1. Rotation 旋轉(Chap. 10) • We are going to consider the rotation of a rigid body (a body of fixed shape and size) about a fixed axis.

2. To begin, we have to define the angular position: • When the body rotates, the angular position θchanges as a function of t. • θplays the role of linear position x.

3. Similarly, we have: angular displacement 角移 angular velocity角速度 with T the period and f the frequency. also • note that the angular velocity is also a vector, i.e. it has a direction (however, we only consider rotation about a fixed axis, so ω has only + and – sign. • the direction is given by the right-hand rule. angular acceleration角加速度

5. Is angular displacement a vector? The answer is no: because the addition of two angular displacements depends on the order, which is different from usual vector addition.

6. Constant angular acceleration • for the case of constant angular acceleration, the equation of motion is equivalent to the equations for constant linear acceleration. • the derivation is identical to those in linear motion.

7. Relation between linear and angular motion: The point P in the rotating body will follow a circular motion. the first derivative gives: the period the point P is: in other words,

8. second derivation of s: Be careful that this value only present the acceleration tangent to the circle. there is another component of the acceleration which is along the radius, i.e. the centripetal acceleration (or radial acceleration) ar: the net acceleration is then: with Note that atis non-zero only when there is an angular acceleration while ar is non-zero even if angular velocity is constant.

9. Kinetic energy of rotating body One can consider rotating body as a collection of particles, and the kinetic energy is then simply the sum of all kinetic energies of the particles: Note that all particles have the same angular velocity. Hence we can define a rotational inertia 轉動慣量I: • I (also called the moment of inertia 慣性矩 ) is a measure of its resistance to change in its angular velocity. • This tells us how the mass is distributed about its axis of rotation. • It is a constant for a particular rigid body and for a particular axis.

10. And now the kinetic energy is given by: This is similar to the kinetic energy of linear motion: • The rotational inertia now plays the role of mass. • But it also related to the mass distribution the axis. e.g. to rotate a rod about the central axis is more easier than to rotate if perpendicular to its length, it means the rotational inertia in (a) is less than that in (b).

11. Which one has largest rotational inertia? The rotational inertia depends on the rotational axis.

12. Find the moment of inertia of the simple molecule along A, B, C and D axes. What is the rotational inertia about the axes A and B?

13. For a continuous body, the rotational inertia can be calculated as:

14. For a thin rod with a uniform linear mass densityλ=M/L:

15. For a disc with density σ=M/(πR2): How to calculate I about an axis at the edge of the disc?

16. Parallel-axis theorem Proof: Choose the origin locate at the center of mass: Expanding the equation: =0 =0 (center of mass)

17. I about an axis at the edge of the disc:

19. (a) How can one tell the difference between a had-boiled egg and a raw egg by trying to spin them? (b) After they are spinning, explain what happens if each is stopped for an instant and then released. • About which axis is the moment of inertia (a) the largest; (b) the smallest? • Two identical cans of concentrated orange juice are released at the top of an incline. One is frozen and the other has defrosted. Which reaches the bottom first?

20. Torque 力矩,轉矩 • A Force acting on a rigid body can be decomposed into two components: one (Fr)is along the vector r and the other (Ft)perpendicular to r. • Frdoes not cause rotation, only Ft will cause rotations about O. The ability of Ft to cause rotations also depends on the distance r. So we define the quantity torque as:

21. Torque, comes from the Latin word meaning “to twist”. • The unit of torque is Nm (Newton - meter), but note that it is different from the work which also has the unit Nm=J. But the torque is never a measure of energy. • Same as forces, torques can adds together the total is the net torque of the body.

22. Newton’s second law for rotation: since

23. Example:

26. The force needed is twice the force in (a).

27. Work and rotational kinetic energy Very similar to the case of linear motion: