily spin on its axis. (c) What percent of the earth’s total kinetic energy is contained in its axial spin motion?

62. Compact discs. When a compact disc (CD) is playing, the angular speed of the turntable is adjusted so that the laser beam, which reads the digital information encoded in the surface of the disc, maintains a constant tangential speed. The laser begins tracking at the inside of the disc and spirals outward as the disc plays, while the angular speed of the disc varies between 500 rpm and 200 rpm. A certain CD (see Figure 9.31) plays for 74 minutes and has a playing area of outer diameter 12.1 cm (which is also essentially the outside diameter of the disc). (a) What is the tangential speed of the laser beam? (b) What is the diameter of the inside of the playing area of this CD? (c) What is the angular acceleration of the CD while it is playing, assuming it to be constant?

63. A vacuum cleaner belt is looped over a shaft of radius 0.45 cm and a wheel of radius 2.00 cm. The motor turns the shaft at 60.0 rev/s and the moving belt turns the wheel, which in turn is connected by another shaft to the roller that beats the dirt out of the rug being vacuumed (see Figure 9.32). Assume

that the belt doesn’t slip on either the shaft or the wheel. (a) What is the speed of a point on the belt? (b) What is the angular velocity of the wheel, in rad/s.

64. A basketball (which can be closely modeled as a hollow spherical shell) rolls down a mountainside into a valley and then up the opposite side, starting from rest at a height H0 above the bottom. In Figure 9.33, the rough part of the terrain prevents slipping while the smooth part has no friction. (a) How high, in terms of H0, will it go up the other side? (b) Why doesn’t the ball return to height H0? Has it lost any of its original potential energy?

65. Human rotational energy. A dancer is spinning at 72 rpm about an axis through her center with her arms outstretched, as shown in Figure 9.34. From biomedical measurements, the typical distribution of mass in a human body is as follows: Head: 7.0%, Arms: 13% (for both), Trunk and legs: 80.0%. Suppose you are this dancer. Using this information plus length measurements on your own body, calculate (a) your moment of inertia about your spin axis and (b) your rotational kinetic energy. Use the figures in Table 9.2 to model reasonable approximations for the pertinent parts of your body.

66. A solid uniform spherical boulder rolls down the hill as shown in Figure 9.35, starting from rest when it is 50.0 m above the bottom. The upper half of the hill is free of ice, so the boulder rolls without slipping. But the lower half is covered with perfectly smooth ice. How fast is the boulder moving when it reaches the bottom of the hill?

67. A thin uniform rod 50.0 cm long with mass 0.320 kg is bent at its center into a V shape, with a 70.0° angle at its vertex. Find the moment of inertia of this V-shaped object about an axis perpendicular to the plane of the V at its vertex.

68. In redesigning a piece of equipment, you need to replace a solid spherical part of mass M with a hollow spherical shell of the same size. If both parts must spin at the same rate about an axis through their center, and the new part must have the same kinetic energy as the old one, what must be the mass of the new part in terms of M?

69. A solid uniform spherical stone starts moving from rest at the top of a hill. At the bottom of the hill the ground curves upward, launching the stone vertically a distance H below its start. How high will the stone go (a) if there is no friction on the hill and (b) if there is enough friction on the hill for the stone to roll without slipping? (c) Why do you get two different answers even though the stone starts with the same gravitational potential energy in both cases?

70. A solid, uniform ball rolls without slipping up a hill, as shown in Figure 9.36. At the top of the hill, it is moving horizontally; then it goes over the vertical cliff. (a) How far from the foot of the cliff does the ball land, and how fast is it moving just before it lands? (b) Notice that when the ball lands, it has a larger translational speed than it had at the bottom of the hill. Does this mean that the ball somehow gained energy by going up the hill? Explain!

71. The kinetic energy of walking. If a person of mass M simply moved forward with speed V, his kinetic energy would be ? MV2. However, in addition to possessing a forward motion, various parts of his body (such as the arms and legs) undergo rotation. Therefore, his total kinetic energy is the sum of the energy from his forward motion plus the rotational kinetic energy of his arms and legs. The purpose of this problem is to see how much this rotational motion contributes to the person’s kinetic energy. Biomedical measurements show that the arms and hands together typically make up 13% of a person’s mass, while the legs and feet together account for 37%. For a rough (but reasonable) calculation, we can model the arms and legs as thin uniform bars pivoting about the shoulder and hip, respectively. In a brisk walk, the arms and legs each move through an angle of about ±30.0° (a total of 60°) from the vertical in approximately 1 second. We shall assume that they are held straight, rather than being bent, which is not quite true. Let us consider a 75 kg person walking at 5.0 km/h, having arms 70 cm long and legs 90 cm long. (a) What is the average angular velocity of his arms and legs? (b) Using the average angular velocity from part (a), calculate the amount of rotational kinetic energy in this person’s arms and legs as he walks. (c) What is the total kinetic energy due to both his forward motion and his rotation? (d) What percentage of his kinetic energy is due to the rotation of his legs and arms?

72. The kinetic energy of running. Using the previous problem as a guide, apply it to a person running at 12 km/h, with his arms and legs each swinging through ±30.0° in ? s. As before, assume that the arms and legs are kept straight.

73. The pulley in Fig. 9.37 has radius R and a moment of inertia I. The rope does not slip over the pulley, and the pulley spins on a frictionless axle. The coefficient of kinetic friction between block A and the tabletop is ?k. The system is released from rest, and block B descends. Block A has mass mA and block B has mass mB. Use energy methods to calculate the speed of block B as a function of the distance d that it has descended.

**Chapter 10**

Multiple-Choice Problems

1. You are designing a wheel that must have a fixed mass and diameter, but that can have its mass distributed in various uniform ways. If the torque you exert on it is also fixed, which of the wheels shown will have the smallest angular acceleration about an axis perpendicular to it at its center?

2. Two equal masses m are connected by a very light string over a frictionless pulley of mass m/2. The system has been given a push to get it moving as shown, but that push is no longer acting. In which segment of the string is the tension greater?

A. The tension in A is greater.

B. The tension in B is greater.

C. The two tensions are the same.

3. The irregular object shown in Figure 10.36 is dropped from rest on the moon, where there is no air. Its center of mass is as shown in the figure. What will it do after it is dropped?

A. It will start rotating clockwise about its center of mass.

B. It will start rotating counterclockwise about its center of mass.

C. It will rotate until point A is directly below its center of mass, and then it will stop turning.

D. It will rotate until point A is directly above its center of mass, and then it will stop turning.

E. It will not rotate.

4. A student is sitting on a frictionless rotating stool with her arms outstretched holding equal heavy weights in each hand. If she suddenly lets go of the weights, her angular speed will

A. increase B. stay the same C. decrease.

5. If the torques on an object balance, then it follows that this object (there could be more than one correct choice)

A. cannot be rotating.

B. cannot have any angular acceleration.

C. cannot be moving.

D. cannot be accelerating.

6. If the forces on an object balance, then it follows that this object (there could be more than one correct choice)

A. cannot be rotating.

B. cannot have any angular acceleration.

C. cannot be moving.

D. cannot be accelerating.

7. A solid uniform ball and a solid uniform cylinder with the same mass and diameter are released from rest and roll without slipping down a hill. Which one will have a greater linear acceleration on the hill?

A. the sphere.

B. the cylinder.

C. They will both have the same acceleration.

8. A uniform beam is suspended horizontally as shown in Figure 10.37. It is attached to the wall by a small hinge. The force that the hinge exerts on the beam has components

A. upward and to the right.

B. upward and to the left.

C. downward and to the right.

D. downward and to the left.

9. Two identical cars are traveling in the same lane 100 m apart on a freeway. The lead car is moving at 88 km/h and the trailing car is moving at 82 km/h. From the point of view of a stationary observer by the road, what is happening to the center of mass of the system consisting of the two cars?

