1. Rolling 滾動, Torque and Angular momentum Rolling is a combination of translation of the center and rotation about the center. Since the rolling distance is: The speed of center of mass is:

2. The upper part of the wheel moves faster than the lower part of the wheel: Angular velocity around the axis at P is the same as the one around the axis at O. One can verify that this by showing that the velocity on the top of the wheel is again 2v com :

3. Kinetic energy of rolling: Consider the kinetic energy of a rolling wheel as measured by a stationary observer. where I P is the moment of inertia about the axis at P. Using parallel-axis theorem, we have: This is simply the sum of kinetic energies of a pure rotation and a pure translation.

4. Forces of rolling When a wheel begins to accelerate, it tends to slide 滑動 at the point P. A frictional force must act on the wheel at P to oppose that tendency. If the wheel does not slide, the static frictional force f s leads to an acceleration a com : Rolling down a ramp When a wheel rolls down a ramp, it tends to slide down the slope and therefore the static frictional force must oppose that tendency and acting up the slope. The equations of motion are: and

5. Torque

6. Angular momentum 角動量 Newton’s second law: For rotational motion:

7. Proof: consider the net force: For a system of particles:

8. Angular momentum of a rigid body: For a small mass element, the angular momentum is : Its direction is shown in figure (b). But we are interested in the component along the z-axis: Using the fact that: Analogous to p=mv

12. Conservation of Angular momentum If there is no external torque acting on a system, the total angular momentum must remain constant. =0 or Example: A person spins with his arms extended can reduce his moment of inertia by pulling in his arms and then increase his angular velocity because of the conservation of angular momentum.

13. A springboard diving can increase the angular velocity by folding her body so that the moment of inertia is reduced. She can then extend her body again to lower the angular velocity before hitting the water. The orientation of a spacecraft with a flywheel installed can be changed by turning the flywheel. The total angular momentum of flywheel and spacecraft must remain zero. Ice-skater always make use of this phenomenon.

15. One can change his angular position without violate the conservation of angular momentum: (a)A person stands on a frictionless platform with arms outstretched. (b)When the arms and upper body are rotated clockwise, the lower body and feet rotated counterclockwise. (c)The arms are brought down to reduce the moment of inertia of the upper body. When the twist at the waist is removed, the whole body has rotated through a finite angle.

16. 1.It is easier to balance a baseball bat with your finger at the thin end. Try it and explain why. 2.Why does spreading out both arms help one to balance on a tightrope? Why is holding a long pole even better? 3.What can happen if only the front brakes are applied on a bicycle? Why does it happen? 4.(a) Why does a helicopter usually have spinning blade at its tail? (b) Why do some helicopters have two rotors?

17. Moment of inertia of the rods: Initial angular momentum: Final angular momentum: By conservation of angular momentum:

18. Precession 進動 of a gyroscope 陀螺儀 Newton’s second law: For a non-spinning gyroscope, its own weight cause a torque in the y direction. The wheel simply falls. However, if the wheel is spinning rapidly that it has an angular momentum L, the axle of the wheel will rotate around the z-axis instead of falling. Figure (c) shows the direction of L changed by the torque due to the weight. since and The precession rate :

19. (a) (b)

21. (a) (b) (c) (d) (e) (f)