轉動力學 ( Rotational Motion)

1. Chapter 10 轉動力學(Rotational Motion) Rotation

5. 10.2 The Rotational Variables A rigid bodyis a body that can rotate with all its parts locked together and without any change in its shape. A fixed axismeans that the rotation occurs about an axis that does not move. Figure skater Sasha Cohen in motion of pure rotation about a vertical axis. (Elsa/Getty Images, Inc.)

6. Rigid Object ( 剛體 ) • A rigid object is one that is nondeformable • The relative locations of all particles making up the object remain constant. • All real objects are deformable to some extent, but the rigid object model is very useful in many situations where the deformation is negligible Q

7. Angular Position • The arc length s and rare related: • s=qr

8. Conversions • Comparing degrees and radians • Converting from degrees to radians

9. Angular Displacement and Angular Velocity • Angular Displacement • Average Angular Speed • Instantaneous Angular Speed ( rad/s or s-1)

10. Angular Acceleration • Average angular acceleration • Instantaneousangular acceleration ( rad/s2 or s-2)

11. Directions of wand a • The directions are given by the right-hand rule

13. 10.4: Rotation with Constant Angular Acceleration Just as in the basic equations for constant linear acceleration, the basic equations for constant angular acceleration can be derived in a similar manner. The constant angular acceleration equations are similar to the constant linear acceleration equations.

14. Comparison Between Rotational and Linear Equations

15. Relating Linear and Angular Variables • Displacements • TangentialSpeeds The period of revolution T for the motion of each point and for the rigidbody itself is given by ;

16. Relating Linear and Angular Variables • TangentialAcceleration • Centripetal Acceleration

17. 10.6: Kinetic Energy of Rotation • An object rotating about some axis with an angular speed, , has rotational kinetic energy even though it may not have any translational kinetic energy • Each particle has a kinetic energy of • Ki = 1/2 mivi2 • Since the tangential velocity depends on the distance, r, from the axis of rotation, we can substitutevi= wi r

18. 10.6: Kinetic Energy of Rotation For an extended rotating rigid body, treat the body as a collection of particles with different speeds, and add up the kinetic energies of all the particles to find the total kinetic energy of the body: (mi is the mass of the ith particle and vi is its speed). (w is the same for all particles). The quantity in parentheses on the right side is called the rotational inertia (or moment of inertiaI) of the body with respect to the axis of rotation. It is a constant for a particular rigid body and a particular rotation axis. (That axis must always be specified.) Therefore,

19. Calculating the Rotational Inertia (moment of inertiaI)

20. r r m m R 質點的轉動慣量 • 此系統的轉動慣量 • I = 2mr2 • 此系統的轉動慣量 ( 環 ) M

21. Moment of Inertia of a Uniform Solid Cylinder • RadiusR, massM and lengthL • 質量均勻分佈 • Divide the cylinder into concentric shells with radius r, thickness dr and length L M

22. Moment of Inertia of a Uniform Solid Cylinder • 整個系統的轉動慣量 M

23. 10.7: Calculating the Rotational Inertia If a rigid body consists of a great many adjacent particles (it is continuous, like a Frisbee), we consider an integral and define the rotational inertia of the body as

24. Moments of Inertia of Various Rigid Objects

25. D C.M. Parallel-Axis Theorem(平行軸定理) • For an arbitrary axis, the parallel-axis theorem often simplifies calculations • Ip = ICM + MD 2 • Ipis about any axis parallel to the axis through the center of mass of the object • ICM is about the axis through the center of mass • D is the distance from the center of mass axis to the arbitrary axis P

26. y  y x  C.M. P x r  R rP Proof of the Parallel-Axis Theorem y  y C.M. r  △m x  C.M. R rP P x

27. y  y x  C.M. P x r  R rP r  △m R rP

28. r2 r1 r2 r1 R y  y x  C.M. P x △m2 r2 r1 △m1 r2 R r1

29. y  y x  C.M. P x 0 R IP = ICM + MR2 (Parallel-Axis Theorem)

30. Perpendicular-Axis Theorem(垂直軸定理) Iz = Ix + Iy ri Prove it Hint :

31. 圓盤 R M 利用垂直軸定理，系統的軸之轉動慣量 = ?

32. Ex 1 A rigid body is made of three identical thin rods, each with length L , fastened together in the form of a letter H (Figure). The body is free to rotate about a horizontal axis that runs along the length of one of the legs of the H. The body is allowed to fall from rest from a position in which the plane of the H is horizontal.What is the angular speed of the body when the plane of the H is vertical ? M O M M lCM L O C.M i lCM= ? IO = ? M L/2 M M f

33. Ex 2 A rigid body is made of three identical thin rods, each with length L , fastened together in the form of a letter H (Figure). The body is free to rotate about a horizontal axis that runs along the length of one of the legs of the H. The body is allowed to fall from rest from a position in which the plane of the H is horizontal.What is the angular speed of the body when the plane of the H is vertical ? mass of this leg is zero O M M lCM L O C.M i lCM= ? IO = ? M L/2 M f

34. The Center of Mass • There is a special point in a system or object, called the center of mass, that moves as if all of the mass of the system is concentrated at that point • The system will move as if an external force were applied to a single particle of mass M located at the center of mass • M is the total mass of the system

35. Center of Mass, position • The center of mass can be located by its position vector, • is the position of the i th particle, defined by

36. Center of Mass, Coordinates • The coordinates of the center of mass are • where M is the total mass of the system

37. Center of Mass, Example • Both masses are on the x-axis • The center of mass is on the x-axis • The center of mass is closer to the particle with the larger mass

38. Center of Mass, Objectwitha continuous mass distribution • An object can be considered a distribution of small mass elements, Dmi • The center of mass is located at position

39. Center of Mass, Objectwitha continuous mass distribution • The coordinates of the center of mass of the object are

40. Forces In a System of Particles • The acceleration can be related to a force • If we sum over all the internal forces, they cancel in pairs and the net force on the system is caused only by the external forces

41. Forces In a System of Particles The system will move as if an external force were applied to a single particle of mass M located at the center of mass

43. Ex 1 A rigid body is made of three identical thin rods, each with length L , fastened together in the form of a letter H (Figure). The body is free to rotate about a horizontal axis that runs along the length of one of the legs of the H. The body is allowed to fall from rest from a position in which the plane of the H is horizontal.What is the angular speed of the body when the plane of the H is vertical ? M O M M lCM L O C.M i lCM= ? IO = ? M L/2 M M f

44. Ex 2 A rigid body is made of three identical thin rods, each with length L , fastened together in the form of a letter H (Figure). The body is free to rotate about a horizontal axis that runs along the length of one of the legs of the H. The body is allowed to fall from rest from a position in which the plane of the H is horizontal.What is the angular speed of the body when the plane of the H is vertical ? mass of this leg is zero O M M lCM L O C.M i lCM= ? IO = ? M L/2 M f

45. Torque ( 力矩 )

46. Torque ( 力矩 ) • Torque,t, is the tendency of a force to rotate an object about some axis. • Torque is a vector • t = r F sin f= F d

47. Vector Product, General • Given any two vectors, and • The vector product is defined as a third vector, whose magnitude is • The direction of C is given by the right-hand rule

48. 10.8: Torque The ability of a force F to rotate the body depends on both the magnitude of its tangential component Ft, and also on just how far from O, the pivot point, the force is applied. To include both these factors, a quantity called torque tis definedas: OR, where is called the moment arm of F.

49. 10.9: Newton’s Law of Rotation For more than one force, we can generalize:

50. Torque and Angular Acceleration on a Particle F sinf Δm F cosf