1. Angular Momentum of a Point Particle and Fixed Axis Rotation 8.01 W11D1 Fall 2006

2. Angular Momentum of a Point Particle Point particle of mass m moving with a velocity Momentum Fix a point S Vector from the point S to the location of the object Angular momentum about the point S SI Unit

3. Cross Product: Angular Momentum of a Point Particle Direction Right Hand Rule

4. Cross Product: Angular Momentum of a Point Particle Magnitude: a)moment arm b)Perpendicular momentum

5. Example Problem: Angular Momentum and Cross Product A particle of mass m = 2 kg moves with a uniform velocity At time t, the position vector of the particle with respect ot the point S is Find the direction and the magnitude of the angular momentum about the origin, ( the point S) at time t.

6. Solution: Angular Momentum and Cross Product The angular momentum vector of the particle about the point S is given by : The direction is in the negative direction, and the magnitude is

7. Group Problem: Angular Momentum and Circular Motion Consider a point particle of mass m moving in a circle of radius R with velocity. Find the direction and magnitude of the angular momentum about the center of a circle in terms of the radius R, mass m, and angular speed .

8. Solution: Angular Momentum and Circular Motion of a Point Particle Fixed axis of rotation: z -axis Angular velocity Velocity Angular momentum about the point S

9. Fixed Axis Rotation : Kinematics Angle variable Angular velocity Angular acceleration Mass element Radius of orbit Moment of inertia Parallel Axis Theorem

10. Angular Momentum for Fixed Axis Rotation Angular Momentum about the point S Tangential component of momentum z -component of angular momentum about S :

11. Concept Question: Angular Momentum A dumbbell is rotating at a constant angular speed about its center. Compared to the dumbbell's angular momentum about its center A, its angular momentum about point B (as shown in the figure) is 1.bigger. 2.the same. 3.smaller.

12. Concept Question: Angular momentum A disk with mass M and radius R is spinning with angular velocity  about an axis that passes through the rim of the disk perpendicular to its plane. The magnitude of its angular momentum is:

13. Dynamics: Newton’s Second Law and Torque about S Tangential force on mass element produces z - component of torque about axis passing through point S Newton’s Second Law z -component of torque about S z -component of total torque about S is the sum over all elements

14. Moment of Inertia and Torque z -component of total torque about S Recall: Moment of Inertia about S : Summary:

15. Torque and Angular Momentum about S Torque about S : Recall: Angular acceleration and angular velocity Angular momentum about S : Summary:

16. Concept Question: change in angular momentum A person spins a tennis ball on a string in a horizontal circle with velocity (so that the axis of rotation is vertical). At the point indicated below, the ball is given a sharp blow (force ) in the forward direction. This causes a change in angular momentum in the 1. direction 2. direction 3. direction

17. Demo: twirling skater A woman,holding dumbbells in her arms, spins on a rotating stool. When she pulls the dumbbells inward, the moment of inertia changes and she spins faster. The magnitude of the angular momentum is 1.the same. 2.larger because she's rotating faster. 3.smaller because her moment of inertia inertia is smaller.

18. Rotational and Translational Comparison QuantityRotationTranslation Force Torque Kinetic Energy Momentum Angular Momentum Kinetic Energy

19. Angular Impulse Angular impulse during interaction Change in angular momentum Rotational dynamics

20. Group Problem: Experiment Angular Momentum A steel washer, is mounted on the shaft of a small motor. The moment of inertia of the motor and washer is I 0. Assume that the frictional torque on the axle remains the same throughout the slowing down. The washer is set into motion. When it reaches an initial angular velocity  0, at t = 0, the power to the motor is shut off, and the washer slows down during the time interval  t 1 = t a until it reaches an angular velocity of  a at time t a. At that instant, a second steel washer with a moment of inertia I w is dropped on top of the first washer. Assume that the second washer is only in contact with the first washer. The collision takes place over a time  t col = t b - t a. Assume the frictional torque on the axle remains the same. The two washers continue to slow down during the time interval  t 2 = t f - t b until they stop at t = t f. Express your answers in terms of I 0, I w,  0,  a,  t 1,  t col, and  t 2. a)What is the angular deceleration  1 while the washer and motor are slowing down during the interval  t 1 = t a ? b)What is the angular impulse during the collision? c)What is the angular velocity of the two washers immediately after the collision is finished? d)What is the angular deceleration  2 after the collision?

21. Rotational Work Tangential force Displacement vector work for a small displacement

22. Rotational Work Newton’s Second Law Tangential acceleration Work for small displacement Summation becomes integration for continuous body

23. Rotational Work Rotational work for a small displacement Torque about S Infinitesimal rotational work Integrate total work

24. Rotational Work-Kinetic Energy Theorem Infinitesimal rotational work Integrate rotational work Kinetic energy of rotation about S

25. Rotational Power Rotational power is the time rate of doing rotational work Product of the applied torque with the angular velocity

26. Group Problem: Rotational Work A steel washer is mounted on the shaft of a small motor. The moment of inertia of the motor and washer is I 0. The washer is set into motion. When it reaches an initial angular speed  0, at t = 0, the power to the motor is shut off, and the washer slows down during an interval  t 1 down until it reaches an angular speed of  a at time t a. At that instant, a second steel washer with a moment of inertia I w is dropped on top of the first washer. Assume that the second washer is only in contact with the first washer. The collision takes place over a time  t int after which the two washers and rotor rotate with the angular speed  b. Assume the frictional torque  f on the axle is independent of speed, and remains the same when the second washer is dropped. a) What angle does the rotor rotate through during the collision? b) What is the work done by the friction torque  f from the bearings during the collision?