A. It is moving backward at 3 km/h

B. It is moving forward at 3 km/h

C. It is moving backward at 85 km/h

D. It is moving forward at 85 km/h

10. A person pushes vertically downward with force P on a lever of length L that is inclined at an angle ? above the horizontal as shown in Figure 10.38. The torque that the person’s push produces about point A is

A. PL sin ?

B. PL cos ?

C. PL.

11. String is wrapped around the outer rim of a solid uniform cylinder that is free to rotate about a frictionless axle through its center. When the string is pulled with a force P tangent to the rim, it gives the cylinder an angular acceleration ?. If the cylinder had twice the radius, but everything else was the same, the angular acceleration would be

A. 4? B. 2? C. ?/2 D. ?/4

12. A weight W swings from a hook in the ceiling by a light string of length L, as shown in Figure 10.39. T is the tension in the string. When the string makes an angle ? with the vertical, the net torque about the hook is

A. WL.

B. (W – T)L

C. WL

D. WL sin ?

E. WL cos ?

13. A ball of mass 0.20 kg is whirled in a horizontal circle at the end of a light string 75 cm long at a speed of 3.0 m/s. If the string is lengthened to 1.5 m while the ball is being twirled, then the speed of the ball will now be

A. 12 m/s B. 6.0 m/s C. 1.5 m/s D. 0.75 m/s

14. A heavy solid disk rotating freely and slowed only by friction applied at its outer edge takes 100 seconds to come to a stop. If the disk had twice the radius and twice the mass, but the frictional force remained the same, the time it would it take the wheel to come to a stop from the same initial rotational speed is

A. 25 s B. 100 s C. 400 s D. 800 s

15. A uniform metal meterstick is balanced as shown in Figure 10.40 with a 1.0 kg rock attached to the left end of the stick. (Pay attention to the scale of the diagram.) What is the mass of the meterstick?

A. 0.60 kg. B. 1.0 kg. C. 2.0 kg. D. 3.0 kg.

Problems

1. Calculate the torque (magnitude and direction) about point O due to the force F in each of the situations sketched in Figure 10.41. In each case, the force F and the rod both lie in the plane of the page, the rod has length 4.00 m, and the force has magnitude 10.0 N.

2. Calculate the net torque about point O for the two forces applied as in Figure 10.42. The rod and both forces are in the plane of the page.

3. Three forces are applied to a wheel of radius 0.350 m, as shown in Fig. 10.43. One force is perpendicular to the rim, one is tangent to it, and the other one makes a 40.0° angle with the radius. What is the net torque on the wheel due to these three forces for an axis perpendicular to the wheel and passing through its center?

4. In Figure 10.44, forces A, B, C and D, each have magnitude 50 N and act at the same point on the object. (a) What torque (magnitude and direction) does each of these forces exert on the object about point P? (b) What is the total torque about point P?

5. A square metal plate 0.180 m on each side is pivoted about an axis through point O at its center and perpendicular to the plate. (See Figure 10.45.) Calculate the net torque about this axis due to the three forces shown in the figure if the magnitudes of the forces are F1 = 18.0 N, F2 = 26.0 N and F3 = 14.0 N The plate and all forces are in the plane of the page.

6. A cord is wrapped around the rim of a wheel 0.250 m in radius, and a steady pull of 40.0 N is exerted on the cord. The wheel is mounted on frictionless bearings on a horizontal shaft through its center. The moment of inertia of the wheel about this shaft is 5.00 kg m2. Compute the angular acceleration of the wheel.

7. A certain type of propeller blade can be modeled as a thin uniform bar 2.50 m long and of mass 24.0 kg that is free to rotate about a frictionless axle perpendicular to the bar at its midpoint. If a technician strikes this blade with a mallet 1.15 m from the center with a 35.0 N force perpendicular to the blade, find the maximum angular acceleration the blade could achieve.

8. A 750 gram grinding wheel 25.0 cm in diameter is in the shape of a uniform solid disk. (We can ignore the small hole at the center.) When it is in use, it turns at a constant 220 rpm about an axle perpendicular to its face through its center. When the power switch is turned off, you observe that the wheel stops in 45.0 s with constant angular acceleration due to friction at the axle. What torque does friction exert while this wheel is slowing down?

9. A grindstone in the shape of a solid disk with diameter 0.520 m and a mass of 50.0 kg is rotating at 850 rev/min. You press an ax against the rim with a normal force of 160 N (see Figure 10.46), and the grindstone comes to rest in 7.50 s. Find the coefficient of kinetic friction between the ax and the grindstone. There is negligible friction in the bearings.

10. A solid, uniform cylinder with mass 8.25 kg and diameter 15.0 cm is spinning at 220 rpm on a thin, frictionless axle that passes along the cylinder axis. You design a simple friction brake to stop the cylinder by pressing the brake against the outer rim with a normal force. The coefficient of kinetic friction between the brake and rim is 0.333. What must the applied normal force be to bring the cylinder to rest after it has turned through 5.25 revolutions?

11. A 2.00 kg stone is tied to a thin, light wire wrapped around the outer edge of the uniform 10.0 kg cylindrical pulley shown in Figure 10.47. The inner diameter of the pulley is 60.0 cm, while the outer diameter is 1.00 m. The system is released from rest, and there is no friction at the axle of the pulley. Find (a) the acceleration of the stone, (b) the tension in the wire, and (c) the angular acceleration of the pulley.

12. A light rope is wrapped several times around a large wheel with a radius of 0.400 m. The wheel rotates in frictionless bearings about a stationary horizontal axis, as shown in Figure 10.48. The free end of the rope is tied to a suitcase with a mass of 15.0 kg. The suitcase is released from rest at a height of 4.00 m above the ground. The suitcase has a speed of 3.50 m/s when it reaches the ground. Calculate (a) the angular velocity of the wheel when the suitcase reaches the ground and (b) the moment of inertia of the wheel.

13. A 22,500 N elevator is to be accelerated upward by connecting it to a counterweight using a light (but strong!) cable passing over a solid uniform disk-shaped pulley. There is no appreciable friction at the axle of the pulley, but its mass is 875 kg and it is 1.50 m in diameter. (a) How heavy should the

counterweight be so that it will accelerate the elevator upward through 6.75 m in the first 3.00 s, starting from rest? (b) Under these conditions, what is the tension in the cable on each side of the pulley?

14. A thin, light string is wrapped around the rim of a 4.00 kg solid uniform disk that is 30.0 cm in diameter. A person pulls on the string with a constant force of 100.0 N tangent to the disk, as shown in Figure 10.49. The disk is not attached to anything and is free to move and turn. (a) Find the angular acceleration of the disk about its center of mass and the linear acceleration of its center of mass. (b) If the disk is replaced by a hollow thin-walled cylinder of the same mass and diameter, what will be the accelerations in part (a)?

15. A uniform, 8.40-kg, spherical shell 50.0 cm in diameter has four small 2.00-kg masses attached to its outer surface and equally spaced around it. This combination is spinning about an axis running through the center of the sphere and two of the small masses (Fig. 10.50). What friction torque is needed to reduce its angular speed from 75.0 rpm to 50.0 rpm in 30.0 s?

16. A hollow spherical shell with mass 2.00 kg rolls without slipping down a 38.0°slope. (a) Find the acceleration of the shell and the friction force on it. Is the friction kinetic or static friction? Why? (b) How would your answers to part (a) change if the mass were doubled to 4.00 kg?

17. A solid disk of radius 8.50 cm and mass 1.25 kg, which is rolling at a speed of 2.50 m/s begins rolling without slipping up a 10.0° slope. How long will it take for the disk to come to a stop?

18. What is the power output in horsepower of an electric motor turning at 4800 rev/min and developing a torque of 4.30 N·m?

19. A playground merry-go-round has a radius of 4.40 m and a moment of inertia of 245 kg·m2 and turns with negligible friction about a vertical axle through its center. (a) A child applies a 25.0 N force tangentially to the edge of the merry-go-round for 20.0 s. If the merry-go-round is initially at rest, what

is its angular velocity after this 20.0 s interval? (b) How much work did the child do on the merry-go-round? (c) What is the average power supplied by the child?

20. The flywheel of a motor has a mass of 300.0 kg and a moment of inertia of 580 kg·m2. The motor develops a constant torque of 2000.0 N·m and the flywheel starts from rest. (a) What is the angular acceleration of the flywheel? (b) What is its angular velocity after it makes 4.00 revolutions? (c) How much work is done by the motor during the first 4.00 revolutions?

21. (a) Compute the torque developed by an industrial motor whose output is 150 kW at an angular speed of 4000.0 rev/min (b) A drum with negligible mass and 0.400 m in diameter is attached to the motor shaft, and the power output of the motor is used to raise a weight hanging from a rope wrapped around the drum. How heavy a weight can the motor lift at constant speed? (c) At what constant speed will the weight rise?

22. Calculate the angular momentum and kinetic energy of a solid uniform sphere with a radius of 0.120 m and a mass of 14.0 kg if it is rotating at 6.00 rad/s about an axis through its center.

23. (a) Calculate the magnitude of the angular momentum of the earth in a circular orbit around the sun. Is it reasonable to model it as a particle? (b) Calculate the magnitude of the angular momentum of the earth due to its rotation around an axis through the north and south poles, modeling it as a uniform sphere.

