1. Rotational Motion Hope you learned polar coordinates

2. Rotation The motion of these objects cannot be easily described by using linear equations of motion. These involve a body that rotates about an axis that is stationary in some inertial frame of reference.

3. Purely Rotational Motion No Translational Motion Rotation of a rigid body about an axis of rotation ◦ Rigid body – object with definite shape. It will not deform.  The tire is not exactly a rigid body but work with me ◦ Axis of Rotation – The line that passes through a point where all other points move around in circles

4. Purely Rotational Motion

5. Velocity of a point Rotating at 500 rpm 0.5m diameter tire P is on the rim

6. Velocity of a Point

7. Velocity of a point Rotating at 500 rpm 0.5m diameter tire Q halfway to the rim

8. Velocity of a Point

9. Angular Quantities Different points on a rotating rigid body have different linear velocities This makes calculations clunky Better to use angular velocities

10. Linear Displacement

11. Angular Displacement

12. Angle measured in radians 1 Radian – angle subtended by an arc whose length is equal to the radius

13. Angular Velocity

15. Instantaneous angular velocity

16. Angular Velocity Convention: Counterclockwise rotation is positive Clockwise rotation is negative All points in a rotating object have the same angular velocity !

17. Angular Velocity Define f as the frequency – number of complete revolutions per second Period T – time to complete one revolution

18. Angular Velocity Vector Direction of given by right hand rule Useful for when the axis of rotation changes

19. Angular Acceleration 500 rpm at time t 0 600 rpm at time t

20. Angular Acceleration Instantaneous acceleration Convention: All points in a rotating object have the same angular acceleration !

21. Angular Acceleration Vector Direction of given by right hand rule Useful for when the axis of rotation changes If and are in the same direction, objects rotation is speeding up. If in opposite direction it’s slowing down

22. Relationship of linear and angular Note: there is still a radial component of the acceleration

23. Example You are asked to design an airplane propeller to turn at 2400 rpm. The forward airspeed of the plane is to be 75.0 m/s, and the speed of the tips of the propeller blades through the air must not exceed 270 m/s (a) What is the maximum radius the propeller can have? (b) With this radius, what is the acceleration of the propeller tip

24. Example v fastest at the tip

25. Example

26. Kinematics of Angular Motion If α is constant

27. Kinematics of Angular Motion If α is constant

28. Energy in Rotational Motion A rotating rigid body, even with no translational motion, has kinetic energy Imagine rigid body is made up of many small particles Where r i is perpendicular distance to axis of rotation

29. Energy in Rotational Motion Total kinetic energy is the sum of all the KE of the particles ω is the same for all particles

30. Moment of Inertia Moment of Inertia is the rotational analogue of mass It is dependent on the distribution of the mass and the distance from the axis of rotation

31. Example An engineer is designing a machine part consisting of three heavy disks linked by lightweight struts. (a) What is the moment of inertia of this body about an axis through the center of disk A? (b) What is the moment of inertia about an axis through the center of disks B and C? (c) If the body rotates about an axis through A perpendicular to the plane of the diagram with angular speed =4.0 rad/s, what is its kinetic energy?

32. Example (a) (b)

33. Example (c) Moment of inertia is not an intrinsic property, it depends on the axis of rotation

34. Moment of Inertia Table

35. Parallel-Axis Theorem Let the moment of inertia about an axis that passes through the center of mass be Then the moment of inertia through an axis parallel to the original axis, but displaced by some distance d is

36. Proof

37. Example A uniform rigid rod has a mass of 3.6kg and length 0.30m. Find the moment of inertia perpendicular through one end?

