1. Chapt. 10 Angular Momentum
Definition of angular momentum
Vector nature of torque
11/17/2018
Phys 201, Spring 2011

2. Angular rotation using vectors
Angular quantities in vector notation Ft
τ = F r sin ϕ
= F L
11/17/2018
Phys 201, Spring 2011

3. The vector product It follows: Non commutative: And distribution rule:
Unit vectors:
11/17/2018
Phys 201, Spring 2011

4. Example: Vector algebra
If and = 12 find . Let Then Finally:
11/17/2018
Phys 201, Spring 2011

5. Linear momentum --> Angular momentum
The linear momentum
The angular momentum with respect to the axis of rotation must scale with r: m
= r m v = m r2 ω = I ω
11/17/2018
Phys 201, Spring 2011

6. Linear momentum --> Angular momentum
The angular momentum,
the vector nature:
11/17/2018
Phys 201, Spring 2011

7. Angular momentum of a rigid body about a fixed axis:
Consider a rigid distribution of point particles rotating in the x-y plane around the z axis, as shown below. The total angular momentum around the origin is the sum of the angular momenta of each particle:
(since ri and vi are perpendicular) v1
We see that L is in the z direction. m2 j
Using vi = ω ri , we get r2 r1 m1 i v2 ϕ r3 m3 v3
11/17/2018
Phys 201, Spring 2011

8. Angular momentum of a rigid body about a fixed axis:
In general, for an object rotating about a fixed (z) axis we can write LZ = I ω
The direction of LZ is given by the right hand rule (same as ω).
We will omit the axis (Z) subscript
for simplicity, and write L = I ω z ω
11/17/2018
Phys 201, Spring 2011

9. Rotational and Linear Quantity Linear Angular Position Velocity
Acceleration
Time
Inertia
Dynamics
Momentum
Kinetic energy
11/17/2018
Phys 201, Spring 2011

10. The 2nd Law in rotation: F = m a
Translational (linear) motion for a system of particles
What is the rotational version of this??
F = m a
The rotational analogue of force F is torque 
Define the rotational analogue of momentum p to be angular momentum
11/17/2018
Phys 201, Spring 2011

11. Definitions & Derivations...
First consider the rate of change of L:
So:
Recall:

τEXT
Finally:
Direct analogue:
11/17/2018
Phys 201, Spring 2011

12. What does it mean? where and In the absence of external torques
Total angular momentum is conserved
11/17/2018
Phys 201, Spring 2011

14. Gyroscope:
L (and therefore the wheel)
moves in a horizontal
circle around O: “precession”

15. The precession frequency
Let’s calculate the precession frequency
(Last figure)
L forms a circular motion:
The precession frequency

16. Question: We repeat the gyroscope experiment on the moon
(g_moon = 1/6 g_Earth)
but with an angular velocity double from the one in the lecture.
Will the precession of the wheel be faster, slower or the same?