Chapter 5 Rotation of a Rigid Body

1. Chapter 5 Rotation of a Rigid Body

2. §5-1 Motion of a Rigid body §5-2 Torque The Law of Rotation rotational Inertia §5-3 Application the Law of Rotation §5-4 Kinetic Energy and Work in Rotational Motion §5-5 Angular Momentum of a rigid Body Conservation of Angular Momentum

3. 1. Rigid body The body has a perfectly definite and unchanged shape and size no matter how much external force acts on it. §5-1 Motion of a Rigid body The distance between two points in a rigid body maintains constant forever 2. Motion forms of rigid body Translation(平动) Can be regarded as a particle

4. Fixed axis Rotating around axis Moving axis Rotating around a point Rotation --Superposition of several rotations around axis axis

5. Translation + rotation

6. P Q axis 3. Rotation of a rigid body around a fixed axis Every point of the rigid body moves in a circle They have the same angular displacement, angular speed & angular acceleration. Select arbitrary point P. P’s rotation can represent the rotation of the rigid body.

7. P Q axis Angular position   Angular displacement  Angular speed Angular acceleration

8. z P 0 x 4. Relation between angular and linear quantities

9.  is located in the plane perpendicular to the axis F M φ r d §5-2 Torque The Law of Rotation Rotational Inertia 1. Torque

10. F F 1  is not placed in the plane perpendicular to the axis F 2 r Only is useful for the rotation around the axis No effect at all to rotate the rigid body around the axis

11. 0` 0 2. The law of rotation

12. Use Newton’s second law to Internal torque External torque The radial component of force passes through the axis and can’t cause the body to rotate. Tangential component Multiply both side byri ,

13. Rotational inertia I resultant external torqueM resultant internal torque=0 For entire rigid body --Law of rotation

14. --the rotational inertia (moment of inertia) of the rigid body about the axis It describes the rotating inertia of a rigid body about an axis. 3. Calculation of moment of inertia The magnitude of moment of inertia depends on the total mass and mass distribution of body, the location and orientation of the axis. discrete particles

15.  Continuous distribution of mass the mass distribution over a line —the mass per unit length the mass distribution over a surface —the mass per unit area the mass distribution over a volume —the mass per unit volume

16. Exa. A slender rod has mass m，lengthl. Find its I about some axis follow as ⑴the axis through O and perpendicular with the rod. ⑵ the axis through an end of the rod and perpendicular with it. ⑶ The axis at arbitrary distance hfrom O Solution l m ⑴ o

17. ⑵ l m

18. ⑶ m l h --Parallel-axis theorem

19. Suppose a thin board is located in xOy plane  Perpendicular-axis theorem -- Perpendicular-axis theorem

20. Example are fastenedtogether by a weightless rope across a fixed pulley ( ) < The pulley has mass and radius . There is a fractional torque exerting on the axis. The rope does not slip over the pulley. §5-3 Application the Law of Rotation Calculate the Acce. of the blocks and the tension of the rope.

22. Solution According to New.’s Second Law ： ： According to Rotational Law Pulley：

23. We get

24. Example A uniform circular plate with mass m and radius R is placed on a roughly horizontal plane. At the beginning, the plate rotates with angular speed 0 around the axis across the center of it. Suppose the fractional coefficient between them is . Calculate how many times does the plate rotate before it stops?

25. dr r R Solution Select mass unit

26. The friction exerts on Frictional torque correspondingly

27. According to We have We get How many revolutions does the plate rotate before it stops? Question

28. 0 The rigid rotating from 0 §5-4 Kinetic Energy and Work in Rotational Motion 1. The work done by torque As The work done by the torque

29. s KE of rotation 2. Kinetic Energy of rotation For the entire rigid body

30. Then -- KE of rotation 3. The theorem of KE of a RB rotating about a fixed axis as

31. -- The theorem of KE of a RB rotating about a fixed axis 4. Gravitational potential energy of RB …can be seen as the GPE of a particle located on the center of mass with the same mass of m.

32. Example A uniform thin rodm, l，One end is fixed. Find its of point A as it rotate angular from horizontal line? m、 l O A  A

33. l O Solution as

34. We get

35. Use conservation law of energy of rigid body, we can get first. Or use We can get then Another solution Then use We can get the Acce. of A finally

36. the angular momentum of : §5-4 Angular Momentum of a rigid Body Conservation of Angular Momentum 1. Angular momentum of a rigid body rotating about a fixed axis Direction

37. Direction: Same as Angular momentum of a rigid body rotating about a fixed axis： 2. Angular momentum theorem As ，we have integration

38. ---impulse torque =Constant when M=0 —Angular momentum theorem of a rigid body 3. Conservation Law of Angular Momentum

39. m m r 2 r 1 I0 ExampleThe rotational inertiaof a person and round slab isI0. The mass of dumbbell is m . Their rotating angular speed is0 and the rotating radius of m is r1 at the beginning. Calculate: The angular speed and the increment of the mechanical energy when the arms of the person contracts fromr1tor2

40. Person + round slab + dumbbells = system Resultant external torque is zero. So its angular momentum is conservative.

41. The increment of the mechanical energy

42. Example A round platform has mass of M and radius of R. It can rotates around a vertical axis through its center. Suppose all resistant force can be neglected. A girl with mass of m stands on the edge of the platform. At the beginning, the platform and the girl are at rest. If the girl runs one revolution, how much degree does the girl and the platform rounds relative to the ground, respectively?

43. m R M

44. Solution Resultant external torque of system =0 Angular momentum is conservative： Let： be rotational inertia of the girl and the platform. be angular speed of the girl and the platform relative to the ground. then： And

45. We get： The girl relative to the platform ： Let t refer to thetime that the girl runs one revolution on the platform, then ：

46. The girl rounds relative to the ground： The platform rounds relative to the ground：

47. Example A uniformlythin rod has M,2l . It can rotate in vertical plane around the horizontal axis through its mass center O. At the beginning, the rod is placed along the horizontal position. A small ball with mass of m and speed of u falls to one end of the rod. If the collision between the ball and the rod is elastic. Find the speed of the ball and the angular speed of the rod after they collide each other.

48. Solution As mg of the ball << the impulse force between the ball and the rod So we can neglect mg during colliding. Then the resultant external torque of the system with respect to O=0.

49. The angular momentum of the system is conservative. As the collision is elastic, then the mechanical energy of the system is conservative And

50. Solve above equations, we get Ball’s Rod’s