38. Example

39. Brief Review AngularLinear Acceleration α a Velocity ω v Displacement Θ x Equations (constant acceleration) Kinetic Energy For a point at distance R away from the axis of rotation x=R θ v=R ω a=R α

40. Dynamics of Rotational Motion Torque, Angular Acceleration and Rotation of Rigid Bodies

41. Rotation of Rigid Bodies Suppose you have a rigid body that is pinned at point C Force F is applied to point P The object will begin to rotate about the axis of rotation C (  to the screen)

42. Torque Torque is the quantitative measure of the tendency of the force to change the rotational motion of a body. Rotational analog of force. Represented by the symbol τ ◦ Define: Torque is positive for counter-clockwise rotation and negative for clockwise rotation Vector form Magnitude Has SI unit N · m eq(1)

43. Torque – Simple Example To loosen a bolt at point O, a force F is applied on the wrench at point P. Calculate for the magnitude and direction of τ. Two Solutions to = rF tan = F l From right hand rule direction is outwards  to the screen F tan = F sin θ = Force perpendicular to r l = r sin θ = lever arm

44. Torque – Practical Application

45. Torque The radial component of the force, F Rad, has no effect on the rotation of the object. The line of action of the force must not pass through the axis of rotation (sin θ ≠ 0). Always define torque with respect to a specific point. [Torque of F relative to point A] A

46. Torque and Angular Acceleration Applying torque changes the rotational motion of a body, a change in its acceleration. Assume the object is made up of small masses For P 1 a pply Newton’s Second law F=ma F tan, 1 =ma=m 1 r 1 α τ 1 =F tan, 1 r=m 1 r 1 2 α 1

47. Torque and Angular Acceleration Torque going through point 1 τ 1 =m 1 r 1 2 α Net Torque τ net = Σ( m i r i 2 ) α τ net = I α where I = Σ( m i r i 2 ) Moment of inertia, I, is the rotational analog for mass and it measures the resistance of an object to a change in angular acceleration. Equation 2 is rotational analog to Newton’s 2 nd law of motion. Note: valid only for rigid bodies. 1 eq(2)

48. Effect of Internal Torque

49. Not so Simple Example 10-16. A 12.0 kg box resting on the horizontal, frictionless surface is attached to a 5.00 kg weight by a thin, light wire that passes over a frictionless pulley. The pulley has the shape of a uniform solid disk of mass 2.00 kg and a diameter of 0.500m. After the system is released, find: ◦ (a) tension in the wire on both sides of the pulley ◦ (b) the acceleration of the box ◦ (c) the horizontal and vertical components of the force that the axle exerts on the pulley. 12 kg 5 kg

50. Problem 10-16 ∑F=ma For box 1 T 1 =m 1 a For box 2 m 2 g-T 2 =m 2 a For pulley (for convenience let clockwise rotation be positive) ∑ τ = I α (T 2 -T 1 )R= m 3 R 2 α /2 (T 2 -T 1 )=m 3 a/2 Adding all three equations m 2 g=(m 1 +m 2 +m 3 /2)a (5)(9.8)=(12+5+2/2)a a=2.72 m/s 2 12 kg 5 kg N m1gm1g T1T1 m2gm2g T2T2 T1T1 T2T2 F m3gm3g

51. Problem 10-16 T 1 =m 1 a=(12)(2.72) =32.6N T 2 =m 2 (g-a)=(5)(9.8- 2.72)=35.4N Force on pulley F x =-T 1 =32.6N F y =-(T 2 +m 3 g) =35.4+(2)(9.8) =55.0N 12 kg 5 kg m1gm1g T1T1 m2gm2g T2T2 T1T1 T2T2 F m3gm3g

52. Rigid Body Rotation about a Moving Axis We now combine rotational and translational motion Motion of a rigid body can always be divided into translation of the center of mass and rotation about the center of mass

53. Rolling without Slipping Linear velocity at the rim has equal magnitude but opposite direction to the velocity of the center of mass. Where R is the radius of the car

54. Rolling without Slipping Velocity at point of contact is zero! We use static friction

55. Example A primitive yoyo is made by wrapping a string around cylinder with mass 1kg and radius 10cm. Holding one end of the string stationary, the cylinder is released from rest. The string unwinds without slipping. Find the speed of the center of mass of the cylinder after it has dropped a distance 0.25m.