24. A small 0.300 kg bird is flying horizontally at 3.50 m/s toward a 0.750 kg thin bar hanging vertically from a hook at its upper end, as shown in Figure 10.51. (a) When the bird is far from the bar, what are the magnitude and direction (clockwise or counterclockwise) of its angular momentum about a horizontal axis perpendicular to the plane of the figure and passing through (i) point A, (ii) point B, and (iii) point C? (b) Repeat part (a) when the bird is just ready to hit the bar, but is still flying horizontally.

25. A small 4.0 kg brick is released from rest 2.5 m above a horizontal seesaw on a fulcrum at its center, as shown in Figure 10.52. Find the angular momentum of this brick about a horizontal axis through the fulcrum and perpendicular to the plane of the figure (a) the instant the brick is released and (b) the instant before it strikes the seesaw.

26. A woman with mass 50.0 kg is standing on the rim of a large disk that is rotating at 0.50 rev/s about an axis perpendicular to it through its center. The disk has a mass of 110 kg and a radius of 4.0 m. Calculate the magnitude of the total angular momentum of the woman-plus-disk system, assuming that you can treat the woman as a point.

27. A certain drawbridge can be modeled as a uniform 15,000 N bar, 12.0 m long, pivoted about its lower end. When this bridge is raised to an angle of 60.0° above the horizontal, the cable holding it suddenly breaks, allowing the bridge to fall. At the instant after the cable breaks, (a) what is the torque on this bridge about the pivot and (b) at what rate is its angular momentum changing?

28. On an old-fashioned rotating piano stool, a woman sits holding a pair of dumbbells at a distance of 0.60 m from the axis of rotation of the stool. She is given an angular velocity of 3.00 rad/s, after which she pulls the dumbbells in until they are only 0.20 m distant from the axis. The woman’s moment of inertia about the axis of rotation is 5.00 kg·m2 and may be considered constant. Each dumbbell has a mass of 5.00 kg and may be considered a point mass. Neglect friction. (a) What is the initial angular momentum of the system? (b) What is the angular velocity of the system after the dumbbells are pulled in toward the axis? (c) Compute the kinetic energy of the system before and after the dumbbells are pulled in. Account for the difference, if any.

29. The spinning figure skater. The outstretched hands and arms of a figure skater preparing for a spin can be considered a slender rod pivoting about an axis through its center. When the skater’s hands and arms are brought in and wrapped around his body to execute the spin, the hands and arms can be considered a thin-walled hollow cylinder. His hands and arms have a combined mass of 8.0 kg. When outstretched, they span 1.8 m; when wrapped, they form a cylinder of radius 25 cm. The moment of inertia about the axis of rotation of the remainder of his body is constant and equal to 0.40 kg·m2. If the skater’s original angular speed is 0.40 rev/s what is his final angular speed?

30. A small block on a frictionless horizontal surface has a mass of 0.0250 kg. It is attached to a massless cord passing through a hole in the surface. The block is originally revolving at a distance of 0.300 m from the hole with an angular speed of 1.75 rad/s. The cord is then pulled from below, shortening the radius of the circle in which the block revolves to 0.150 m. You may treat the block as a particle. (a) Is angular momentum conserved? Why or why not? (b) What is the new angular speed? (c) Find the change in kinetic energy of the block. (d) How much work was done in pulling the cord?

31. A uniform 4.5 kg square solid wooden gate 1.5 m on each side hangs vertically from a frictionless pivot at the center of its upper edge. A 1.1 kg raven flying horizontally at 5.0 m/s flies into this gate at its center and bounces back at 2.0 m/s in the opposite direction. (a) What is the angular speed of the gate just after it is struck by the unfortunate raven? (b) During the collision, why is the angular momentum conserved, but not the linear momentum?

32. A diver comes off a board with arms straight up and legs straight down, giving her a moment of inertia about her rotation axis of 18 kg·m2. She then tucks into a small ball, decreasing this moment of inertia to 3.6 kg·m2. While tucked, she makes two complete revolutions in 1.0 s. If she hadn’t tucked at all, how many revolutions would she have made in the 1.5 s from board to water?

33. A large turntable rotates about a fixed vertical axis, making one revolution in 6.00 s. The moment of inertia of the turntable about this axis is 1200 kg·m2. A child of mass 40.0 kg, initially standing at the center of the turntable, runs out along a radius. What is the angular speed of the turntable when the child is 2.00 m from the center, assuming that you can treat the child as a particle?

34. A large wooden turntable in the shape of a flat disk has a radius of 2.00 m and a total mass of 120 kg. The turntable is initially rotating at 3.00 rad/s about a vertical axis through its center. Suddenly, a 70.0 kg parachutist makes a soft landing on the turntable at a point on its outer edge. Find the angular speed of the turntable after the parachutist lands.

35. Which of the objects shown in Figure 10.55 are in only translational equilibrium, only rotational equilibrium (about the axis A), both translational and rotational equilibrium (about A), or neither equilibrium?

36. (a) In each of the objects indicated in Figure 10.56, what magnitude of force F (if any) is needed to put the object into rotational equilibrium about the axis A shown? (b) After you have found the F required to put the object into rotational equilibrium, find out which (if any) of these objects is also in translational equilibrium.

37. Supporting a broken leg. A therapist tells a 74 kg patient with a broken leg that he must have his leg in a cast suspended horizontally. For minimum discomfort, the leg should be supported by a vertical strap attached at the center of mass of the leg–cast system. In order to comply with these instructions, the patient consults a table of typical mass distributions and finds that both upper legs (thighs) together typically account for 21.5% of body weight and the center of mass of each thigh is 18.0 cm from the hip joint. The patient also reads that the two lower legs (including the feet) are 14.0% of body weight, with a center of mass 69.0 cm from the hip joint. The cast has a mass of 5.50 kg, and its center of mass is 78.0 cm from the hip joint. How far from the hip joint should the supporting strap be attached to the cast?

38. Two people are carrying a uniform wooden board that is 3.00 m long and weighs 160 N. If one person applies an upward force equal to 60 N at one end, at what point and with what force does the other person lift? Start with a free-body diagram of the board.

39. Push-ups. To strengthen his arm and chest muscles, an 82 kg athlete 2.0 m tall is doing a series of push-ups as shown in Figure 10.58. His center of mass is 1.15 m from the bottom of his feet, and the centers of his palms are 30.0 cm from the top of his head. Find the force that the floor exerts on each of

his feet and on each hand, assuming that both feet exert the same force and both palms do likewise. Begin with a free-body diagram of the athlete.

40. Two people carry a heavy electric motor by placing it on a light board 2.00 m long. One person lifts at one end with a force of 400.0 N, and the other lifts at the opposite end with a force of 600.0 N. (a) Start by making a free-body diagram of the motor. (b) What is the weight of the motor? (c) Where along the board is its center of gravity located?

41. A 60.0-cm, uniform, 50.0-N shelf is supported horizontally by two vertical wires attached to the sloping ceiling (Fig. 10.59). A very small 25.0-N tool is placed on the shelf midway between the points where the wires are attached to it. Find the tension in each wire. Begin by making a free-body diagram of the shelf.

42. The horizontal beam in Figure 10.60 weighs 150 N, and its center of gravity is at its center. First make a free-body diagram of the beam. Then find (a) the tension in the cable and (b) the horizontal and vertical components of the force exerted on the beam at the wall.

43. The boom in Figure 10.61 weighs 2600 N and is attached to a frictionless pivot at its lower end. It is not uniform; the distance of its center of gravity from the pivot is 35% of its length. Find (a) the tension in the guy wire and (b) the horizontal and vertical components of the force exerted on the boom at its lower end. Start with a free-body diagram of the boom.

44. A uniform 250 N ladder rests against a perfectly smooth wall, making a 35° angle with the wall. (a) Draw a free-body diagram of the ladder. (b) Find the normal forces that the wall and the floor exert on the ladder. (c) What is the friction force on the ladder at the floor?

45. A uniform ladder 7.0 m long weighing 450 N rests with one end on the ground and the other end against a perfectly smooth vertical wall. The ladder rises at 60.0° above the horizontal floor. A 750 N painter finds that she can climb 2.75 m up the ladder, measured along its length, before it begins to slip. (a) Make a free-body diagram of the ladder. (b) What force does the wall exert on the ladder? (c) Find the friction force and normal force that the floor exerts on the ladder.

