Queens College, CUNY - PHYSICS 152

Chapter 10—Rotational Motion

MULTIPLE CHOICE

1)At t = 0, a wheel rotating about a fixed axis at a constant angular acceleration has an angular velocity of 2.0 rad/s. Two seconds later it has turned through 5.0 complete revolutions. What is the angular acceleration of this wheel?

1. 1. 17 rad/s²
   2. 14 rad/s²
   3. 20 rad/s²
   4. 23 rad/s²
   5. 13 rad/s²

2. At t = 0, a wheel rotating about a fixed axis at a constant angular acceleration of ?0.40 rad/s² has an angular velocity of 1.5 rad/s and an angular position of 2.3 rad. What is the angular position of the wheel at t = 2.0 s?
   1. 4.9 rad
   2. 4.7 rad
   3. 4.5 rad
   4. 4.3 rad
   5. 4.1 rad

3. A wheel rotating about a fixed axis has an angular position given by ? = 3.0 ? 2.0t³, where ? is measured in radians and t in seconds. What is the angular acceleration of the wheel at t = 2.0 s?
   1. ?1.0 rad/s²
   2. ?24 rad/s²
   3. ?2.0 rad/s²
   4. ?4.0 rad/s²
   5. ?3.5 rad/s²

4. A wheel rotating about a fixed axis with a constant angular acceleration of 2.0 rad/s² turns through 2.4 revolutions during a 2.0-s time interval. What is the angular velocity at the end of this time interval?
   1. 9.5 rad/s
   2. 9.7 rad/s
   3. 9.3 rad/s
   4. 9.1 rad/s
   5. 8.8 rad/s

5. The turntable of a record player has an angular velocity of 8.0 rad/s when it is turned off. The turntable comes to rest 2.5 s after being turned off. Through how many radians does the turntable rotate after being turned off? Assume constant angular acceleration.
   1. 12 rad
   2. 8.0 rad
   3. 10 rad
   4. 16 rad

1. 5. 6.8 rad

6. A wheel rotates about a fixed axis with an initial angular velocity of 20 rad/s. During a 5.0-s interval the angular velocity increases to 40 rad/s. Assume that the angular acceleration was constant during the 5.0-s interval. How many revolutions does the wheel turn through during the 5.0-s interval?
   1. 20 rev
   2. 24 rev
   3. 32 rev
   4. 28 rev
   5. 39 rev

7. A wheel rotates about a fixed axis with an initial angular velocity of 20 rad/s. During a 5.0-s interval the angular velocity decreases to 10 rad/s. Assume that the angular acceleration is constant during the 5.0-s interval. How many radians does the wheel turn through during the 5.0-s interval?
   1. 95 rad
   2. 85 rad
   3. 65 rad
   4. 75 rad
   5. 125 rad

8. A wheel starts from rest and rotates with a constant angular acceleration about a fixed axis. It completes the first revolution 6.0 s after it started. How long after it started will the wheel complete the second revolution?
   1. 9.9 s
   2. 7.8 s
   3. 8.5 s
   4. 9.2 s
   5. 6.4 s

9. A wheel rotating about a fixed axis has a constant angular acceleration of 4.0 rad/s². In a 4.0-s interval the wheel turns through an angle of 80 radians. Assuming the wheel started from rest, how long had it been in motion at the start of the 4.0-s interval?
   1. 2.5 s
   2. 4.0 s
   3. 3.5 s
   4. 3.0 s
   5. 4.5 s

10. The figure below shows a graph of angular velocity as a function of time for a car driving around a circular track. Through how many radians does the car travel in the first 10 minutes?

1. 1. 30
   2. 50
   3. 70
   4. 90

e.   100

11. The graphs below show angular velocity as a function of time. In which one is the magnitude of the angular acceleration constantly decreasing?
    1. b.                         c.                         d.                         e.

                                 : Exhibit 10-01

Exhibit 10-1

The figure below shows a graph of angular velocity versus time for a woman bicycling around a circular track.

Use this exhibit to answer the following question(s).

12. Refer to Exhibit 10-1. What is her angular displacement (in rad) in the first 8 minutes?
    1. 0
    2. ?

1. 3. 4?
   4. 8?

e.   16?

13. Refer to Exhibit 10-1. What is her angular displacement (in rad) in the first 12 minutes?
    1. 0
    2. 2?
    3. 4?

d.   16?

e.   32?

14. Refer to Exhibit 10-1. What is her angular displacement (in rad) in the 16 minute period shown in the graph?
    1. 0

b.   16?

c.   32?

d.   40?

e.   64?

15. Refer to Exhibit 10-1. How many revolutions does she complete in the first 12 minutes?
    1. 4
    2. 8
    3. 12
    4. 16
    5. 32

16. Refer to Exhibit 10-1. How many revolutions does she complete in the 16 minute period?
    1. 8
    2. 12
    3. 16
    4. 20
    5. 40

17. The angular speed of the minute hand of a clock, in rad/s, is a.

.

b.

.

c.

.

d. ?.

e.   120?.

18. The angular speed of the hour hand of a clock, in rad/s, is a.

.

b.

.

c.

.

d.   1 800?.

e.   7 200?.

19. The angular speed of the hour hand of a clock, in rad/min, is a.

.

b.

.

c.

.

d. ?.

e.   120?.

: Exhibit 10-02

Exhibit 10-2

The figure below shows a graph of angular velocity versus time for a man bicycling around a circular track.

