chapter 8
10. Anika and Sam are astronauts floating in space outside their spaceship. They are tethered together by a rope of length 10.5 m. They both begin pulling on the rope at the same time. Anika has mass of 80 kg, and Sam 100 kg. Because her mass is less, Anika reaches the middle of the rope first. Assuming the center of the rope is at the origin and that Anika and Sam start at 5.25 m and -5.25 m, respectively, where is Sam when Anika reaches the middle?
a. 1.05 m
19. A 650-N billboard worker stands on a 4.00-m scaffold weighing 600 N and supported by vertical ropes at each end. How far would the worker stand from one of the supporting ropes to produce a tension of 660 N in that rope?
a. 1.78 m
63. A hoop (I = MR2) is rolling along the ground at 10 m/s. It comes to a hill and starts going up. Assuming no losses to friction, how far does it travel vertically up the hill before it stops?
a. 10 m
27. A 2.00-kg mass is placed at (3.00, 4.00) m, and a 3.50-kg mass is placed at (3.00, −4.00) m. What is the moment of inertia of this system of masses about the z axis?
a. 138 kg·m2
54. A bus is designed to draw its power from a rotating flywheel that is brought up to its maximum speed (3000 rpm) by an electric motor. The flywheel is a solid cylinder of mass 250 kg and radius 0.500 m (Icylinder = MR2/2). If the bus requires an average power of 10.0 kW, how long will the flywheel rotate?
a. 154 s
53. An initially installed flywheel can store 106 J of kinetic energy when rotating at 300 rad/s. It is replaced by another flywheel of the same size but made of a stronger material. If its mass is the same as that of the original, and it is now capable of achieving a rotational speed of 425 rad/s, what maximum energy can be stored?
a. 2.01E+6 J
30. A majorette takes two batons and fastens them together in the middle at right angles to make an "x" shape. Each baton was 0.70 m long and each ball on the end is 0.10 kg. (Ignore the mass of the rods.) What is the moment of inertia if the arrangement is spun around an axis formed by one of the batons?
a. 2.5E-2 kg·m2
76. A turntable has a moment of inertia of 3.00 × 10−2 kg·m2 and spins freely on a frictionless bearing at 25.0 rev/min. A 0.600-kg ball of putty is dropped vertically onto the turntable and sticks at a point 0.100 m from the center. What is the new rate of rotation of the system?
a. 20.8 rev/min
21. A uniform, horizontal beam of length 8.0 m and weight 42.4 N is attached at one end to a wall by a pin connection (so that it may rotate). A cable attached to the wall above the pin supports the opposite end. The cable makes an angle of 60.0° with the horizontal. What is the tension in the cable needed to maintain the beam in equilibrium?
a. 24.5 N
8. A "double satellite" is launched into space. It consists of a satellite of mass m and another satellite of mass 5m. The connecting mechanism of these satellites is an explosive bolt and a compressed spring, both of negligible mass. The bolt is exploded and the satellites drift apart having been pushed by the spring. When the less massive satellite has moved a distance of 2.5 m from the center of mass of the double satellite, how far away from it is the satellite of greater mass?
a. 3.0 m
35. A non-uniform rod of length L and mass M is pivoted about one end. The moment of inertia of this rod about its pivoted end is 1ML2/4. Attached to the rod is a mass 3M at its midpoint and at the end opposite the pivot is another mass 2M. What is the moment of inertia of this configuration about the pivot?
a. 3^ML^2
61. A solid sphere of mass 2.50 kg and radius 0.120 m starts from rest at the top of a ramp inclined 15.0° and rolls to the bottom. The upper end of the ramp is 1.60 m higher than the lower end. What is the linear speed of the sphere when it reaches the bottom of the ramp? (Note: I = 2/5 MR2 for a solid sphere, and g = 9.80 m/s2.)
a. 4.73 m/s
64. A meter stick is hinged at its lower end and allowed to fall from a vertical position. If its moment of inertia is ML2/3, with what angular speed does it hit the table?
a. 5.42 rad/s
47. A bowling ball has a mass of 7.00 kg, a moment of inertia of 2.80 × 10−2 kg·m2, and a radius of 0.100 m. If it rolls down the lane without slipping at a linear speed of 4.00 m/s, what is its total kinetic energy?
a. 78.4 J
33. An object consists of a rod (of length 3.00 m and negligible moment of inertia) to which four small 3.50-kg masses are attached, one at each end and one at each point on the rod spaced at 1.00-m intervals. The moment of inertia of this object about an axis perpendicular to the rod and through one of the end masses:
a. is 49 kg·m2.