46. A 9.0 m uniform beam is hinged to a vertical wall and held horizontally by a 5.0 m cable attached to the wall 4.0 m above the hinge, as shown in Figure 10.62. The metal of this cable has a test strength of 1.00 kN, which means that it will break if the tension in it exceeds that amount. (a) Draw a free-body diagram of the beam. (b) What is the heaviest beam that the cable can support with the given configuration? (c) Find the horizontal and vertical components of the force the hinge exerts on the beam.

47. A uniform beam 4.0 m long and weighing 2500 N carries a 3500 N weight 1.50 m from the far end, as shown in Figure 10.63. It is supported horizontally by a hinge at the wall and a metal wire at the far end. (a) Make a free-body diagram of the beam. (b) How strong does the wire have to be? That is, what is the minimum tension it must be able to support without breaking? (c) What are the horizontal and vertical components of the force that the hinge exerts on the beam?

48. Leg raises. In a simplified version of the musculature action in leg raises, the abdominal muscles pull on the femur (thigh bone) to raise the leg by pivoting it about one end, as shown in Figure 10.64. When you are lying horizontally, these muscles make an angle of approximately 5° with the femur, and if you raise your legs, the muscles remain approximately horizontal, so the angle θ increases. We shall assume for simplicity that these muscles attach to the femur in only one place, 10 cm from the hip joint (although, in reality, the situation is more complicated). For a certain 80 kg person having a leg 90 cm long, the mass of the leg is 15 kg and its center of mass is 44 cm from his hip joint as measured along the leg. If the person raises his leg to 60° above the horizontal, the angle between the abdominal muscles and his femur would also be about 60° (a) With his leg raised to 60°, find the tension in the abdominal muscle on each leg. As usual, begin your solution with a free-body diagram. (b) When is the tension in this muscle greater, when the leg is raised to 60° or when the person just starts to raise it off the ground? Why? (Try this yourself to check your answer.) (c) If the abdominal muscles attached to the femur were perfectly horizontal when a person was lying down, could the person raise his leg? Why or why not?

49. A diving board 3.00 m long is supported at a point 1.00 m from the end, and a diver weighing 500 N stands at the free end (Fig. 10.65). The diving board is of uniform cross section and weighs 280 N. Find (a) the force at the support point and (b) the force at the left-hand end.

50. Two people carry a heavy electric motor by placing it on a light board 2.00 m long. One person lifts at one end with a force of 400 N, and the other lifts the opposite end with a force of 600 N. (a) What is the weight of the motor, and where along the board is its center of gravity located? (b) Suppose the board is not light but weighs 200 N, with its center of gravity at its center, and the two people each exert the same forces as before. What is the weight of the motor in this case, and where is its center of gravity located?

51. Pumping iron. A 72.0 kg weight lifter is doing arm raises using a 7.50 kg weight in her hand. Her arm pivots around the elbow joint, starting 40.0° below the horizontal. Biometric measurements have shown that both forearms and the hands together account for 6.00% of a person’s weight. Since the upper arm is held vertically, the biceps muscle always acts vertically and is attached to the bones of the forearm 5.50 cm from the elbow joint. The center of mass of this person’s forearm–hand combination is 16.0 cm from the elbow joint, along the bones of the forearm, and the weight is held 38.0 cm from the elbow joint. (a) Make a free-body diagram of the forearm. (b) What force does the biceps muscle exert on the forearm? (c) Find the magnitude and direction of the force that the elbow joint exerts on the forearm. (d) As the weight lifter raises her arm toward a horizontal position, will the force in the biceps muscle increase, decrease, or stay the same? Why?

52. The deltoid muscle. The deltoid muscle is the main muscle that allows you to raise your arm or even hold it out. It is connected to the humerus of the upper arm. The person shown is holding his arm out horizontally with a 2.50 kg weight in his hand. This weight is 60.0 cm from the shoulder joint. His forearm (including his hand) has a mass of 2.44 kg and is 34.0 cm long; its center of mass is 43 cm from the shoulder joint, measured along the arm. His upper arm is 26.0 cm long and has a mass of 2.63 kg; its center of mass is 13.0 cm from the shoulder joint. The deltoid muscle is attached to the humerus 15.0 cm from the shoulder joint and makes a 14.0° angle with the humerus. (a) Make a free-body diagram of the arm. (b) What is the tension in the deltoid muscle?

53. A uniform, 90.0-N table is 3.6 m long, 1.0 m high, and 1.2 m wide. A 1500-N weight is placed 0.50 m from one end of the table, a distance of 0.60 m from each of the two legs at that end. Draw a free-body diagram for the table and find the force that each of the four legs exerts on the floor.

54. The rotor (flywheel) of a toy gyroscope has a mass of 0.140 kg. Its moment of inertia about its axis is 1.20 × 10–4 kg·m2. The mass of the frame is 0.0250 kg. The gyroscope is supported on a single pivot with its center of mass a horizontal distance of 4.00 cm from the pivot. The gyroscope is precessing in a horizontal plane at the rate of 1 revolution in 2.20 s. (a) Find the upward force exerted by the pivot. (b) Find the angular speed with which the rotor is spinning about its axis, expressed in rev/min (c) Copy the diagram and draw vectors to show the angular momentum of the rotor and the torque acting on it.

55. For each of the following rotating objects, describe the direction of the angular momentum vector: (a) the minute hand of a clock; (b) the right front tire of a car moving backwards; (c) an ice skater spinning clockwise; (d) the earth, rotating on its axis.

56. Back pains during pregnancy. Women often suffer from back pains during pregnancy. Let us investigate the cause of these pains, assuming that the woman’s mass is 60 kg before pregnancy. Typically, women gain about 10 kg during pregnancy, due to the weight of the fetus, placenta, amniotic fluid, etc. To make the calculations easy, but still realistic, we shall model the unpregnant woman as a uniform cylinder of diameter 30 cm. We can model the added mass due to the fetus as a 10 kg sphere 25 cm in diameter and centered about 5 cm outside the woman’s original front surface. (a) By how much does her pregnancy change the horizontal location of the woman’s center of mass? (b) How does the change in part (a) affect the way the pregnant woman must stand and walk? In other words, what must she do to her posture to make up for her shifted center of mass? (c) Can you now explain why she might have backaches?

57. You are asked to design the decorative mobile shown in Figure 10.69. The strings and rods have negligible weight, and the rods are to hang horizontally. (a) Draw a free-body diagram for each rod. (b) Find the weights of the balls A, B, and C. Find the tensions S1, S2, and S3 in the strings. (c) What can you say about the horizontal location of the mobile’s center of gravity? Explain.

58. A good workout. You are doing exercises on a Nautilus machine in a gym to strengthen your deltoid (shoulder) muscles. Your arms are raised vertically and can pivot around the shoulder joint, and you grasp the cable of the machine in your hand 64.0 cm from your shoulder joint. The deltoid muscle is attached to the humerus 15.0 cm from the shoulder joint and makes a 12.0° angle with that bone. If you have set the tension in the cable of the machine to 36.0 N on each arm, what is the tension in each deltoid muscle if you simply hold your outstretched arms in place?

59. Prior to being placed in its hole, a 5700-N, 9.0-m-long, uniform utility pole makes some nonzero angle with the vertical. A vertical cable attached 2.0 m below its upper end holds it in place while its lower end rests on the ground. (a) Find the tension in the cable and the magnitude and direction of the force exerted by the ground on the pole. (b) Why don’t we need to know the angle the pole makes with the vertical, as long as it is not zero?

60. A uniform drawbridge must be held at a 37° angle above the horizontal to allow ships to pass underneath. The drawbridge weighs 45,000 N, is 14.0 m long, and pivots about a hinge at its lower end. A cable is connected 3.5 m from the hinge, as measured along the bridge, and pulls horizontally on the bridge to hold it in place. (a) What is the tension in the cable? (b) Find the magnitude and direction of the force the hinge exerts on the bridge. (c) If the cable suddenly breaks, what is the initial angular acceleration of the bridge?

61. Pyramid builders. Ancient pyramid builders are balancing a uniform rectangular slab of stone tipped at an angle above the horizontal using a rope. The rope is held by five workers who share the force equally. (a) If θ = 20.0° what force does each worker exert on the rope? (b) As θ increases, does each worker have to exert more or less force than in part (a), assuming they do not change the angle of the rope? Why? (c) At what angle do the workers need to exert no force to balance the slab? What happens if θ exceeds this value?

62. The farmyard gate. A gate 4.00 m wide and 2.00 m high weighs 500 N. Its center of gravity is at its center, and it is hinged at A and B. To relieve the strain on the top hinge, a wire CD is connected as shown in Fig. 10.72. The tension in CD is increased until the horizontal force at hinge A is zero. (a) What is the tension in the wire CD? (b) What is the magnitude of the horizontal component of the force at hinge B? (c) What is the combined vertical force exerted by hinges A and B?