Use this exhibit to answer the following question(s).

20. Refer to Exhibit 10-2. What is his average angular acceleration, in rad/s², in the first 10 minutes?
    1. 0 b.

c.

d.

e.

21. Refer to Exhibit 10-2. What is his average angular acceleration, in rad/s², in the period from t = 6 min to t = 8 min?
    1. 0 b.

c.

d.

e.

22. A small sphere attached to a light rigid rod rotates about an axis perpendicular to and fixed to the other end of the rod. Relative to the positive direction of the axis of rotation, the angular positions of the sphere are positive, its angular velocity is positive, and its angular acceleration is negative. The sphere is
    1. rotating clockwise and slowing down.
    2. rotating counterclockwise and slowing down.
    3. rotating clockwise and speeding up.
    4. rotating counterclockwise and speeding up.
    5. first rotating clockwise and then counterclockwise.
23. A small sphere attached to a light rigid rod rotates about an axis perpendicular to and fixed to the other end of the rod. Relative to the positive direction of the axis of rotation, the angular positions of the sphere are negative, its angular velocity is negative, and its angular acceleration is positive. The sphere is
    1. rotating clockwise and slowing down.
    2. rotating counterclockwise and slowing down.
    3. rotating clockwise and speeding up.
    4. rotating counterclockwise and speeding up.
    5. first rotating counterclockwise and then clockwise.
24. A small sphere attached to a light rigid rod rotates about an axis perpendicular to and fixed to the other end of the rod. Relative to the positive direction of the axis of rotation, the angular positions of the sphere are negative, its angular velocity is negative, and its angular acceleration is negative. The sphere is
    1. rotating clockwise and slowing down.
    2. rotating counterclockwise and slowing down.
    3. rotating clockwise and speeding up.
    4. rotating counterclockwise and speeding up.
    5. first rotating counterclockwise and then clockwise.     

: Exhibit 10-03

Exhibit 10-3

The graph below shows a plot of angular velocity in rad/s versus time in s from t = 0 s to t = 7 s.

Use this exhibit to answer the following question(s).

25. Refer to Exhibit 10-3. The change in angular position, ??, during the 7-second period is
    1. 21 rad, CW.
    2. 21 rad, CCW.
    3. 30 rad, CW.
    4. 30 rad, CCW.
    5. 39 rad, CCW.

26. Refer to Exhibit 10-3. The angular position, ?, at t = 0 s is 3.0 rad, clockwise. The angular position, ?, at t = 7 s is
    1. 27 rad, CW.
    2. 27 rad, CCW.
    3. 33 rad, CW.
    4. 33 rad, CCW.
    5. 36 rad, CCW.

27. The graph below shows a plot of angular acceleration in rad/s² versus time from t = 0 s to t = 8 s. The change in angular velocity, ??, during this 8-second period is

a.

, CW.

b.

, CCW.

c.

, CW.

d.

, CCW.

e.

, CW.

28. The graph below shows a plot of angular acceleration in rad/s² versus time from t = 0 s to t = 8 s. The angular velocity at t = 0 s is       , CCW. The angular velocity, ?, at t = 8 s is

a.

, CW.

b.

, CCW.

c.

, CW.

d.

, CCW.

e.

, CW.

29. A rigid rod of length l rotates about an axis perpendicular to the rod, with one end of the rod fixed to the axis. Which of the following are equal at all points on the rod?

1. the angular position
2. the angular velocity
3. the angular acceleration
4. the centripetal acceleration
5. the tangential acceleration

1. 1. I and II
   2. I, II, and III
   3. I, II, III and IV
   4. I, II, III, IV and V
   5. I, II and IV.

30. A wheel rotating about a fixed axis with a constant angular acceleration of 2.0 rad/s² starts from rest at t = 0. The wheel has a diameter of 20 cm. What is the magnitude of the total linear acceleration of a point on the outer edge of the wheel at t = 0.60 s?

a.   0.24 m/s²

b.   0.50 m/s²

c.   0.14 m/s²

d.   0.34 m/s²

e.   0.20 m/s²

31. A wheel rotates about a fixed axis with a constant angular acceleration of 4.0 rad/s². The diameter of the wheel is 40 cm. What is the linear speed of a point on the rim of this wheel at an instant when that point has a total linear acceleration with a magnitude of 1.2 m/s²?
    1. 39 cm/s
    2. 42 cm/s
    3. 45 cm/s
    4. 35 cm/s
    5. 53 cm/s

32. A disk (radius = 8.0 cm) that rotates about a fixed axis starts from rest and accelerates at a constant rate to an angular velocity of 4.0 rad/s in 2.0 s. What is the magnitude of the total linear acceleration of a point on the rim of the disk at the instant when the angular velocity of the disk is 1.5 rad/s?
    1. 24 cm/s²

33. A horizontal disk with a radius of 10 cm rotates about a vertical axis through its center. The disk starts from rest at t = 0 and has a constant angular acceleration of 2.1 rad/s². At what value of t will the radial and tangential components of the linear acceleration of a point on the rim of the disk be equal in magnitude?

a.   0.55 s

b.   0.63 s

c.   0.69 s

d.   0.59 s

e.   0.47 s

34. Two particles (m₁ = 0.20 kg, m₂ = 0.30 kg) are positioned at the ends of a 2.0-m long rod of negligible mass. What is the moment of inertia of this rigid body about an axis perpendicular to the rod and through the center of mass?