68. A hoop, a solid sphere, and a solid cylinder each have the same mass, radius, and initial speed and are going up the same inclined plane. Which one goes the farthest?
a. the hoop
82. An object of mass m and moment of inertia I has rotational kinetic energy KER. Its angular momentum is:
b. (2 IKER)^1/2.
81. An object with mass m and moment of inertia I is spinning with an angular momentum L. Its kinetic energy is:
b. 0.5L2/I.
79. A tetherball is attached to a pole with a 2.0-m rope. It is circling at 0.20 rev/s. As the rope wraps around the pole, it shortens. How long is the rope when the ball is moving at 6.0 m/s?
b. 0.83 m
15. Tasha has mass 20 kg and wants to use a 5.0-m board of mass 12 kg as a seesaw. Her friends are busy, so Tasha seesaws by herself by putting the support at the system's center of gravity when she sits on one end of the board. How far is she from the support point?
b. 0.94 m
72. A ventilation fan with a moment of inertia of 0.034 kg·m2 has a net torque of 0.11 N·m applied to it. If it starts from rest, what angular momentum will it have 9.0 s later?
b. 0.99 kg·m2/s
77. A turntable has a moment of inertia of 3.00 × 10−2 kg·m2 and spins freely on a frictionless bearing at 25.0 rev/min. A 0.600-kg ball of putty is dropped vertically on the turntable and sticks at a point 0.100 m from the center. By what factor does the angular momentum of the system change after the putty is dropped onto the turntable?
b. 1.00 (no change)
3. A uniform meterstick balances on a fulcrum placed at the 70.0-cm mark when a weight w is placed at the 90.0-cm mark. What is the weight of the meterstick?
b. 1.0w
48. A bucket of water with total mass 21.0 kg is attached to a rope, which in turn is wound around a 0.050-m radius cylinder, with crank, at the top of a well. The moment of inertia of the cylinder and crank is 0.21 kg m2. The bucket is raised to the top of the well and released to fall back into the well. What is the kinetic energy of the cylinder and crank at the instant the bucket is moving with a speed of 5.4 m/s?
b. 1.2E+3 J
16. A 60-kg man is one fourth of the way up a 10-m ladder that is resting against a smooth, frictionless wall. If the ladder has a mass of 25 kg and it makes an angle of 60° with the ground, find the force of friction of the ground on the foot of the ladder.
b. 1.6E+2 N
29. A baseball pitcher loosens up his pitching arm. He tosses a 0.20-kg ball using only the rotation of his forearm, 0.28 m in length, to accelerate the ball. What is the moment of inertia of the ball alone as it moves in a circular arc with a radius of 0.28 m?
b. 1.6E-2 kg·m2
75. A figure skater on ice with arms extended, spins at a rate of 3.5 rev/s. After he draws his arms in, he spins at 6.0 rev/s. By what factor does the skater's kinetic energy change when he draws his arms in?
b. 1.7
49. A cylinder with its mass concentrated toward the center has a moment of inertia of 0.2MR2. If this cylinder is rolling without slipping along a level surface with a linear speed v, what is the ratio of its rotational kinetic energy to its linear kinetic energy?
b. 1/5
57. A non-uniform round object of mass M and radius R is found to have its total kinetic energy to be given by KE = 3/5 Mv2 when rolling down an incline. What is this object's moment of inertia?
b. 1/5 MR2
46. A ventilation fan with a moment of inertia of 0.0340 kg·m2 has a net torque of 0.110 N m applied to it. If it starts from rest, what kinetic energy will it have 10.4 s later?
b. 19.2 J
7. Three masses are positioned along the x-axis. Their masses and positions are m1 = m with x1 = 1.0 m, m2 = 2m with x2 = 2.0 m, and m3 = 9.0m with x3 = 3.0 m. What is the x-coordinate of the center of mass of this system, to the nearest tenth?
b. 2.7 m
41. A bucket of water with total mass 26 kg is attached to a rope, which in turn is wound around a 0.04-m radius cylinder at the top of a well. A crank with a turning radius of 0.20 m is attached to the end of the cylinder and the moment of inertia of cylinder and crank is 0.11 kg·m2. If the bucket is raised to the top of the well and released, what is the acceleration of the bucket as it falls toward the bottom of the well? (Assume rope's mass is negligible, that cylinder turns on frictionless bearings and that g = 9.8 m/s2.)
