# Chapter 12. Rotation of a Rigid Body

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1 Chapter 12. Rotation of a Rigid Body Not all motion can be described as that of a particle. Rotation requires the idea of an extended object. This diver is moving toward the water along a parabolic trajectory, and she s rotating rapidly around her center of mass. Chapter Goal: To understand the physics of rotating objects.

2 Chapter 12. Rotation of a Rigid Body Topics: Rotational Motion Rotation About the Center of Mass Rotational Energy Calculating Moment of Inertia Torque Rotational Dynamics Rotation About a Fixed Axis Static Equilibrium Rolling Motion The Vector Description of Rotational Motion Angular Momentum of a Rigid Body

4 A new way of multiplying two vectors is introduced in this chapter. What is it called? A. Dot Product B. Scalar Product C. Tensor Product D. Cross Product E. Angular Product

5 A new way of multiplying two vectors is introduced in this chapter. What is it called? A. Dot Product B. Scalar Product C. Tensor Product D. Cross Product E. Angular Product

6 Moment of inertia is A. the rotational equivalent of mass. B. the point at which all forces appear to act. C. the time at which inertia occurs. D. an alternative term for moment arm.

7 Moment of inertia is A. the rotational equivalent of mass. B. the point at which all forces appear to act. C. the time at which inertia occurs. D. an alternative term for moment arm.

8 A rigid body is in equilibrium if A. B. C. neither A nor B. D. either A or B. E. both A and B.

9 A rigid body is in equilibrium if A. B. C. neither A nor B. D. either A or B. E. both A and B.

10 Chapter 12. Basic Content and Examples

11 Rotational Motion The figure shows a wheel rotating on an axle. Its angular velocity is The units of ω are rad/s. If the wheel is speeding up or slowing down, its angular acceleration is The units of α are rad/s2.

12 Rotational Motion

13 EXAMPLE 12.1 A rotating crankshaft QUESTION:

14 EXAMPLE 12.1 A rotating crankshaft

15 EXAMPLE 12.1 A rotating crankshaft

16 EXAMPLE 12.1 A rotating crankshaft

17 EXAMPLE 12.1 A rotating crankshaft

18 Rotation About the Center of Mass An unconstrained object (i.e., one not on an axle or a pivot) on which there is no net force rotates about a point called the center of mass. The center of mass remains motionless while every other point in the object undergoes circular motion around it.

19 Rotation About the Center of Mass The center of mass is the mass-weighted center of the object.

20 Rotational Energy A rotating rigid body has kinetic energy because all atoms in the object are in motion. The kinetic energy due to rotation is called rotational kinetic energy. Here the quantity I is called the object s moment of inertia. The units of moment of inertia are kg m2. An object s moment of inertia depends on the axis of rotation.

21

22 EXAMPLE 12.5 The speed of a rotating rod QUESTION:

23 EXAMPLE 12.5 The speed of a rotating rod

24 EXAMPLE 12.5 The speed of a rotating rod

25 EXAMPLE 12.5 The speed of a rotating rod

26 EXAMPLE 12.5 The speed of a rotating rod

27 EXAMPLE 12.5 The speed of a rotating rod

28 Torque Consider the common experience of pushing open a door. Shown is a top view of a door hinged on the left. Four pushing forces are shown, all of equal strength. Which of these will be most effective at opening the door? The ability of a force to cause a rotation depends on three factors: 1. the magnitude F of the force. 2. the distance r from the point of application to the pivot. 3. the angle at which the force is applied.

29 Torque Let s define a new quantity called torque τ (Greek tau) as

30 EXAMPLE 12.9 Applying a torque QUESTION:

31 EXAMPLE 12.9 Applying a torque

32 EXAMPLE 12.9 Applying a torque

33 EXAMPLE 12.9 Applying a torque

34 Analogies between Linear and Rotational Dynamics In the absence of a net torque (τnet = 0), the object either does not rotate (ω = 0) or rotates with constant angular velocity (ω = constant).

35 Problem-Solving Strategy: Rotational Dynamics Problems

36 Problem-Solving Strategy: Rotational Dynamics Problems

37 Problem-Solving Strategy: Rotational Dynamics Problems

38 Problem-Solving Strategy: Rotational Dynamics Problems

39 EXAMPLE Starting an airplane engine QUESTION:

40 EXAMPLE Starting an airplane engine

41 EXAMPLE Starting an airplane engine

42 EXAMPLE Starting an airplane engine

43 EXAMPLE Starting an airplane engine

44 Static Equilibrium The condition for a rigid body to be in static equilibrium is that there is no net force and no net torque. An important branch of engineering called statics analyzes buildings, dams, bridges, and other structures in total static equilibrium. No matter which pivot point you choose, an object that is not rotating is not rotating about that point. For a rigid body in total equilibrium, there is no net torque about any point. This is the basis of a problem-solving strategy.