a.   0.48 kg?m²

b. 0.50 kg?m²

c.   1.2 kg?m²

d. 0.80 kg?m²

e.   0.70 kg?m²

35. Four identical particles (mass of each = 0.24 kg) are placed at the vertices of a rectangle (2.0 m ? 3.0 m) and held in those positions by four light rods which form the sides of the rectangle. What is the moment of inertia of this rigid body about an axis that passes through the center of mass of the body and is parallel to the shorter sides of the rectangle?
    1. 2.4 kg?m²
    2. 2.2 kg?m²
    3. 1.9 kg?m²
    4. 2.7 kg?m²
    5. 8.6 kg?m²

36. A thin uniform rod (length = 1.2 m, mass = 2.0 kg) is pivoted about a horizontal, frictionless pin through one end of the rod. (The moment of inertia of the rod about this axis is ML²/3.) The rod is released when it makes an angle of 37? with the horizontal. What is the angular acceleration of the rod at the instant it is released?
    1. 9.8 rad/s²
    2. 7.4 rad/s²
    3. 8.4 rad/s²
    4. 5.9 rad/s²
    5. 6.5 rad/s²

37. Four identical particles (mass of each = 0.40 kg) are placed at the vertices of a rectangle (2.0 m ? 3.0 m) and held in those positions by four light rods which form the sides of the rectangle. What is the moment of inertia of this rigid body about an axis that passes through the mid-points of the longer sides and is parallel to the shorter sides?
    1. 2.7 kg?m²
    2. 3.6 kg?m²
    3. 3.1 kg?m²
    4. 4.1 kg?m²
    5. 1.6 kg?m²

38. The rigid object shown is rotated about an axis perpendicular to the paper and through point P. The total kinetic energy of the object as it rotates is equal to 1.4 J. If M = 1.3 kg and L = 0.50 m, what is the angular velocity of the object? Neglect the mass of the connecting rods and treat the masses as particles.

1. 1. 1.3 rad/s
   2. 1.5 rad/s
   3. 1.7 rad/s
   4. 1.2 rad/s
   5. 2.1 rad/s

39. If M = 0.50 kg, L = 1.2 m, and the mass of each connecting rod shown is negligible, what is the moment of inertia about an axis perpendicular to the paper through the center of mass? Treat the mass as particles.

1. 1. 3.7 kg?m²
   2. 2.8 kg?m²
   3. 3.2 kg?m²
   4. 2.3 kg?m²
   5. 3.9 kg?m²

40. Three particles, each of which has a mass of 80 g, are positioned at the vertices of an equilateral triangle with sides of length 60 cm. The particles are connected by rods of negligible mass. What is the moment of inertia of this rigid body about an axis that is parallel to one side of the triangle and passes through the respective midpoints of the other two sides?

a.   0.018 kg?m²

b. 0.020 kg?m²

c.   0.016 kg?m²

d. 0.022 kg?m²

e.   0.032 kg?m²

41. The rigid body shown rotates about an axis through its center of mass and perpendicular to the paper. If M = 2.0 kg and L = 80 cm, what is the kinetic energy of this object when its angular speed about this axis is equal to 5.0 rad/s? Neglect the mass of the connecting rod and treat the masses as particles.

1. 1. 18 J
   2. 15 J
   3. 12 J
   4. 23 J
   5. 26 J

42. The rigid body shown is rotated about an axis perpendicular to the paper and through the point P. If M

= 0.40 kg, a = 30 cm, and b = 50 cm, how much work is required to take the body from rest to an angular speed of 5.0 rad/s? Neglect the mass of the connecting rods and treat the masses as particles.

43. A uniform sphere of radius R and mass M rotates freely about a horizontal axis that is tangent to an equatorial plane of the sphere, as shown below. The moment of inertia of the sphere about this axis is

a.

.

b.

.

c.

.

d.

.

e.

.

44. A uniform cylinder of radius R, mass M, and length L rotates freely about a horizontal axis parallel and tangent to the cylinder, as shown below. The moment of inertia of the cylinder about this axis is

a.

.

b.

.

c.   MR². d.

.

e.

.

45. You throw a Frisbee of mass m and radius r so that it is spinning about a horizontal axis perpendicular to the plane of the Frisbee. Ignoring air resistance, the torque exerted about its center of mass by gravity is
    1. 0.
    2. mgr.
    3. 2mgr.
    4. a function of the angular velocity.
    5. small at first, then increasing as the Frisbee loses the torque given it by your hand.
46. Which of the following diagrams shows the greatest magnitude net torque with a zero net force? All the rods, of length 2r, rotate about an axis that is perpendicular to the rod and fixed in the center of the rod. All the forces are of magnitude F or 2F and all distances from the axis are r or r/2.
    1. c.                                            e.

1. 2. d.

47. Two vectors lying in the xy plane are given by the equations          and                 . The value of is
    1. 19

b.   ?11

c.   ?19

4. 11
5. 10

48. Two vectors lying in the xz plane are given by the equations          and                  . The value of is

a.

b.

3. 7
4. ?7

e.  