b. 2.7 m/s2
51. A rotating flywheel can be used as a method to store energy. If it is required that such a device be able to store up to a maximum of 2.00 × 106 J when rotating at 400 rad/s, what moment of inertia is required?
b. 25.0 kg m2
4. Two children seat themselves on a seesaw. The one on the left has a weight of 300 N while the one on the right weighs 200 N. The fulcrum is at the midpoint of the seesaw. If the child on the right is not at the end but is 1.50 m from the fulcrum and the seesaw is balanced, what is the torque provided by the weight of the child on the left? (Take ccw as positive.)
b. 300 N·m
37. A ventilation fan with a moment of inertia of 0.040 kg·m2 has a net torque of 0.160 N·m applied to it. What angular acceleration does it experience?
b. 4.0 rad/s2
62. A solid cylinder of mass 3.0 kg and radius 0.2 m starts from rest at the top of a ramp, inclined 15°, and rolls to the bottom without slipping. (For a cylinder I = MR2.) The upper end of the ramp is 1.5 m higher than the lower end. Find the linear speed of the cylinder when it reaches the bottom of the ramp. (g = 9.8 m/s2)
b. 4.4 m/s
52. A rotating flywheel can be used as a method to store energy. If it has 2.00 × 106 J of kinetic energy when rotating at 400 rad/s, and if a frictional torque of 4.00 N m acts on the system, in what interval of time would the flywheel come to rest?
b. 41.7 min
28. A bowling ball has a mass of 7.00 kg, a moment of inertia of 2.8 × 10−2 kg·m2, and a radius of 0.100 m. If it rolls down the lane without slipping at a linear speed of 4.30 m/s, what is its angular speed?
b. 43.0 rad/s
44. A solid sphere of radius R is placed on an incline that makes an angle θ with the horizontal. The sphere rolls down the incline without slipping. What is its acceleration?
b. 5/7 g sinθ
80. An astronaut is on a 100-m lifeline outside a spaceship, circling the ship with an angular speed of 0.100 rad/s. How far inward can she be pulled before the centripetal acceleration reaches 4g = 39.2 m/s2?
b. 70.6 m
5. A bucket filled with water has a mass of 20.0 kg and is attached to a rope, which in turn, is wound around a 0.050-m radius cylinder at the top of a well. What torque does the weight of water and bucket produce on the cylinder if the cylinder is not permitted to rotate? (g = 9.80 m/s2)
b. 9.8 N·m
71. The quantity "angular momentum" (in terms of the fundamental quantities of mass, length, and time) is equivalent to:
b. ML2T−1.
73. A figure skater with arms initially extended starts spinning on the ice at 3 rad/s. She then pulls her arms in close to her body. Which of the following results?
b. a greater rotational rate
36. If a nonzero torque is applied to an object, that object will experience:
b. an angular acceleration.
74. An ice skater spins at 3.5 rev/s with his arms are extended. He draws his arms in and spins at 8.3 rev/s. By what factor does his moment of inertia change in the process?
c. 0.42
78. A turntable has a moment of inertia of 3.0 × 10−2 kg·m2 and spins freely on a frictionless bearing at 25 rev/min. A 0.60-kg ball of putty is dropped vertically on the turntable and sticks at a point 0.10 m from the center. By what factor does the kinetic energy of the system change after the putty is dropped onto the turntable?
c. 0.83
50. A hoop with mass, M, and radius, R, rolls along a level surface without slipping with a linear speed, v. What is the ratio of rotational to linear kinetic energy? (For a hoop, I = MR2.)
c. 1/1
25. A 5.0-kg mass is placed at (3.0, 4.0) m, and a 6.0-kg mass is placed at (3.0, −4.0) m. What is the moment of inertia of this system of masses about the x axis?
c. 176 kg·m2
40. A baseball pitcher loosens up his pitching arm. He tosses a 0.140-kg ball using only the rotation of his forearm, 0.270 m in length, to accelerate the ball. If the ball starts at rest and is released with a speed of 24.0 m/s in a time of 0.425 s, what torque is applied to the ball while being held by the pitcher's hand to produce the angular acceleration?