63. Atwood’s machine. Figure 10.73 illustrates an Atwood’s machine. Find the linear accelerations of blocks A and B, the angular acceleration of the wheel C, and the tension in each side of the cord if there is no slipping between the cord and the surface of the wheel. Let the masses of blocks A and B be 4.00 kg and 2.00 kg, respectively, the moment of inertia of the wheel about its axis be 0.300 kg·m2 and the radius of the wheel be 0.120 m.

64. Neck muscles. A student bends her head at 40.0° from the vertical while intently reading her physics book, pivoting the head around the upper vertebra (point P in Figure 10.74). Her head has a mass of 4.50 kg (which is typical), and its center of mass is 11.0 cm from the pivot point P. Her neck muscles are 1.50 cm from point P, as measured perpendicular to these muscles. The neck itself and the vertebrae are held vertical. (a) Draw a free-body diagram of the student’s head. (b) Find the tension in her neck muscles.

65. Russell traction apparatus. The device shown in Figure 10.75 is one version of a Russell traction apparatus. It has two functions: to support the injured leg horizontally and at the same time provide a horizontal traction force on it. This can be done by adjusting the weight W and the angle θ. For this patient, his leg (including his foot) is 95.0 cm long (measured from the hip joint) and has a mass of 14.2 kg. Its center of mass is 41.0 cm from the hip joint. A support strap is attached to the patient’s ankle 13.0 cm from the bottom of his foot. (a) What weight W is needed to support the leg horizontally? (b) If the therapist specifies that the traction force must be 12.0 N horizontally, what must be the angle θ? (c) What is the greatest traction force that this apparatus could supply to this patient’s leg, and what is θ in that case?

66. Supporting an injured arm: I. A 650 N person must have her injured arm supported, with the upper arm horizontal and the forearm vertical. According to biomedical tables and direct measurements, her upper arm is 26 cm long (measured from the shoulder joint), accounts for 3.50% of her body weight, and has a center of mass 13.0 cm from her shoulder joint. Her forearm (including the hand) is 34.0 cm long, makes up 3.25% of her body weight, and has a center of mass 43.0 cm from her shoulder joint. (a) Where is the center of mass of the person’s arm when it is supported as shown? (b) What weight W is needed to support her arm? (c) Find the horizontal and vertical components of the force that the shoulder joint exerts on her arm.

67. Supporting an injured arm: II. As part of therapy, the person’s arm in the previous problem is later to be supported with the upper arm horizontal, but with the forearm at an angle of 55° above the horizontal, as shown in Figure 10.77. (a) Where is the center of mass of the person’s arm? (b) What weight W is needed to support her arm this way? (c) Find the horizontal and vertical components of the force that the shoulder joint exerts on her arm.

68. The forces on the foot. A 750 N athlete standing on one foot on a very smooth gym floor lifts his body by pivoting his foot upward through a 30.0° angle, balancing all of his weight on the ball of the foot. The forces on the foot bones from the rest of his body are due to the Achilles tendon and the ankle joint. The Achilles tendon acts perpendicular to a line through the heel and toes; it is this line that has rotated upward through 30.0°. Assume that the weight of the foot is negligible compared with that of the rest of the body, and begin by making a free-body diagram of the foot. (a) What are the magnitude and direction of the force that the floor exerts on the athlete’s foot? (b) What is the tension in the Achilles tendon? (c) Find the horizontal and vertical components of the force that the ankle joint exerts on the foot. Then use these components to find the magnitude of this force.

69. A uniform solid cylinder of mass M is supported on a ramp that rises at an angle θ above the horizontal by a wire that is wrapped around its rim and pulls on it tangentially parallel to the ramp (Fig. 10.79). (a) Show that there must be friction on the surface for the cylinder to balance this way. (b) Show that the tension in the wire must be equal to the friction force, and find this tension.

70. A uniform 8.0 m, 1500 kg beam is hinged to a wall and supported by a thin cable attached 2.0 m from the free end of the beam, as shown in Figure 10.80. The beam is supported at an angle of 30.0° above the horizontal. (a) Make a free-body diagram of the beam. (b) Find the tension in the cable. (c) How hard does the beam push inward on the wall?

71. You are trying to raise a bicycle wheel of mass m and radius R up over a curb of height h. To do this, you apply a horizontal force F (Fig. 10.81). What is the smallest magnitude of the force F that will succeed in raising the wheel onto the curb when the force is applied (a) at the center of the wheel, and (b) at the top of the wheel? (c) In which case is less force required?

72. An experimental bicycle wheel is placed on a test stand so that it is free to turn on its axle. If a constant net torque of 5.00 N·m is applied to the tire for 2.00 s, the angular speed of the tire increases from zero to 100 rev/min. The external torque is then removed, and the wheel is brought to rest in 125 s by friction in its bearings. Compute (a) the moment of inertia of the wheel about the axis of rotation, (b) the friction torque, and (c) the total number of revolutions made by the wheel in the 125 s time interval.

73. Under some circumstances, a star can collapse into an extremely dense object made mostly of neutrons and called a neutron star. The density of a neutron star is roughly times as great as that of ordinary solid matter. Suppose we represent the star as a uniform, solid, rigid sphere, both before and after the collapse. The star's initial radius was 7.0 × 105 km (comparable to our sun); its final radius is 16 km. If the original star rotated once in 30 days, find the angular speed of the neutron star.

74. Disks A and B are mounted on shaft SS and may be connected or disconnected by clutch C. Disk A is made of a lighter material than disk B, so the moment of inertia of disk A about the shaft is one-third that of disk B. The moments of inertia of the shaft and clutch are negligible. With the clutch disconnected, A is brought up to an angular speed ω0. The accelerating torque is then removed from A, and A is coupled to disk B by the clutch. (You can ignore bearing friction.) It is found that 2400 J of thermal energy is developed in the clutch when the connection is made. What was the original kinetic energy of disk A?

75. While exploring a castle, Exena the Exterminator is spotted by a dragon who chases her down a hallway. Exena runs into a room and attempts to swing the heavy door shut before the dragon gets her. The door is initially perpendicular to the wall, so it must be turned through 90° to close. The door is 3.00 m tall and 1.25 m wide, and it weighs 750 N. You can ignore the friction at the hinges. If Exena applies a force of 220 N at the edge of the door and perpendicular to it, how much time does it take her to close the door?

76. Downward-Facing Dog. One yoga exercise, known as the “Downward-Facing Dog,” requires stretching your hands straight out above your head and bending down to lean against the floor. This exercise is performed by a certain 750 N person, as shown in the simplified model in Figure 10.83. When he bends his body at the hip to a 90° angle between his legs and trunk, his legs, trunk, head and arms have the dimensions indicated. Furthermore, his legs and feet weigh a total of 277 N, and their center of mass is 41 cm from his hip, measured along his legs. The person’s trunk, head, and arms weigh 473 N, and their center of gravity is 65 cm from his hip, measured along the upper body. (a) Find the normal force that the floor exerts on each foot and on each hand, assuming that the person does not favor either hand or either foot. (b) Find the friction force on each foot and on each hand, assuming that it is the same on both feet and on both hands (but not necessarily the same on the feet as on the hands).

77. A uniform, 7.5-m-long beam weighing 9000 N is hinged to a wall and supported by a thin cable attached 1.5 m from the free end of the beam. The cable runs between the beam and the wall and makes a 40° angle with the beam. What is the tension in the cable when the beam is at an angle of 30° above the horizontal?

**Chapter 11**

Multiple-Choice Problems

1. A spring–mass system is undergoing simple harmonic motion of amplitude 2.0 cm and angular frequency 10.0 s–1. The acceleration of the mass as it passes through the equilibrium position is

A. 0.00 m/s2 B. 0.20 m/s2 C. 2.0 m/s2 D. 31.4 m/s2

2. In a design for a piece of medical apparatus, you need a material that is easily compressed when a pressure is applied to it.

A. This material should have a large bulk modulus.

B. This material should have a small bulk modulus.

C. The bulk modulus is not relevant to this situation.

3. If he force F in Figure 11.32 is constant over the area A, the pressure on that area is

A. 2.50 Pa B. 4.33 Pa C. 5.00 Pa

4. A box with a mass of 5 kg, whose bottom measures 10 × 10 cm, sits on a table. The pressure the box exerts on the tabletop is approximately

A. 0.5 Pa B. 50 Pa C. 500 Pa D. 5000 Pa

5. When a mass attached to a spring is released from rest 3.0 cm from its equilibrium position, it oscillates with frequency f. If this mass were instead released from rest 6.0 cm from its equilibrium position, it would oscillate with frequency

A. 2f B. √2f C. f D. f/2

6. As the bob on a pendulum swings down toward its lowest point, its angular frequency ω

A. increases.

B. decreases.

C. does not change.

7. Suppose you increase the amplitude of oscillation of a mass vibrating on a spring. Which of the following statements about this mass are correct?