49. A particle located at the position vector           m has a force                    N acting on it. The torque about the origin is
    1. (1 )N?m
    2. (5 )N?m

c.   (?1 )N?m

d.   (?5 )N?m

e.   (2 + 3 )N?m

50. When the sum of the external forces and the sum of the external torques on a body are both zero, we can conclude that
    1. the body is moving at constant velocity but is not rotating.
    2. the body is rotating at constant angular velocity but has no linear velocity.
    3. the body has neither linear nor angular velocity.
    4. the body may have constant linear or angular velocity, but not both simultaneously.
    5. the body may have constant linear or constant angular velocity, or both simultaneously.       

51. A mass (M₁ = 5.0 kg) is connected by a light cord to a mass (M₂ = 4.0 kg) which slides on a smooth surface, as shown in the figure. The pulley (radius = 0.20 m) rotates about a frictionless axle. The acceleration of M₂ is 3.5 m/s². What is the moment of inertia of the pulley?

a.   0.29 kg?m²

b. 0.42 kg?m²

c.   0.20 kg?m²

d. 0.62 kg?m²

e.   0.60 kg?m²

52. A wheel (radius = 0.20 m) is mounted on a frictionless, horizontal axis. A light cord wrapped around the wheel supports a 0.50-kg object, as shown in the figure. When released from rest the object falls with a downward acceleration of 5.0 m/s². What is the moment of inertia of the wheel?

a.   0.023 kg?m²

b. 0.027 kg?m²

c.   0.016 kg?m²

d. 0.019 kg?m²

e.   0.032 kg?m²

53. A wheel (radius = 0.25 m) is mounted on a frictionless, horizontal axis. The moment of inertia of the wheel about the axis is 0.040 kg?m². A light cord wrapped around the wheel supports a 0.50-kg object as shown in the figure. The object is released from rest. What is the magnitude of the acceleration of the 0.50-kg object?

a.   3.0 m/s²

b.   3.4 m/s²

c.   4.3 m/s²

d.   3.8 m/s²

e.   2.7 m/s²

54. A mass m = 4.0 kg is connected, as shown, by a light cord to a mass M = 6.0 kg, which slides on a smooth horizontal surface. The pulley rotates about a frictionless axle and has a radius R = 0.12 m and a moment of inertia I = 0.090 kg?m². The cord does not slip on the pulley. What is the magnitude of the acceleration of m?

55. A cylinder rotating about its axis with a constant angular acceleration of 1.6 rad/s² starts from rest at t

= 0. At the instant when it has turned through 0.40 radian, what is the magnitude of the total linear acceleration of a point on the rim (radius = 13 cm)?

a.   0.31 m/s²

b.   0.27 m/s²

c.   0.35 m/s²

d.   0.39 m/s²

e.   0.45 m/s²

56. A wheel (radius = 0.20 m) starts from rest and rotates with a constant angular acceleration of 2.0 rad/s². At the instant when the angular velocity is equal to 1.2 rad/s, what is the magnitude of the total linear acceleration of a point on the rim of the wheel?

a.   0.40 m/s²

b.   0.29 m/s²

c.   0.69 m/s²

d.   0.49 m/s²

e.   0.35 m/s²

57. Four identical particles (mass of each = 0.40 kg) are placed at the vertices of a rectangle (2.5 m ? 4.0 m) and held in those positions by four light rods which form the sides of the rectangle. What is the moment of inertia of this rigid body about an axis that passes through the mid-points of the shorter sides and is parallel to the longer sides?
    1. 2.2 kg?m²
    2. 2.8 kg?m²
    3. 2.5 kg?m²
    4. 3.1 kg?m²
    5. 1.6 kg?m²

58. A uniform rod (mass = 2.0 kg, length = 0.60 m) is free to rotate about a frictionless pivot at one end. The rod is released from rest in the horizontal position. What is the magnitude of the angular acceleration of the rod at the instant it is 60? below the horizontal?
    1. 15 rad/s²
    2. 12 rad/s²
    3. 18 rad/s²
    4. 29 rad/s²
    5. 23 rad/s²

59. Particles (mass of each = 0.20 kg) are placed at the 40-cm and 100-cm marks of a meter stick of negligible mass. This rigid body is free to rotate about a frictionless pivot at the 0-cm end. The body is released from rest in the horizontal position. What is the initial angular acceleration of the body?
    1. 12 rad/s²
    2. 5.9 rad/s²
    3. 8.4 rad/s²
    4. 5.4 rad/s²
    5. 17 rad/s²

60. Particles (mass of each = 0.40 kg) are placed at the 60-cm and 100-cm marks of a meter stick of negligible mass. This rigid body is free to rotate about a frictionless pivot at the 0-cm end. The body is released from rest in the horizontal position. What is the magnitude of the initial linear acceleration of the end of the body opposite the pivot?