c. 2.13 N·m
45. A bucket of water with total mass 20.0 kg is attached to a rope, which in turn is wound around a 4.00-cm radius cylinder at the top of a well. The bucket is raised to the top of the well and released. The bucket is moving with a speed of 8.00 m/s upon hitting the water surface in the well. What is the angular speed of the cylinder at this instant?
c. 200 rad/s
60. A solid sphere of mass 2.50 kg and radius 0.120 m is at rest at the top of a ramp inclined 15.0°. It rolls to the bottom without slipping. The upper end of the ramp is 1.20 m higher than the lower end. Find the sphere's total kinetic energy when it reaches the bottom.
c. 29.4 J
6. A bucket of water with total mass 24.0 kg is attached to a rope, which in turn, is wound around a 0.050-m radius cylinder at the top of a well. A crank with a turning radius of 0.35 m is attached to the end of the cylinder. What minimum force directed perpendicular to the crank handle is required to just raise the bucket? (Assume the rope's mass is negligible, that the cylinder turns on frictionless bearings, and that g = 9.8 m/s2.)
c. 34 N
42. A rod of length L is hinged at one end. The moment of inertia as the rod rotates around that hinge is ML2/3. Suppose a 2.00-m rod with a mass of 6.00 kg is hinged at one end and is held in a horizontal position. The rod is released as the free end is allowed to fall. What is the angular acceleration as it is released?
c. 7.35 rad/s2
2. A 3.00-m rod is pivoted about its left end. A force of 7.80 N is applied perpendicular to the rod at a distance of 1.60 m from the pivot causing a ccw torque, and a force of 2.60 N is applied at the end of the rod 3.00 m from the pivot. The 2.60-N force is at an angle of 30.0o to the rod and causes a cw torque. What is the net torque about the pivot? (Take ccw as positive.)
c. 8.58 N·m
56. The Earth's density increases with depth. If instead the Earth had uniform density, but the same angular velocity and mass, how would its rotational kinetic energy compare to the actual value?
c. It would be greater than the actual value.
67. A solid sphere and a solid cylinder both have the same radius and mass. If the sphere is spun about an axis through it center with angular velocity sphere, what angular velocity must the cylinder have to the same rotational kinetic energy as the sphere? Round your answer to 2 significant figures.
c. cylinder = 0.89 sphere
A rod system with a uniform linear density of l = 4.40 kg/m has the shape shown in the figure, with rod 1 the vertical rod and rod 2 the horizontal rod. Rod 1 has length L1 = 8.00 m, and rod 2 has length L2 = 7.50 m. Neglect the diameter of the rods. 12. What is the center of mass of each rod?
c. rod 1: (2.50, 4.00), rod 2: (3.75 m, 0 m)
55. A solid sphere, a solid cylinder, and a hoop each have the same mass and radius. If they are spinning at the same angular velocity, which one has the greatest rotational kinetic energy?
c. the hoop
69. A solid sphere, a solid cylinder, and a hoop each have the same mass and radius. If they are spinning at the same angular velocity, which one has the greatest angular momentum?
c. the hoop
70. A 40-kg boy is standing on the edge of a stationary 30-kg circular platform that is free to rotate. The boy tries to walk around the platform in a counterclockwise direction. As he does:
c. the platform rotates in a clockwise direction while the boy goes around in a counterclockwise direction relative to the ground.
23. The torque caused by a force F applied at a distance r from the axis of rotation, is given by τ = rFsinθ, where θ is the angle between and . If the magnitudes r and F remain the same, what other angle θ' will produce the same torque as was produced at angle θ if θ was less than 90°?
c. θ' = 180° - θ
A rod system with a uniform linear density of l = 4.40 kg/m has the shape shown in the figure, with rod 1 the vertical rod and rod 2 the horizontal rod. Rod 1 has length L1 = 8.00 m, and rod 2 has length L2 = 7.50 m. Neglect the diameter of the rods. 13. What is the center of mass of the system?
d. (3.10 m, 2.06 m)
9. A system of 3 masses is located as follows: m1 = m0 at (0, L), m2 = 2m0 at the origin, and m3 = 3m0 at (3L, 0). Where is the center of mass of these masses?
d. (3/2 L, 1/6 L)
43. A bucket in an old well is hoisted upward by a rope which winds up on a cylinder having a radius of 0.050 m. How many rev/s must the cylinder turn if the bucket is raised at a speed of 0.250 m/s?