A. Its maximum speed increases.

B. Its period of oscillation increases.

C. Its maximum acceleration increases.

D. Its maximum kinetic energy increases.

8. An object of mass M suspended by a spring vibrates with period T. If this object is replaced by one of mass 4M, the new object vibrates with a period

A. T. B. 2T. C. 4T. D. 16T.

9. When two wires of identical dimensions are used to hang 25 kg weights, wire A is observed to stretch twice as much as wire B. From this observation, you can make the following conclusions about the Young’s moduli of these wires:

A. YA = 4YB B. YA = 2YB C. YA = ½YB D. YA = ¼YB

10. A mass on a spring oscillates with a period T. If both the mass and the force constant of the spring are doubled, the new period will be

A. 4T. B. √2T C. T. D. T/√2 E. T/4

11. A pendulum oscillates with a period T. If both the mass of the bob and the length of the pendulum are doubled, the new period will be

A. 4T. B. √2T C. T. D. T/√2 E. T/4

12. When a 100-kg mass is hung from a cable made of a certain material, the cable is observed to lengthen by 1%. If the cable is now replaced by one with twice the cross-sectional area and made of a material with twice the Young’s modulus of the original one, and the mass hung from it is also doubled, how much stretching will be observed?

A. 1% B. 0.5% C. 4% D. 8%

13. An object with mass M suspended by a spring vibrates with frequency f. When a second object is attached to the first, the system now vibrates with frequency f/2. The mass of the second object is

A. 4M. B. 3M. C. 2M. D. M.

14. A pendulum on earth swings with angular frequency ω. On an unknown planet, it swings with angular frequency ω/2. The acceleration due to gravity on this planet is

A. 4g. B. 2g. C. g/2 D. g/4

15. A mass oscillates with simple harmonic motion of amplitude A. The kinetic energy of the mass will equal the potential energy of the spring when the position is

A. x = 0 B. x = A/2 C. x = A/√2 D. x = A/4

**Problems**

1. A thin, light wire 75.0 cm long having a circular cross section 0.550 mm in diameter has a 25.0 kg weight attached to it, causing it to stretch by 1.10 mm. (a) What is the stress in this wire? (b) What is the strain of the wire? (c) Find Young’s modulus for the material of the wire.

2. A petite young woman distributes her 500 N weight equally over the heels of her high-heeled shoes. Each heel has an area of 0.750 cm2 (a) What pressure is exerted on the floor by each heel? (b) With the same pressure, how much weight could be supported by two flat-bottomed sandals, each of area 200 cm2.

3. Two circular rods, one steel and the other copper, are joined end to end. Each rod is 0.750 m long and 1.50 cm in diameter. The combination is subjected to a tensile force with magnitude 4000 N. For each rod, what are (a) the strain and (b) the elongation?

4. A 5.0 kg mass is hung by a vertical steel wire 0.500 m long and 6.0 × 10–3 cm2 in cross-sectional area. Hanging from the bottom of this mass is a similar steel wire, from which in turn hangs a 10.0 kg mass. For each wire, compute (a) the tensile strain and (b) the elongation.

5. Biceps muscle. A relaxed biceps muscle requires a force of 25.0 N for an elongation of 3.0 cm; under maximum tension, the same muscle requires a force of 500 N for the same elongation. Find Young’s modulus for the muscle tissue under each of these conditions if the muscle can be modeled as a uniform cylinder with an initial length of 0.200 m and a cross-sectional area of 50.0 cm2.

6. Stress on a mountaineer’s rope. A nylon rope used by mountaineers elongates 1.10 m under the weight of a 65.0 kg climber. If the rope is 45.0 m in length and 7.0 mm in diameter, what is Young’s modulus for this nylon?

7. A steel wire 2.00 m long with circular cross section must stretch no more than 0.25 cm when a 400.0 N weight is hung from one of its ends. What minimum diameter must this wire have?

8. Achilles tendon. The Achilles tendon, which connects the calf muscles to the heel, is the thickest and strongest tendon in the body. In extreme activities, such as sprinting, it can be subjected to forces as high as 13 times a person’s weight. According to one set of experiments, the average area of the Achilles tendon is 78.1 mm2, its average length is 25 cm, and its average Young’s modulus is 1474 MPa. (a) How much tensile stress is required to stretch this muscle by 5.0% of its length? (b) If we model the tendon as a spring, what is its force constant? (c) If a 75 kg sprinter exerts a force of 13 times his weight on his Achilles tendon, by how much will it stretch?

9. Artificial tendons. The largest Young’s modulus measured for any biological material is 30 GPa, for the S-layer of cell walls. Suppose that in the future we are able to use this material to construct an artificial Achilles tendon (see previous problem) for an athlete who has damaged his. The artificial Achilles tendon should be just as long as the original one so that it can connect at the proper places. (a) If the artificial tendon has a circular cross section, what should be its diameter be so that it will have the same force constant as the natural Achilles tendon? (b) How does this diameter compare with that of a natural Achilles tendon?

10. Human hair. According to one set of measurements, the tensile strength of hair is 196 MPa, which produces a maximum strain of 0.40 in the hair. The thickness of hair varies considerably, but let’s use a diameter of 50 μm (a) What is the magnitude of the force giving this tensile stress? (b) If the length of a strand of the hair is 12 cm at its breaking point, what was its unstressed length?

11. The effect of jogging on the knees. High-impact activities such as jogging can cause considerable damage to the cartilage at the knee joints. Peak loads on each knee can be eight times body weight during jogging. The bones at the knee are separated by cartilage called the medial and lateral meniscus. Although it varies considerably, the force at impact acts over approximately 10 cm2 of this cartilage. Human cartilage has a Young’s modulus of about 24 MPa (although that also varies). (a) By what percent does the peak load impact of jogging compress the knee cartilage of a 75 kg person? (b) What would be the percentage for a lower-impact activity, such as power walking, for which the peak load is about four times body weight?

12. A solid gold bar is pulled up from the hold of the sunken RMS Titanic. (a) What happens to its volume as it goes from the pressure at the ship to the lower pressure at the ocean’s surface? (b) The pressure difference is proportional to the depth. How many times greater would the volume change have been had the ship been twice as deep? (c) The bulk modulus of lead is one-fourth that of gold. Find the ratio of the volume change of a solid lead bar to that of a gold bar of equal volume for the same pressure change.

13. In the Challenger Deep of the Marianas Trench, the depth of seawater is 10.9 km and the pressure is 1.16 × 108 Pa (about 1150 atmospheres). (a) If a cubic meter of water is taken to this depth from the surface (where the normal atmospheric pressure is about 1.0 × 105 Pa) what is the change in its volume? Assume that the bulk modulus for seawater is the same as for freshwater (2.2 × 109 Pa) (b) At the surface, seawater has a density of 1.03 × 103 kg/m3. What is the density of seawater at the depth of the Challenger Deep?

14. Effect of diving on blood. It is reasonable to assume that the bulk modulus of blood is about the same as that of water (2.2 GPa). As one goes deeper and deeper in the ocean, the pressure increases by 1.0 × 104 Pa for every meter below the surface. (a) If a diver goes down 33 m (a bit over 100 ft) in the ocean, by how much does each cubic centimeter of her blood change in volume? (b) How deep must a diver go so that each drop of blood compresses to half its volume at the surface? Is the ocean deep enough to have this effect on the diver?

15. Shear forces are applied to a rectangular solid. The same forces are applied to another rectangular solid of the same material, but with three times each edge length. In each case the forces are small enough that Hooke’s law is obeyed. What is the ratio of the shear strain for the larger object to that of the smaller object?

16. Compression of human bone. The bulk modulus for bone is 15 GPa. (a) If a diver-in-training is put into a pressurized suit, by how much would the pressure have to be raised (in atmospheres) above atmospheric pressure to compress her bones by 0.10% of their original volume? (b) Given that the pressure in the ocean increases by 1.0 × 104 Pa for every meter of depth below the surface, how deep would this diver have to go for her bones to compress by 0.10%? Does it seem that bone compression is a problem she needs to be concerned with when diving?