61. A wheel (radius = 12 cm) is mounted on a frictionless, horizontal axle that is perpendicular to the wheel and passes through the center of mass of the wheel. A light cord wrapped around the wheel supports a 0.40-kg object. If released from rest with the string taut, the object is observed to fall with a downward acceleration of 3.0 m/s². What is the moment of inertia (of the wheel) about the given axle? a.   0.023 kg?m²

b. 0.013 kg?m²

c.   0.020 kg?m²

d. 0.016 kg?m²

e.   0.035 kg?m²

62. A uniform rod is 2.0 m long. The rod is pivoted about a horizontal, frictionless pin through one end. The rod is released from rest at an angle of 30? above the horizontal. What is the angular acceleration of the rod at the instant it is released?
    1. 4.7 rad/s²
    2. 6.9 rad/s²
    3. 6.4 rad/s²
    4. 5.6 rad/s²
    5. 4.2 rad/s²

63. A uniform rod is 2.0 m long. The rod is pivoted about a horizontal, frictionless pin through one end. The rod is released from rest at the horizontal position. What is the angular acceleration of the rod at the instant the rod makes an angle of 70? with the horizontal?
    1. 3.7 rad/s²
    2. 1.3 rad/s²
    3. 2.5 rad/s²
    4. 4.9 rad/s²
    5. 1.9 rad/s²

64. A uniform rod of mass M = 1.2 kg and length L = 0.80 m, lying on a frictionless horizontal plane, is free to pivot about a vertical axis through one end, as shown. The moment of inertia of the rod about this axis is given by (1/3)ML². If a force (F = 5.0 N, ? = 40?) acts as shown, what is the resulting angular acceleration about the pivot point?

1. 1. 16 rad/s²
   2. 12 rad/s²

1. 3. 14 rad/s²
   4. 10 rad/s²
   5. 33 rad/s²

65. A uniform meter stick is pivoted to rotate about a horizontal axis through the 25-cm mark on the stick. The stick is released from rest in a horizontal position. The moment of inertia of a uniform rod about an axis perpendicular to the rod and through the center of mass of the rod is given by (1/12)ML². Determine the magnitude of the initial angular acceleration of the stick.
    1. 17 rad/s²
    2. 13 rad/s²
    3. 15 rad/s²
    4. 19 rad/s²
    5. 23 rad/s²

66. Two forces of magnitude 50 N, as shown in the figure below, act on a cylinder of radius 4 m and mass

6.25 kg. The cylinder, which is initially at rest, sits on a frictionless surface. After 1 second, the velocity and angular velocity of the cylinder in m/s and rad/s are respectively

a.   v = 0; ? = 0.

b.   v = 0; ? = 4.

c.   v = 0; ? = 8.

d.   v = 8; ? = 8.

e.   v = 16; ? = 8.

67. A massless rope is wrapped around a uniform cylinder that has radius R and mass M, as shown in the figure. Initially, the unwrapped portion of the rope is vertical and the cylinder is horizontal. The linear acceleration of the cylinder is

a.   (2/3)g

b.   (1/2)g

c.   (1/3)g

d.   (1/6)g

e.   (5/6)g

68. Two blocks, m₁ = 1.0 kg and m₂ = 2.0 kg, are connected by a light string as shown in the figure. If the radius of the pulley is 1.0 m and its moment of inertia is 5.0 kg?m², the acceleration of the system is

a.   (1/6)g

b.   (3/8)g

c.   (1/8)g

d.   (1/2)g

e.   (5/8)g

69. Two objects of mass m₁ = 2m and m₂ = m move around a rotation axis A in parallel circles of radii r₁ = r and r₂ = 2r with equal tangential speeds. As they rotate, forces of equal magnitude are applied opposite to their velocities to stop them. Which statement is correct?

69. 1. m₂ will stop first because it has the larger initial angular velocity.
    2. m₁ will stop first because it has the smaller radius.
    3. m₂ will stop first because the torque on it is greater.
    4. m₁ will stop first because it has the smaller moment of inertia.
    5. Both objects will stop at the same time because the angular accelerations are equal.                      
70. A torque can be exerted on a body with a fixed axis of rotation
    1. only by a centripetal force.
    2. only by a force directed radially outwards.
    3. only by a tangential force.

70. 4. only by a force with a component directed radially outwards.
    5. by any force not pointing directly toward or away from the axis of rotation.                  
71. Five identical cylinders are each acted on by forces of equal magnitude. Which force exerts the biggest torque about the central axes of the cylinders?

a.

b.

c.

d.

e.

72. The diagram below shows five 20-kg rods of the same 2.0-m length free to rotate about axes through the rods, as indicated. Which rod experiences the greatest magnitude gravitational torque?

a.

b.

c.

d.

e.

73. A force    is applied to a cylindrical roll of paper of radius R and mass M by pulling on the paper as shown. The acceleration of the center of mass of the roll of paper (when it rolls without slipping) is

a.

.

b.

.

c.

.

d.

.

e.

.

74. A uniform rod (length = 2.0 m) is mounted to rotate freely about a horizontal axis that is perpendicular to the rod and that passes through the rod at a point 0.50 m from one end of the rod. If the rod is released from rest in a horizontal position, what is the angular speed of the rod as it rotates through its lowest position?
    1. 3.5 rad/s
    2. 3.8 rad/s
    3. 4.1 rad/s
    4. 2.0 rad/s
    5. 5.6 rad/s

75. Identical particles are placed at the 50-cm and 80-cm marks on a meter stick of negligible mass. This rigid body is then mounted so as to rotate freely about a pivot at the 0-cm mark on the meter stick. If this body is released from rest in a horizontal position, what is the angular speed of the meter stick as it swings through its lowest position?
    1. 4.2 rad/s
    2. 5.4 rad/s
    3. 4.6 rad/s
    4. 5.0 rad/s
    5. 1.7 rad/s

76. A uniform rod (mass = 1.5 kg) is 2.0 m long. The rod is pivoted about a horizontal, frictionless pin through one end. The rod is released from rest in a horizontal position. What is the angular speed of the rod when the rod makes an angle of 30? with the horizontal? (The moment of inertia of the rod about the pin is 2.0 kg?m²).
    1. 2.2 rad/s
    2. 3.6 rad/s
    3. 2.7 rad/s
    4. 3.1 rad/s
    5. 1.8 rad/s