56. Example at all times

57. Example A uniform rod of length 1.0m with weight 2.0 kg is free to rotate around a frictionless pin going through one end. (a)What is its angular speed when it reaches its lowest position? (b) What is the tangential speed of the center of mass and the tangential speed of the lowest point when the rod is in the vertical position?

58. Example

59. Conceptual Problem Several objects roll without slipping down a ramp of height H, all starting from rest at the same moment. You have a hoop, a basketball, a solid marble, a solid cylinder and a greased box that slides without friction. In what order will they reach the bottom of the ramp?

60. Conceptual Problem

61. Where c is a coefficient dependent on the objects shape

62. Conceptual Problem Larger c means lower v. Box c=0 Solid Sphere c=2/5 Solid Cylinder c=1/2 Spherical Shell c=2/3 Thin Cylinder c=1

63. Work and Power Similar to linear, a force applied on a rigid object will cause it to rotate a certain distance θ. If torque applied is constant

64. Work and Power Work-energy theorem for rotation Power for rotation

65. Example An electric motor exerts a constant torque of 10 N.m on a grindstone. The moment of inertia of the grindstone about its shaft is around 2.0 kg.m 2. If the system starts from rest, find the work done by the motor in 8 seconds and the kinetic energy at 8 s. What was the average power delivered by the motor?

66. Example

67. Angular Momentum Rotational analog to linear momentum Magnitude Where l is the perpendicular distance

68. Angular Momentum For a slice of an extended body

69. Angular Momentum We then compute the angular momentum for other slices, computation gets complicated for irregularly shaped objects. If the body rotates about an axis of symmetry If the axis of rotation is not a line of symmetry, is not parallel to the axis of rotation

70. Example Estimate the angular momentum of a 6.0 kg bowling ball with a radius of 12cm spinning at 10 rev/s?

71. Example

72. Conservation of Angular Momentum When the net external torque of a system is zero, the total angular momentum of the system is conserved (constant).

73. Example A man on a turntable holds two 5.0 kg weights at arms length away while he is spinning at one revolution per second. What is his angular velocity when he tucks the weights nearer to his body? The dumbbells are 1.0m away from the axis of rotation initially and 0.20m at the end.

74. Example

75. Example A horizontal circular platform rotates about a vertical frictionless axle. The platform has a mass M=100.0 kg and a radius of R=2.0m. A student of m=60.0 kg walks from the rim towards the center. If the angular speed when the student is at the rim is 2.0 rad/s, what is the angular speed when the student is at r=0.50m

76. Example

77. Earths Rotation

78. Leap Seconds The Earths Rotation is Slowing down (0.5 billion years ago a day was 22 hours long) Tidal drag one of main reasons

79. Leap Seconds Glacial Rebound helps reduce the slowing But it won’t be there forever Source: http://what-if.xkcd.com/26/

80. Problem – Giancoli 8-20 A small rubber wheel is used to drive a large pottery wheel, and they are mounted so that their circular edges touch. The small wheel has a radius of 2.0cm and accelerates at a rate of 7.2rad/s 2 and is in contact with the pottery wheel (radius 25.0cm) without slipping. Calculate (a) the angular acceleration of the pottery wheel, and (b) the time it takes the pottery wheel to reach its required speed of 65rpm.

81. Young and Freedman 9.59 A thin uniform rod of mass M and length L is bent at it’s center so that the two segments are perpendicular to each other. Find its moment of inertia about an axis perpendicular to its plane and passing through (a) the point where the two segments meet and (b) the mid point of the line connecting the two ends.

82. Serway 10-31 Find the net torque on the wheel about the axle O if a= 10.0 cm and b=25.0 cm

83. Giancoli 8-65 An asteroid of mass 1.0x10 5 kg travelling at a speed of 30 km/s relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision. M E =5.97x10 24 kg R E =6.38x10 6 m