45 Problem-Solving Strategy: Static Equilibrium Problems

46 Problem-Solving Strategy: Static Equilibrium Problems

47 Problem-Solving Strategy: Static Equilibrium Problems

48 Problem-Solving Strategy: Static Equilibrium Problems

49 EXAMPLE Will the ladder slip? QUESTION:

50 EXAMPLE Will the ladder slip?

51 EXAMPLE Will the ladder slip?

52 EXAMPLE Will the ladder slip?

53 EXAMPLE Will the ladder slip?

54 EXAMPLE Will the ladder slip?

55 EXAMPLE Will the ladder slip?

56 Balance and Stability

57 Rolling Without Slipping For an object that is rolling without slipping, there is a rolling constraint that links translation and rotation:

58 Rolling Without Slipping We know from the rolling constraint that Rω is the center-of-mass velocity vcm. Thus the kinetic energy of a rolling object is In other words, the rolling motion of a rigid body can be described as a translation of the center of mass (with kinetic energy Kcm) plus a rotation about the center of mass (with kinetic energy Krot).

59 The Angular Velocity Vector The magnitude of the angular velocity vector is ω. The angular velocity vector points along the axis of rotation in the direction given by the right-hand rule as illustrated above.

60 Angular Momentum of a Particle A particle is moving along a trajectory as shown. At this instant of time, the particle s momentum vector, tangent to the trajectory, makes an angle β with the position vector.

61 Angular Momentum of a Particle We define the particle s angular momentum vector relative to the origin to be

62 Analogies between Linear and Angular Momentum and Energy

63

64 Chapter 12. Summary Slides

65 General Principles

66 General Principles

67 Important Concepts

68 Important Concepts

69 Important Concepts

70 Important Concepts

71 Applications

72 Applications

73 Applications

74 Chapter 12. Clicker Questions

75 The fan blade is speeding up. What are the signs of ω and α? A. B. C. D. ω is positive and α is positive. ω is positive and α is negative. ω is negative and α is positive. ω is negative and α is negative.

76 The fan blade is speeding up. What are the signs of ω and α? A. B. C. D. ω is positive and α is positive. ω is positive and α is negative. ω is negative and α is positive. ω is negative and α is negative.

77 Four Ts are made from two identical rods of equal mass and length. Rank in order, from largest to smallest, the moments of inertia Ia to Id for rotation about the dotted line. (a) (b) (c) A. Ia > Id > Ib > Ic B. Ic = Id > Ia = Ib C. Ia = Ib > Ic = Id D. Ia > Ib > Id > Ic E. I > I > I > I Copyright 2008 Pearson Education, Inc., publishing c as Pearson b Addison-Wesley. d a (d)

78 Four Ts are made from two identical rods of equal mass and length. Rank in order, from largest to smallest, the moments of inertia Ia to Id for rotation about the dotted line. (a) (b) (c) A. Ia > Id > Ib > Ic B. Ic = Id > Ia = Ib C. Ia = Ib > Ic = Id D. Ia > Ib > Id > Ic E. I > I > I > I Copyright 2008 Pearson Education, Inc., publishing c as Pearson b Addison-Wesley. d a (d)

79 Rank in order, from largest to smallest, the five torques τa τe. The rods all have the same length and are pivoted at the dot. (a) (b) (c) (d) Α. A. B. C. D. (e)

80 Rank in order, from largest to smallest, the five torques τa τe. The rods all have the same length and are pivoted at the dot. (a) (b) (c) (d) Α. A. B. C. D. (e)

81 Rank in order, from largest to smallest, the angular accelerations αa to αe. A. B. C. D. E.

82 Rank in order, from largest to smallest, the angular accelerations αa to αe. A. B. C. D. E.

83 A student holds a meter stick straight out with one or more masses dangling from it. Rank in order, from most difficult to least difficult, how hard it will be for the student to keep the meter stick from rotating. (a) (b) (c) A. c > b > d > a B. b = c = d > a C. c > d > b > a D. c > d > a = b E. b > d > c > a (d)

84 A student holds a meter stick straight out with one or more masses dangling from it. Rank in order, from most difficult to least difficult, how hard it will be for the student to keep the meter stick from rotating. (a) (b) (c) A. c > b > d > a B. b = c = d > a C. c > d > b > a D. c > d > a = b E. b > d > c > a (d)

85 Two buckets spin around in a horizontal circle on frictionless bearings. Suddenly, it starts to rain. As a result, A. The buckets speed up because the potential energy of the rain is transformed into kinetic energy. B. The buckets continue to rotate at constant angular velocity because the rain is falling vertically while the buckets move in a horizontal plane. C. The buckets slow down because the angular momentum of the bucket + rain system is conserved. D. The buckets continue to rotate at constant angular velocity because the total mechanical energy of the bucket + rain system is conserved. E. None of the above.

86 Two buckets spin around in a horizontal circle on frictionless bearings. Suddenly, it starts to rain. As a result, A. The buckets speed up because the potential energy of the rain is transformed into kinetic energy. B. The buckets continue to rotate at constant angular velocity because the rain is falling vertically while the buckets move in a horizontal plane. C. The buckets slow down because the angular momentum of the bucket + rain system is conserved. D. The buckets continue to rotate at constant angular velocity because the total mechanical energy of the bucket + rain system is conserved. E. None of the above.

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