77. A uniform rod is 3.0 m long. The rod is pivoted about a horizontal, frictionless pin through one end. The rod is released from rest at an angle of 27? above the horizontal. What is the angular speed of the rod as it passes through the horizontal position?
    1. 3.0 rad/s
    2. 2.8 rad/s
    3. 2.1 rad/s
    4. 2.5 rad/s
    5. 3.4 rad/s

78. A uniform rod of length (L = 2.0 m) and mass (M = 1.5 kg) is pivoted about a horizontal frictionless pin through one end. The rod is released from rest at an angle of 30? below the horizontal. What is the angular speed of the rod when it passes through the vertical position? (The moment of inertia of the rod about the pin is 2.0 kg?m².)
    1. 3.5 rad/s
    2. 2.7 rad/s
    3. 3.1 rad/s
    4. 2.3 rad/s
    5. 1.6 rad/s

79. A nonuniform 2.0-kg rod is 2.0 m long. The rod is mounted to rotate freely about a horizontal axis perpendicular to the rod that passes through one end of the rod. The moment of inertia of the rod about this axis is 4.0 kg?m². The center of mass of the rod is 1.2 m from the axis. If the rod is released from rest in the horizontal position, what is its angular speed as it swings through the vertical position?
    1. 3.4 rad/s
    2. 4.4 rad/s
    3. 4.3 rad/s
    4. 5.8 rad/s
    5. 6.8 rad/s

80. A uniform rod (length = 2.4 m) of negligible mass has a 1.0-kg point mass attached to one end and a 2.0-kg point mass attached to the other end. The rod is mounted to rotate freely about a horizontal axis that is perpendicular to the rod and that passes through a point 1.0 m from the 2.0-kg mass. The rod is released from rest when it is horizontal. What is the angular velocity of the rod at the instant the 2.0-kg mass passes through its low point?
    1. 1.7 rad/s
    2. 2.2 rad/s
    3. 2.0 rad/s
    4. 1.5 rad/s
    5. 3.1 rad/s

81. A car of mass 1 000 kg moves with a speed of 50 m/s on a circular track of radius 100 m. What is the magnitude of its angular momentum (in kg?m²/s) relative to the center of the race track?

a.   5.0 ? 10²

b.   5.0 ? 10⁶

c.   2.5 ? 10⁴

d.   2.5 ? 10⁶

e.   5.0 ? 10³

82. A solid cylinder of radius R = 1.0 m and mass 10 kg rotates about its axis. When its angular velocity is 10 rad/s, its angular momentum (in kg?m²/s) is

a.   50.

b.   20.

c.   40.

d.   25.

e.   70.

83. A particle whose mass is 2 kg moves in the xy plane with a constant speed of 3 m/s in the x direction along the line y = 5. What is its angular momentum (in kg?m²/s) relative to the origin?

a.   ?30

b. 30

c.   ?15

4. 15
5. 45

84. A particle whose mass is 2 kg moves in the xy plane with a constant speed of 3 m/s along the direction

. What is its angular momentum (in kg?m²/s) relative to the origin?

1. 1. 0 b.

c.

4. 6
5. ?6

85. A particle whose mass is 2.0 kg moves in the xy plane with a constant speed of 3.0 m/s along the direction     . What is its angular momentum (in kg?m²/s) relative to the point (0, 5.0) meters?
    1. 12
    2. 11
    3. 13
    4. 14
    5. 21

86. In the figure, a 1.6-kg weight swings in a vertical circle at the end of a string having negligible weight. The string is 2 m long. If the weight is released with zero initial velocity from a horizontal position, its angular momentum (in kg?m²/s) at the lowest point of its path relative to the center of the circle is approximately

1. 1. 40
   2. 10
   3. 30
   4. 20
   5. 50

87. A puck on a frictionless air hockey table has a mass of 5.0 kg and is attached to a cord passing through a hole in the surface as in the figure. The puck is revolving at a distance 2.0 m from the hole with an angular velocity of 3.0 rad/s. The angular momentum of the puck (in kg?m²/s) is

1. 1. 80
   2. 20
   3. 30
   4. 60
   5. 50

88. A pendulum bob of mass m is set into motion in a circular path in a horizontal plane as shown in the figure. The square of the angular momentum of the bob about the vertical axis through the point P is

1. 1. m² gl³ sin⁴ ?/cos ?

1. 2. m² gl³ sin³ ?/cos ?
   3. m² gl³ sin² ?/cos ?
   4. m² gl³ sin ?/cos ?
   5. m² gl³ sin² ?

89. Five objects of mass m move at velocity       at a distance r from an axis of rotation perpendicular to the page through point A, as shown below. The one that has zero angular momentum about that axis is

a.

b.

c.

d.

e.

90. The object shown below has mass m and velocity    . The direction of its angular momentum vector with respect to an axis perpendicular to the page through point O is

1. 1. downwards.
   2. to the right.
   3. into the page.
   4. up out of the page.
   5. counterclockwise.

91. The diagram below shows five cylinders, each cylinder rotating with constant angular velocity about its central axis. The magnitude of the tangential speed of one point of each cylinder is shown, along with each cylinder's radius and mass. Which cylinder has the largest angular momentum?

