1. Angular Momentum Lecturer: Professor Stephen T. Thornton

2. Reading Quiz: Can an object moving in a straight line ever have a nonzero angular momentum?
A) Always
B) Never
C) Sometimes

3. Sometimes, because it depends upon the axis of rotation around which you want to find the angular momentum. There is no angular momentum when the object passes through the rotation axis, because the moment arm is zero. There is angular momentum when the moment arm is nonzero (see left sketch).
Answer: C

4. Last Time Rotational kinetic energy Objects rolling – energy, speed
Rotational free-body diagram
Rotational work

5. Today Angular momentum Vector (cross) products
Torque again with vectors
Unbalanced torque

6. Angular Momentum of Circular Motion
This particle has linear momentum. We can also say it has an angular momentum with respect to a given point, in this case the center of the circle.

7. In the case of the particle moving around the circle, let’s look more carefully at the angular momentum.
This is another way to determine angular momentum.

8. The Angular Momentum of Non-Tangential Motion

9. A particle moving in any direction can have angular momentum about any point.

10. Angular Momentum in Linear and Circular Motion
The L in each view is constant. If are the same, then L is the same.

11. Change in angular momentum
L = I
L = I, divide by t
This equation looks similar to Newton’s 2nd law. It is sometimes called Newton’s 2nd law for rotation.

12. Conservation of angular momentum
Note what happens when there is no torque. L = 0, and angular momentum is constant.
Note similarity to conservation of linear momentum when Fnet,ext = 0.

13. Conceptual Quiz: A figure skater stands on one spot on the ice (assumed frictionless) and spins around with her arms extended. When she pulls in her arms, how do her rotational inertia, her angular momentum and her rotational kinetic energy change?   A) They all increase. B) They all remain the same. C) They all decrease. D) Rot inertia decreases, angular momentum remains constant, and her KE increases. E) Rotational inertia and angular momentum decrease, KE decreases.

14. Answer: D
Angular momentum must be conserved. No torque. Rotational inertia decreases, because radius decreases. Only D is possible.
How does KE increase?
I goes down,  goes up, L constant.
But K will increase because of .

15. Vector Cross Product; Torque as a Vector
The vector cross product is defined as:
The direction of the cross product is defined by a right-hand rule:
Figure The vector C = A x B is perpendicular to the plane containing A and B; its direction is given by the right-hand rule.

16. The vector (cross) product can also be written in determinant form:

17. Some properties of the cross product:
3rd one is called distributive property

18. Conceptual Quiz The direction of the vector cross product is along the directionB) C) D) E)

19. Answer: E

20. For a particle, the torque can be defined around a point O:
Here, is the position vector to the point of application of force relative to O.
Figure τ = r x F, where r is the position vector.

21. Torque can be defined as the vector product of the position vector from the axis of rotation to the point of action of the force with the force itself:
Figure The torque due to the force F (in the plane of the wheel) starts the wheel rotating counterclockwise so ω and α point out of the page.

22. Torque
A right-hand rule gives the direction of the torque.

23. The Right-Hand Rule for Torque

24. Yo-yo demo
torque in
torque out O O

25. Angular Momentum of a Particle
The angular momentum of a particle about a specified axis (or point) is given by:
Figure The angular momentum of a particle of mass m is given by L = r x p = r x mv.

26. The Right-Hand Rule for Angular Momentump

27. Let’s do this demo!

28. Let’s do this demo!
No torque: L is conserved. I decreases, therefore ω must increase.

29. A Rotational Collision – Angular momentum will be conserved here.

30. Angular Momentum of a Particle
If we take the derivative of , we find:
Since
we have:

31. Opposite Particles. Two identical particles have equal but opposite momenta, and , but they are not traveling along the same line. Show that the total angular momentum of this system does not depend on the choice of origin.
Giancoli, 4th ed, Problem 11-35

32. A) Friction of gas particles
Conceptual Quiz: When a large star burns up its fuel, the gravitational force contracts it to a small size, even a few km. This is called a neutron star. When neutron stars rotate at high speed, even 100 rev/sec, they are called pulsars. They have more mass than our sun. What causes the high rotational angular velocity?
A) Friction of gas particles
B) Conservation of angular
momentum
C) The dark force
D) Conservation of energy

33. Answer: B
Just like our own sun, these stars rotate about their own axis. As gravity contracts the particles closer and closer, the density becomes huge. There are no torques, so angular momentum must be conserved. L=I, so as I decreases,  must increase.