17. In Figure 11.33, suppose the object is a square steel plate, 10.0 cm on a side and 1.00 cm thick. Find the magnitude of force required on each of the four sides to cause a shear strain of 0.0400.

18. In lab tests on a 9.25 cm cube of a certain material, a force of 1375 N directed at 8.50° to the cube, as shown in Figure 11.34, causes the cube to deform through an angle of 1.24°. What is the shear modulus of the material?

19. Downhill hiking. During vigorous downhill hiking, the force on the knee cartilage (the medial and lateral meniscus) can be up to eight times body weight. Depending on the angle of descent, this force can cause a large shear force on the cartilage and deform it. The cartilage has an area of about 10 cm2

and a shear modulus of 12 MPa. If the hiker plus his pack have a combined mass of 110 kg (not unreasonable), and if the maximum force at impact is 8 times his body weight (which, of course, includes the weight of his pack) at an angle of 12° with the cartilage (see Figure 11.35), through what angle (in degrees) will his knee cartilage be deformed?

20. A steel wire has the following properties:

Length = 5.00 m

Cross-sectional area = 0.040 cm2

Young’s modulus = 2.0 × 1011 Pa

Shear modulus = 0.84 × 1011 Pa

Proportional limit = 3.6 × 108 Pa

Breaking stress 11.0 × 108 Pa

The wire is fastened at its upper end and hangs vertically. (a) How great a weight can be hung from the wire without exceeding the proportional limit? (b) How much does the wire stretch under this load? (c) What is the maximum weight that can be supported?

21. A steel cable with cross-sectional area of 3.00 cm2 has an elastic limit of 2.4 × 108 Pa. Find the maximum upward acceleration that can be given to a 1200 kg elevator supported by the cable if the stress is not to exceed one-third of the elastic limit.

22. Weight lifting. The legs of a weight lifter must ultimately support the weights he has lifted. A human tibia (shinbone) has a circular cross section of approximately 3.6 cm outer diameter and 2.5 cm inner diameter. (The hollow portion contains marrow.) If a 90 kg lifter stands on both legs, what is the heaviest weight he can lift without breaking his legs, assuming that the breaking stress of the bone is 200 MPa?

24. Find the period, frequency, and angular frequency of (a) the second hand and (b) the minute hand of a wall clock.

25. If an object on a horizontal frictionless surface is attached to a spring, displaced, and then released, it oscillates. Suppose it is displaced 0.120 m from its equilibrium position and released with zero initial speed. After 0.800 s, its displacement is found to be 0.120 m on the opposite side and it has passed the equilibrium position once during this interval. Find (a) the amplitude, (b) the period, and (c) the frequency of the motion.

26. The graph shown in Figure 11.36 closely approximates the displacement x of a tuning fork as a function of time t as it is playing a single note. What are (a) the amplitude, (b) period, (c) frequency, and (d) angular frequency of this fork’s motion?

27. The wings of the Blue-throated Hummingbird (Lampornis clemenciae), which inhabits Mexico and the southwestern United States, beat at a rate of up to 900 times per minute. Calculate (a) the period of vibration of the bird’s wings, (b) the frequency of the wings’ vibration, and (c) the angular frequency of the bird’s wingbeats.

28. A 0.500 kg glider on an air track is attached to the end of an ideal spring with force constant 450 N/m; it undergoes simple harmonic motion with an amplitude of 0.040 m. Compute (a) the maximum speed of the glider, (b) the speed of the glider when it is at x = –0.015 m (c) the magnitude of the maximum acceleration of the glider, (d) the acceleration of the glider at x = –0.015 m and (e) the total mechanical energy of the glider at any point in its motion.

29. A 0.150 kg toy is undergoing SHM on the end of a horizontal spring with force constant 300.0 N/m. When the object is 0.0120 m from its equilibrium position, it is observed to have a speed of 0.300 m/s. Find (a) the total energy of the object at any point in its motion, (b) the amplitude of the motion, and (c) the maximum speed attained by the object during its motion.

30. A 2.00 kg frictionless block is attached to an ideal spring with force constant 315 N/m. Initially the spring is neither stretched nor compressed, but the block is moving in the negative direction at 12.0 m/s. Find (a) the amplitude of the motion, (b) the maximum acceleration of the block, and (c) the maximum force the spring exerts on the block.

31. Repeat the previous problem, but assume that initially the block has velocity –4.00 m/s and displacement +0.200 m.

32. You are watching an object that is moving in SHM. When the object is displaced 0.600 m to the right of its equilibrium position, it has a velocity of 2.20 m/s to the right and an acceleration of 8.40 m/s2 to the left. How much farther from this point will the object move before it stops momentarily and then starts to move back to the left?

33. A mass is oscillating with amplitude A at the end of a spring. How far (in terms of A) is this mass from the equilibrium position of the spring when the elastic potential energy equals the kinetic energy?

34. (a) If a vibrating system has total energy E0 what will its total energy be (in terms of E0) if you double the amplitude of vibration? (b) If you want to triple the total energy of a vibrating system with amplitude A0 what should its new amplitude be (in terms of A0)?

35. A 2.40 kg ball is attached to an unknown spring and allowed to oscillate. Figure 11.37 shows a graph of the ball’s position x as a function of time t. For this motion, what are (a) the period, (b) the frequency, (c) the angular frequency, and (d) the amplitude? (e) What is the force constant of the spring?

36. A proud deep-sea fisherman hangs a 65.0 kg fish from an ideal spring having negligible mass. The fish stretches the spring 0.120 m. (a) What is the force constant of the spring? (b) What is the period of oscillation of the fish if it is pulled down 3.50 cm and released?

37. One end of a stretched ideal spring is attached to an airtrack and the other is attached to a glider with a mass of 0.355 kg. The glider is released and allowed to oscillate in SHM. If the distance of the glider from the fixed end of the spring varies between 1.80 m and 1.06 m, and the period of the oscillation is 2.15 s, find (a) the force constant of the spring, (b) the maximum speed of the glider, and (c) the magnitude of the maximum acceleration of the glider.

38. A mass of 0.20 kg on the end of a spring oscillates with a period of 0.45 s and an amplitude of 0.15 m. Find (a) the velocity when it passes the equilibrium point, (b) the total energy of the system, and (c) the equation describing the motion of the mass, assuming that x was a maximum at time t = 0.

39. A harmonic oscillator is made by using a 0.600 kg frictionless block and an ideal spring of unknown force constant. The oscillator is found to have a period of 0.150 s. Find the force constant of the spring.

40. Weighing astronauts. In order to study the long-term effects of weightlessness, astronauts in space must be weighed (or at least “massed”). One way in which this is done is to seat them in a chair of known mass attached to a spring of known force constant and measure the period of the oscillations of this system. If the 35.4 kg chair alone oscillates with a period of 1.25 s, and the period with the astronaut sitting in the chair is 2.23 s, find (a) the force constant of the spring and (b) the mass of the astronaut.

41. A mass m is attached to a spring of force constant 75 N/m and allowed to oscillate. Figure 11.38 shows a graph of its velocity v as a function of time t. Find (a) the period, (b) the frequency, and (c) the angular frequency of this motion. (d) What is the amplitude (in cm), and at what times does the mass reach this position? (e) Find the maximum acceleration of the mass and the times at which it occurs. (f) What is the mass m?

42. An object of unknown mass is attached to an ideal spring with force constant 120 N/m and is found to vibrate with a frequency of 6.00 Hz. Find (a) the period, (b) the angular frequency, and (c) the mass of this object.

43. Weighing a virus. In February 2004, scientists at Purdue University used a highly sensitive technique to measure the mass of a vaccinia virus (the kind used in smallpox vaccine). The procedure involved measuring the frequency of oscillation of a tiny sliver of silicon (just 30 nm long) with a laser, first without the virus and then after the virus had attached itself to the silicon. The difference in mass caused a change in the frequency. We can model such a process as a mass on a spring. (a) Show that the ratio of the frequency with the virus attached (fS+V) to the frequency without the virus (fS) is given by the formula fS+V/fS = 1/√(1 + mV/mS) where mV is the mass of the virus and mS is the mass of the silicon sliver. Notice that it is not necessary to know or measure the force constant of the spring. (b) In some data, the silicon sliver has a mass of 2.10 × 10–16 g and a frequency of 2.00 × 1015 Hz without the virus and 2.87 × 1014 Hz with the virus. What is the mass of the virus, in grams and femtograms?

44. A science museum has asked you to design a simple pendulum that will make 25.0 complete swings in 85.0 s. What length should you specify for this pendulum?

45. A simple pendulum in a science museum entry hall is 3.50 m long, has a 1.25 kg bob at its lower end, and swings with an amplitude of 11.0°. How much time does the pendulum take to swing from its extreme right side to its extreme left side?