92. A 0.5 kg fish, hooked as shown below, starts to swim away at a speed of 3 m/s. The angular momentum of the fish relative to the hand holding the fishing rod is about

a.

.

b.

.

c.

.

d.

.

e.

.

: example 1

Exhibit 11-1

Two blocks of masses m₁ and m₂ are connected by a light cord that passes over a pulley of mass M, as shown. Block m₂ slides on a frictionless horizontal surface. The blocks and pulley are initially at rest. When m₁ is released, the blocks accelerate and the pulley rotates. The total angular momentum of the system of the two blocks and the pulley relative to the axis of rotation of the pulley is

Use this exhibit to answer the following question(s).

93. Refer to Exhibit 11-1. The total angular momentum of the system of the two blocks and the pulley relative to the axis of rotation of the pulley is
    1. the same at all times.
    2. proportional to l₁, the length of string from the pulley to m₁.
    3. proportional to l₂, the length of string from the pulley to m₂.
    4. conserved because the Earth doesn't move.
    5. proportional to the speed of the blocks.

94. Refer to Exhibit 11-1. The total angular momentum of the system of the two blocks and the pulley relative to the axis of rotation of the pulley is
    1. proportional to the radius of the pulley.
    2. proportional to the speed of the blocks.
    3. proportional to the length of the string.
    4. to all of the above.
    5. only to (a) and (b) above.

95. A 3.0-kg particle has a position vector given by                        where    is in meters and t is in seconds. What is the angular momentum of the particle, in kg?m²/s, about the origin at t = 2 s?
    1. 72

b. ?72

c.   24

d. ?24

e.   22

96. If L represents angular momentum, I represents moment of inertia, p represents linear momentum, m

represents mass, and r represents a distance, which of the following can represent kinetic energy?

1. 1. p²/2m
   2. L²/2I
   3. rpI
   4. all of the above
   5. both (a) and (b)

97. A puck on a frictionless air hockey table has a mass of 5.0 g and is attached to a cord passing through a hole in the surface as in the figure. The puck is revolving at a distance 2.0 m from the hole with an angular velocity of 3.0 rad/s. The cord is then pulled from below, shortening the radius to 1.0 m. The new angular velocity (in rad/s) is

a.   4.0

b.   6.0

3. 12

d. 2.0

e.   8.0

98. A thin rod of mass M and length L is struck at one end by a ball of clay of mass m, moving with speed v as shown in the figure. The ball sticks to the rod. After the collision, the angular momentum of the clay-rod system about A, the midpoint of the rod, is

a.   (m + M/3)(vL/2)

b. (m + M/12)(vL/2)

c.   (m + M/6)(vL/2)

4. mvL/2
5. mvL

99. A particle of mass m = 0.10 kg and speed v₀ = 5.0 m/s collides and sticks to the end of a uniform solid cylinder of mass M = 1.0 kg and radius R = 20 cm. If the cylinder is initially at rest and is pivoted about a frictionless axle through its center, what is the final angular velocity (in rad/s) of the system after the collision?

a.   8.1

b.   2.0

c.   6.1

d.   4.2

e.   10

100. A skater extends her arms horizontally, holding a 5-kg mass in each hand. She is rotating about a vertical axis with an angular velocity of one revolution per second. If she drops her hands to her sides, what will the final angular velocity (in rev/s) be if her moment of inertia remains approximately constant at 5 kg?m², and the distance of the masses from the axis changes from 1 m to 0.1 m?
     1. 6
     2. 3
     3. 9
     4. 4
     5. 7

101. A merry-go-round of radius R = 2.0 m has a moment of inertia I = 250 kg?m², and is rotating at 10 rpm. A child whose mass is 25 kg jumps onto the edge of the merry-go-round, heading directly toward the center at 6.0 m/s. The new angular speed (in rpm) of the merry-go-round is approximately
     1. 10

b.   9.2

c.   8.5

d.   7.1

e.   6.4

102. Stars originate as large bodies of slowly rotating gas. Because of gravity, these clumps of gas slowly decrease in size. The angular velocity of a star increases as it shrinks because of
     1. conservation of angular momentum
     2. conservation of linear momentum
     3. conservation of energy
     4. the law of universal gravitation
     5. conservation of mass

103. When an object is effectively isolated from external torques, like an ice skater twirling on the tip of one skate, the angular momentum of the object
     1. can be increased by shifting mass out away from the axis of rotation.
     2. can be decreased by shifting mass out away from the axis of rotation.
     3. can be increased by shifting mass in toward the axis of rotation.
     4. can be decreased by shifting mass in toward the axis of rotation.
     5. cannot be changed except by friction at the point of contact.     
104. A hockey puck traveling at speed v on essentially frictionless ice collides elastically with one end of a straight stick lying flat on the ice. In this collision
     1. momentum is conserved.
     2. angular momentum is conserved.
     3. energy is conserved.
     4. all of the above are conserved.
     5. only momentum and angular momentum are conserved.     
105. A hockey puck traveling at speed v on essentially frictionless ice collides with one end of a straight stick lying flat on the ice and sticks to that end. In this collision
     1. momentum is conserved.
     2. angular momentum is conserved.
     3. energy is conserved.
     4. all of the above are conserved.
     5. only momentum and angular momentum are conserved.     

