1. 今日課程內容 CH10 轉動 角位移、角速度、角加速度 等角加速度運動 轉動與移動關係 轉動動能 轉動慣量 力矩 轉動牛頓第二運動定律

2. 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.

3. 10.5: Relating Linear and Angular Variables If a reference line on a rigid body rotates through an angle , a point within the body at a position r from the rotation axis moves a distance s along a circular arc, where s is given by: Differentiating the above equation with respect to time—with r held constant—leads to The period of revolution T for the motion of each point and for the rigid body itself is given by Substituting for v we find also that

4. Differentiating the velocity relation with respect to time—again with r held constant—leads to Here,  =d  /dt. Note that dv/dt =a t represents only the part of the linear acceleration that is responsible for changes in the magnitude v of the linear velocity. Like v, that part of the linear acceleration is tangent to the path of the point in question. Also, the radial part of the acceleration is the centripetal acceleration given by 10.5: Relating Linear and Angular Variables

5. 10.6: Kinetic Energy of Rotation( 轉動動能 ) 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: (m i is the mass of the i th particle and v i is its speed). (  is the same for all particles). The quantity in parentheses on the right side is called the rotational inertia( 轉動慣量 ) (or moment of inertia) I 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,

6. 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

7. 10.7: Calculating the Rotational Inertia Parallel Axis Theorem( 平行軸定理 ) If h is a perpendicular distance between a given axis and the axis through the center of mass (these two axes being parallel).Then the rotational inertia I about the given axis is

8. Sample problem: Rotational Inertia

9. 10.8: Torque( 力矩 ) The ability of a force F to rotate the body depends on both the magnitude of its tangential component F t, and also on just how far from O, the pivot point, the force is applied. To include both these factors, a quantity called torque  is defined  as: OR, where is called the moment arm of F.

10. 11.6: Torque Revisited Fig. 11-10 (a) A force F, lying in an x-y plane, acts on a particle at point A. (b) This force produces a torque  = r x F on the particle with respect to the origin O. By the right-hand rule for vector (cross) products, the torque vector points in the positive direction of z. Its magnitude is given by in (b) and by in (c).

11. 10.9: Newton’s Law of Rotation For more than one force, we can generalize:

12. Sample problem: Newton’s 2 nd Law in Rotational Motion

13. 10.10: Work and Rotational Kinetic Energy( 轉動動能 ) where  is the torque doing the work W, and  i and  f are the body’s angular positions before and after the work is done, respectively. When  is constant, The rate at which the work is done is the power

14. 10.10: Work and Rotational Kinetic Energy

15. Sample problem: Work, Rotational KE, Torque

16. Chapter 11 Rolling, Torque, and Angular Momentum

17. 11.2 Rolling( 滾動 ) as Translational and Rotation Combined Although the center of the object moves in a straight line parallel to the surface, a point on the rim certainly does not. This motion can be studied by treating it as a combination of translation of the center of mass and rotation of the rest of the object around that center.

18. 11.2 Rolling

19. 11.3 The Kinetic Energy of Rolling If we view the rolling as pure rotation about an axis through P, then (  is the angular speed of the wheel and I P is the rotational inertia of the wheel about the axis through P). Using the parallel-axis theorem (I P = I com +Mh 2 ), (M is the mass of the wheel, I com is its rotational inertia about an axis through its center of mass, and R is the wheel’s radius, at a perpendicular distance h). Using the relation v com =  R, we get: A rolling object, therefore has two types of kinetic energy: 1.a rotational kinetic energy due to its rotation about its center of mass (=½ I com  2 ), 相對於質心轉動的轉動動能 2.2. and a translational kinetic energy due to translation of its center of mass (=½ Mv 2 com ) ，質心的移動動能

20. 11.4: The Forces of Rolling: Friction and Rolling A wheel rolls horizontally without sliding( 滑動 ) while accelerating with linear acceleration a com. A static frictional force f s acts on the wheel at P, opposing its tendency to slide. The magnitudes of the linear acceleration a com, and the angular acceleration  can be related by: where R is the radius of the wheel. If the wheel does slide when the net force acts on it, the frictional force that acts at P in Fig. 11-3 is a kinetic frictional force, f k.The motion then is not smooth rolling, and the above relation does not apply to the motion.

21. 11.4: The Forces of Rolling: Rolling Down a Ramp( 斜面 )

22. Sample problem: Rolling Down a Ramp A uniform ball, of mass M=6.00 kg and radius R, rolls smoothly from rest down a ramp at angle  =30.0°. a) The ball descends a vertical height h=1.20 m to reach the bottom of the ramp. What is its speed at the bottom? b) What are the magnitude and direction of the frictional force on the ball as it rolls down the ramp?

23. Solid Cylinder Hoop Empty Can Box Sphere

25. 11.5: The Yo-Yo

26. 習題 Ch10: 6, 13, 28, 30, 38, 41, 52, 53, 64