1. Angular Momentum & Torque for Systems of Particles Lecturer: Professor Stephen T. Thornton

2. Reading Quiz
A particle is located in the xy-plane at a location x = 1 and y = 1 and is moving parallel to the +y axis. A force is exerted on the particle along the +x axis. L and are in what directions about the origin?
L and are along the +z axis.
L and are along the -z axis.
L is along the +z axis; is along the –z axis.
L is along the -z axis; is along the +z axis.
L is along the +y axis; is along the +x axis.

3. z y x

4. Angular momentum Vector (cross) products Torque again with vectors
Last Time
Angular momentum
Vector (cross) products
Torque again with vectors

5. Today Angular momentum and torque system of particles rigid objects
Unbalanced torque
Kepler’s 2nd law

6. Atwood Machine. An Atwood machine consists of two masses, and connected by a cord that passes over a pulley free to rotate about a fixed axis. The pulley is a solid cylinder of radius and mass 0.80 kg. (a) Determine the acceleration a of each mass. (b) What percentage of error in a would be made if the moment of inertia of the pulley were ignored? Ignore friction in the pulley bearings.
Giancoli, 4th ed, Problem 11-38

7. System of Particles
The angular momentum of a system of particles can change only if there is an external torque—torques due to internal forces cancel.
This equation is valid in any inertial reference frame. It is also valid about a point uniformly moving in an inertial frame of reference. We are starting to get very technical!
This equation is in general not valid if L and torque are calculated about a point that is accelerating, unless that point is the center of mass.

8. System of Particles
The equation above is not valid in general about a point accelerating in an inertial frame of reference.
But the center of mass is special! The equation is true even for an accelerating center of mass of a system of particles or for a rigid object:

9. Angular Momentum for a Rigid Object
rotating
For a rigid object, we can show that its angular momentum when rotating around a particular axis is given by:
Add up all the particles. If L is along a symmetry axis (z here) through CM, particles on one side of symmetry axis cancel L on the other side.
Figure Calculating Lω = Lz = ΣLiz. Note that Li is perpendicular to ri and Ri is perpendicular to the z axis, so the three angles marked φ are equal. All components of L cancel except along the z axis. L is along omega – the symmetry axis and through the CM.

10. So we finally have these equations for a rigid object.
The values must be calculated about
Origin or axis fixed in an inertial frame. or
2) An origin at the CM or about an axis passing through the CM.
If we do not have this, then things get real complicated! We have reached our limit here!!
Figure Calculating Lω = Lz = ΣLiz. Note that Li is perpendicular to ri and Ri is perpendicular to the z axis, so the three angles marked φ are equal.

11. Torque and Angular Momentum Vectors

12. Torque Gravity and Extended Objects
Gravitational torque acts at the center of mass, as if all mass were concentrated there:

13. Torque Gravity and Extended Objects
Gravitational torque acts at the center of mass, as if all mass were concentrated there.
Do the Falling Rigid Body demo again.

14. Conceptual Quiz You are looking at a bicycle wheel along its axis
Conceptual Quiz You are looking at a bicycle wheel along its axis. The wheel rotates CCW and is supported by a string attached to the rear of the handle. When the wheel is released, the end of the handle closest to you will   A)  move up B)  move to the left C)  move to the right D)  move down

15. Do bicycle wheel demo.

16. Move to the right. The picture below is looking from above.
Answer: C
Move to the right. The picture below is looking from above.

19. Conceptual Quiz A man sits at rest on a frictionless rotating stool
Conceptual Quiz A man sits at rest on a frictionless rotating stool. He holds a rotating bicycle wheel that has an angular momentum L directed up. When he flips the wheel over, so that it has L directed down, the angular momentum of the system (man + stool + wheel) is   A)   zero. B)   L, up. C)   L, down. D)   2L, up. E) 2L, down.

20. Answer: B
Angular momentum has to be conserved. There is no torque to change it.
Do experiment.

21. Angular Momentum and Torque for a Rigid Object
A system that is rotationally imbalanced will not have its angular momentum and angular velocity vectors in the same direction. A torque is required to keep an unbalanced system rotating.
Figure In this system L and ω are not parallel. This is an example of rotational imbalance.

22. An unbalanced car wheel will cause problems on your wheel bearings
An unbalanced car wheel will cause problems on your wheel bearings. We need to keep our wheels well balanced, dynamically not just statically.

23. Kepler’s 2nd Law
There is no torque so L is constant, and Kepler’s second law states that each planet moves so that a line from the Sun to the planet sweeps out equal areas in equal times.
Figure Kepler’s second law of planetary motion.
Solution: From the figure, dA = ½ (r) (v dt sin θ), so dA/dt = L/2m = constant if L is constant.

24. Conceptual Quiz A) remain stationary
You are holding a spinning bicycle wheel while standing on a stationary turntable. If you suddenly flip the wheel over so that it is spinning in the opposite direction, the turntable will:
A) remain stationary
B) start to spin in the same direction as before flipping
C) to spin in the same direction as after flipping
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25. Conceptual Quiz A) remain stationary
You are holding a spinning bicycle wheel while standing on a stationary turntable. If you suddenly flip the wheel over so that it is spinning in the opposite direction, the turntable will:
A) remain stationary
B) start to spin in the same direction as before flipping
C) start to spin in the same direction as after flipping
The total angular momentum of the system is L upward, and it is conserved. So if the wheel has
−L downward, you and the table must have +2L upward.

26. Conceptual Quiz See hint on next slide.
Two different spinning disks have the same angular momentum, but disk 1 has more kinetic energy than disk 2. Which one has the bigger moment of inertia?
A) disk 1
B) disk 2
C) not enough info L L
See hint on next slide.
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Disk 1
Disk 2

27. Conceptual Quiz / A) disk 1 B) disk 2 C) not enough info
Two different spinning disks have the same angular momentum, but disk 1 has more kinetic energy than disk 2. Which one has the bigger moment of inertia?
A) disk 1
B) disk 2
C) not enough info L L
KE = I 2 = L2 (2 I)
(used L = I ). /
Disk 1
Disk 2

28. Conceptual Quiz / A) disk 1 B) disk 2 C) not enough info
Two different spinning disks have the same angular momentum, but disk 1 has more kinetic energy than disk 2. Which one has the bigger moment of inertia?
A) disk 1
B) disk 2
C) not enough info L L
KE = I 2 = L2 (2 I)
(used L = I ).
Because L is the same, bigger I means smaller KE. /
Disk 1
Disk 2