d. 0.80 rev/s
20. A woman who weighs 500 N stands on a 6.0-m-long board that weighs 100 N. The board is supported at each end. The support force at the right end is 3 times the support force at the left end. How far from the right end is the woman standing?
d. 1.2 m
1. A bank vault is opened by applying a force of 140 N perpendicular to the plane of the door, 1.20 m from the hinges. Find the torque due to this force about an axis through the hinges.
d. 168 N·m
39. A baseball pitcher, loosening up his arm before a game, tosses a 0.140-kg ball using only the rotation of his forearm, 0.290 m in length, to accelerate the ball. If the ball starts at rest and is released with a speed of 24.0 m/s in a time of 0.450 s, what is the average angular acceleration of the arm and ball?
d. 184 rad/s2
34. A uniform rod of length L and mass M is pivoted about one end. Attached to the rod is a mass M at its midpoint and at the end opposite the pivot is another mass M. What is the moment of inertia of this configuration about the pivot?
d. 19ML2/12
38. A disk has a moment of inertia of 3.0 × 10−4 kg·m2 and rotates with an angular speed of 3.6 rad/s. What net torque must be applied to bring it to rest within 3.6 s?
d. 3.00E-4 N·m
26. A 3.00-kg mass is placed at (2.00, 3.00) m, and a 4.50-kg mass is placed at (2.00, −3.00) m. What is the moment of inertia of this system of masses about the y axis?
d. 30.0 kg·m2
18. A meterstick is supported by a knife-edge at the 50.0-cm mark. Doug hangs masses of 0.48 and 0.76 kg from the 10cm and 80-cm marks, respectively. Where should Doug hang a third mass of 0.25 kg to keep the stick balanced?
d. 35.6 cm
31. A majorette takes two batons and fastens them together in the middle at right angles to make an "x" shape. Each baton was 0.70 m long and each ball on the end is 0.10 kg. (Ignore the mass of the rods.) What is the moment of inertia if the arrangement is spun around an axis through the center perpendicular to both rods?
d. 4.9E-2 kg·m2
14. A uniform bridge span weighs 70.0 kN and is 40.0 m long. An automobile weighing 15.0 kN is parked with its center of gravity located 15.0 m from the right pier. What upward support force does the right pier provide?
d. 44.4 kN
17. A 274-N uniform ladder, 11.0 m long, rests against a smooth vertical wall. The coefficient of static friction between ladder and floor is 0.400. What minimum angle can the ladder make with the floor before it slips?
d. 51.3°
32. Two hoops or rings (I = MR2) are centered, lying on a turntable. The smaller ring has radius = 0.05 m; the larger has radius = 0.10 m. Both have a mass of 5.0 kg. What is the total moment of inertia as the turntable spins? Ignore the mass of the turntable.
d. 6.3E-2 kg·m2
59. A box slides down a frictionless incline, and a hoop rolls down another incline. Both inclines have the same height, and both the box and the hoop have the same mass. If both objects start from rest, upon reaching the bottom of the incline which one will have the greater kinetic energy and which one will have the greater speed?
d. Both will have the same kinetic energy, but the box will have the greater speed.
58. Three disks have the same mass and the same total kinetic energy. Disk #1 has radius R, disk #2 has radius 2R, and disk #3 has radius 3R. Which one has the highest vtranslational?
d. The all have the same vtranslational.
22. A hoop, solid sphere, and solid cylinder each have the same mass and same radius. When rolling down the same incline, which has the greatest torque from gravity on it?
d. They all have the same torque.
65. An object of radius R and moment of inertia I rolls down an incline of height H after starting from rest. Its total kinetic energy at the bottom of the incline:
d. cannot be found from the given information alone.
66. A uniform solid sphere rolls down an incline of height 3 m after starting from rest. In order to calculate its speed at the bottom of the incline, one needs to know:
d. no more than is given in the problem
A rod system with a uniform linear density of l = 4.40 kg/m has the shape shown in the figure, with rod 1 the vertical rod and rod 2 the horizontal rod. Rod 1 has length L1 = 8.00 m, and rod 2 has length L2 = 7.50 m. Neglect the diameter of the rods. 11. What is the mass of each rod?
d. rod 1: 35.2 kg, rod 2: 33.0 kg
24. An 8.4-kg mass is placed at (3.0, 4.0) m. Where can a 4.2-kg mass be placed so that the moment of inertia about the z axis is zero?
e. There is no position giving this result