46. You’ve made a simple pendulum with a length of 1.55 m, and you also have a (very light) spring with force constant 2.45 N/m. What mass should you add to the spring so that its period will be the same as that of your pendulum?

47. A pendulum on Mars. A certain simple pendulum has a period on earth of 1.60 s. What is its period on the surface of Mars, where the acceleration due to gravity is 3.71 m/s2.

48. In the laboratory, a student studies a pendulum by graphing the angle θ that the string makes with the vertical as a function of time t, obtaining the graph shown in Figure 11.39. (a) What are the period, frequency, angular frequency, and amplitude of the pendulum’s motion? (b) How long is the pendulum? (c) Is it possible to determine the mass of the bob?

49. If a pendulum has period T and you double its length, what is its new period in terms of T? (b) If a pendulum has a length L and you want to triple its frequency, what should be its length in terms of L? (c) Suppose a pendulum has a length L and period T on earth. If you take it to a planet where the acceleration of freely falling objects is ten times what it is on earth, what should you do to the length to keep the period the same as on earth? (d) If you do not change the pendulum’s length in part (c), what is its period on that planet in terms of T? (e) If a pendulum has a period T and you triple the mass of its bob, what happens to the period (in terms of T)?

50. A 1.35 kg object is attached to a horizontal spring of force constant 2.5 N/cm and is started oscillating by pulling it 6.0 cm from its equilibrium position and releasing it so that it is free to oscillate on a frictionless horizontal air track. You observe that after eight cycles its maximum displacement from equilibrium is only 3.5 cm. (a) How much energy has this system lost to damping during these eight cycles? (b) Where did the “lost” energy go? Explain physically how the system could have lost energy.

51. A 2.50 kg rock is attached at the end of a thin, very light rope 1.45 m long and is started swinging by releasing it when the rope makes an 11° angle with the vertical. You record the observation that it rises only to an angle of 4.5° with the vertical after 10.5 swings. (a) How much energy has this system lost during that time? (b) What happened to the “lost” energy? Explain how it could have been “lost.”

52. A mass is vibrating at the end of a spring of force constant 225 N/m. Figure 11.40 shows a graph of its position x as a function of time t. (a) At what times is the mass not moving? (b) How much energy did this system originally contain? (c) How much energy did the system lose between t = 1.0 s and t = 4.0 s? Where did this energy go?

53. What is the maximum kinetic energy of the vibrating ball in Problem 35, and when does it occur?

54. Inside a NASA test vehicle, a 3.50-kg ball is pulled along by a horizontal ideal spring fixed to a friction-free table. The force constant of the spring is 225 N/m. The vehicle has a steady acceleration of 5.00 m/s2 and the ball is not oscillating. Suddenly, when the vehicle’s speed has reached 45.0 m/s its engines turn off, thus eliminating its acceleration but not its velocity. Find (a) the amplitude and (b) the frequency of the resulting oscillations of the ball. (c) What will be the ball’s maximum speed relative to the vehicle?

55. Four passengers with a combined mass of 250 kg compress the springs of a car with worn-out shock absorbers by 4.00 cm when they enter it. Model the car and passengers as a single body on a single ideal spring. If the loaded car has a period of vibration of 1.08 s, what is the period of vibration of the empty car?

56. An astronaut notices that a pendulum which took 2.50 s for a complete cycle of swing when the rocket was waiting on the launch pad takes 1.25 s for the same cycle of swing during liftoff. What is the acceleration of the rocket?

57. An object suspended from a spring vibrates with simple harmonic motion. At an instant when the displacement of the object is equal to one-half the amplitude, what fraction of the total energy of the system is kinetic and what fraction is potential?

58. On the planet Newtonia, a simple pendulum having a bob with a mass of 1.25 kg and a length of 185.0 cm takes 1.42 s, when released from rest, to swing through an angle of 12.5°, where it again has zero speed. The circumference of Newtonia is measured to be 51,400 km. What is the mass of the planet Newtonia?

59. An apple weighs 1.00 N. When you hang it from the end of a long spring of force constant 1.50 N/m and negligible mass, it bounces up and down in SHM. If you stop the bouncing and let the apple swing from side to side through a small angle, the frequency of this simple pendulum is half the bounce frequency. (Because the angle is small, the back-and-forth swings do not cause any appreciable change in the length of the spring.) What is the unstretched length of the spring (with the apple removed)?

60. A block with mass M rests on a frictionless surface and is connected to a horizontal spring of force constant k, the other end of which is attached to a wall. A second block with mass m rests on top of the first block. The coefficient of static friction between the blocks is μS . Find the maximum amplitude of oscillation such that the top block will not slip on the bottom block.

61. In Fig. 11.42 the upper ball is released from rest, collides with the stationary lower ball, and sticks to it. The strings are both 50.0 cm long. The upper ball has mass 2.00 kg, and it is initially 10.0 cm higher than the lower ball, which has mass 3.00 kg. Find the frequency and maximum angular displacement of the motion after the collision.

62. A 15.0 kg mass fastened to the end of a steel wire with an unstretched length of 0.50 m is whirled in a vertical circle with angular velocity 2.00 rev/s at the bottom of the circle. The cross-sectional area of the wire is 0.010 cm2. Calculate the elongation of the wire when the mass is at the lowest point of the path.

63. Stress on the shinbone. The compressive strength of our bones is important in everyday life. Young’s modulus for bone is approximately 14 GPa. Bone can take only about a 1.0% change in its length before fracturing. If Hooke’s law were to hold up to fracture: (a) What is the maximum force that can be applied to a bone whose minimum cross-sectional area is (This is approximately the cross-sectional area of a tibia, or shinbone, at its narrowest point.) (b) Estimate the maximum height from which a 70 kg man can jump and not fracture the tibia. Take the time between when he first touches the floor and when he has stopped to be 0.030 s, and assume that the stress is distributed equally between his legs.

64. You hang a floodlamp from the end of a vertical steel wire. The floodlamp stretches the wire 0.18 mm and the stress is proportional to the strain. How much would it have stretched (a) if the wire were twice as long? (b) If the wire had the same length but twice the diameter? (c) For a copper wire of the original length and diameter?

65. Tendon-stretching exercises. As part of an exercise program, a 75 kg person does toe raises in which he raises his entire body weight on the ball of one foot, as shown in Figure 11.43. The Achilles tendon pulls straight upward on the heel bone of his foot. This tendon is 25 cm long and has a cross-sectional area of 78 mm2 and a Young’s modulus of 1470 MPa. (a) Make a free-body diagram of the person’s foot (everything below the ankle joint).You can neglect the weight of the foot. (b) What force does the Achilles tendon exert on the heel during this exercise? Express your answer in newtons and in multiples of his weight. (c) By how many millimeters does the exercise stretch his Achilles tendon?

66. A 100 kg mass suspended from a wire whose unstretched length is 4.00 m is found to stretch the wire by 6.0 mm. The wire has a uniform cross-sectional area of 0.10 cm2 (a) If the load is pulled down a small additional distance and released, find the frequency at which it vibrates. (b) Compute Young’s modulus for the wire.

67. A brass rod with a length of 1.40 m and a cross-sectional area of 2.00 cm2 is fastened end to end to a nickel rod with length L and cross-sectional area 1.00 cm2. The compound rod is subjected to equal and opposite pulls of magnitude 4.00 × 104 N at its ends. (a) Find the length L of the nickel rod if the elongations of the two rods are equal. (b) What is the stress in each rod? (c) What is the strain in each rod?

68. Rapunzel, Rapunzel, let down your golden hair. In the Grimms’ fairy tale Rapunzel, she lets down her golden hair to a length of 20 yards (we’ll use 20 m, which is not much different) so that the prince can climb up to her room. Human hair has a Young’s modulus of about 490 MPa, and we can assume that Rapunzel’s hair can be squeezed into a rope about 2.0 cm in cross-sectional diameter. The prince is described as young and handsome, so we can estimate a mass of 60 kg for him. (a) Just after the prince has started to climb at constant speed, while he is still near the bottom of the hair, by how many centimeters does he stretch Rapunzel’s hair? (b) What is the mass of the heaviest prince that could climb up, given that the maximum tensile stress hair can support is 196 MPa?

69. Crude oil with a bulk modulus of 2.35 GPa is leaking from a deep-sea well 2250 m below the surface of the ocean, where the water pressure is Pa. Suppose 35,600 barrels of oil leak from the wellhead; assuming all that oil reaches the surface, how many barrels will it be on the surface?

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