106. A space station out beyond the solar system is rotating with constant angular velocity. A spaceship heading into the station along a diameter of the station, uses its rockets to brake, and then docks inside the station at its center. When the spaceship docks, the angular momentum of the system consisting of the station and ship
     1. is less than the original angular momentum of the station.
     2. is the same as the original angular momentum of the station.
     3. is greater than the original angular momentum of the station.
     4. is less than the original angular momentum of the station, but the angular velocity increases.
     5. is greater than the original angular momentum of the station, but the angular velocity decreases.

107. A top is set spinning so that the rotation is counterclockwise around its axis when viewed from above. When the top is placed on a level surface it happens that its axis of rotation is not quite vertical. Viewed from above, which way does the rotational axis of the top precess?
     1. clockwise
     2. counterclockwise
     3. It’s random, if it starts clockwise it will continue clockwise, and vice versa, i.e., a 50% chance either way.
     4. The direction depends on the little shove given to the axis when the top is placed on the surface.
     5. In the northern hemisphere it will be clockwise, in the southern hemisphere it will be counterclockwise.

108. Two cylinders made of the same material roll down a plane inclined at an angle ? with the horizontal. Each travels the same distance. The radius of cylinder B is twice the radius of cylinder A. In what order do they reach the bottom?
     1. A reaches the bottom first because it has the greater acceleration.
     2. A reaches the bottom first because it has a smaller moment of inertia.
     3. B reaches the bottom first because is experiences a larger torque.
     4. B reaches the bottom first because it travels a larger distance in one rotation.
     5. They both reach the bottom at the same time, because each has the same linear acceleration.

109. When a wheel is rolling without slipping, the magnitude of its velocity relative to the ground is greatest at
     1. the point in contact with the ground.
     2. the point at the center of the wheel.
     3. the point at the top of the wheel opposite to the point in contact with the ground.
     4. the point farthest forward from the center of mass of the wheel.
     5. the point farthest behind the center of mass of the wheel.     
110. When the center of a bicycle wheel has linear velocity        relative to the ground, the velocity relative to the ground of point P' at the top of the wheel is

1. 1. 0.
   2. .
   3. .
   4. .
   5. .

111. A solid sphere, a solid cylinder, and a hoop all have the same mass and radius. Each are sent down identical inclined planes starting from rest. Their kinetic energies at the bottom of the incline are K_sphere, K_cylinder, and K_hoop. Which of the following is true?
     1. Ksphere > Kcylinder
     2. Khoop > Ksphere
     3. Khoop > Kcylinder
     4. Kcylinder > Khoop
     5. No answer above is correct.

112. A solid sphere, a solid cylinder, a spherical shell, and a hoop all have the same mass and radius. Each are rolling on a horizontal surface with the same center of mass speed, and then they roll up identical inclines. Which one goes the greatest distance up its incline?
     1. the hoop
     2. the solid sphere
     3. the spherical shell
     4. the cylinder
     5. They all go the same distance up their inclines.

113. A solid sphere (radius R, mass M) rolls without slipping down an incline as shown in the figure. The linear acceleration of its center of mass is

1. 1. (5/7)g sin ?
   2. (3/5)g sin ?
   3. (2/3)g sin ?

1. 4. (1/2)g sin ?
   5. (4/5)g sin ?

114. A solid cylinder rolls without slipping down an incline as shown in the figure. The linear acceleration of its center of mass is

1. 1. (5/7)g sin ?
   2. (1/2)g sin ?
   3. (2/3)g sin ?
   4. (3/5)g sin ?
   5. (4/5)g sin ?

115. A cylindrical shell rolls without slipping down an incline as shown in the figure. The linear acceleration of its center of mass is

1. 1. (5/7)g sin ?
   2. (1/2)g sin ?
   3. (3/5)g sin ?
   4. (2/3)g sin ?
   5. (4/5)g sin ?

116. A solid sphere, spherical shell, solid cylinder and a cylindrical shell all have the same mass m and radius R. If they are all released from rest at the same elevation and roll without slipping, which reaches the bottom of an inclined plane first?
     1. solid sphere
     2. spherical shell
     3. solid cylinder

1. 4. cylindrical shell
   5. all take the same time

PROBLEM

117. The net work done in accelerating a propeller from rest to an angular velocity of 200 rad/s is 3 000 J. What is the moment of inertia of the propeller?

118. A rod of length 1.00 m has a mass per unit length given by                  , where   is in kg/m. The rod is placed on the x axis going from x = 0.00 m to x = 1.00 m. What is the moment of inertia of the rod in kg m² about the y axis?

119. A horizontal force of magnitude 6.5 N is exerted tangentially on a Frisbee of mass 32 grams and radius

14.3 cm. Assuming the Frisbee, a uniform disk, is originally at rest and the force is exerted for 0.08 s, determine the angular velocity of rotation about the central axis when the Frisbee is released.

120. What is the angular momentum of the moon about the Earth? The mass of the moon is 7.35 ? 10²² kg, the center-to-center separation of the Earth and the moon is 3.84 ? 10⁵ km, and the orbital period of the moon is 27.3 days. Ignore the small offset of the center of mass of the system from the center of the Earth in your calculation.

121. A celestial object called a pulsar emits its light in short bursts that are synchronized with its rotation. A pulsar in the Crab Nebula is rotating at a rate of 30 revolutions/second. What is the maximum radius of the pulsar, if no part of its surface can move faster than the speed of light (3 ? 10⁸ m/s)?