1. Angular Momentum Angular momentum of rigid bodies
Newton’s 2nd Law for rotational motion
Torques and angular momentum in 3-D
Text sections
Physics 1D03 - Lecture 30

2. “Angular momentum” is the rotational analogue of linear momentum.
Recall linear momentum: for a particle, p = mv .
Newton’s 2nd Law: The net external force on a particle is equal to the rate of change of its momentum.
To get the corresponding angular relations for a rigid body, replace:
m I
v w
F t
p L (“angular momentum”)
Physics 1D03 - Lecture 30

3. Angular momentum of a rotating rigid body:
Angular momentum, L, is the product of the moment of inertia and the angular velocity.
L = Iw
Units: kg m2/s (no special name). Note similarity to: p=mv
Newton’s 2nd Law for rotation: the torque due to external forces is equal to the rate of change of L.
For a rigid body (constant I ),
So, sometimes
(but not always).
Physics 1D03 - Lecture 30

4. Conservation of Angular momentum
There are three great conservation laws in classical mechanics:
Conservation of Energy
Conservation of linear momentum
and now, Conservation of Angular momentum:
In an isolated system (no external torques), the total angular momentum is constant.
Physics 1D03 - Lecture 30

5. Angular Momentum Vectorw
For a symmetrical, rotating, rigid body, the vector L will be along the axis of rotation, parallel to the vector w, and
L = I w. L
(In general L is not parallel to w, but Iw is still equal to the component of L along the rotation axis.)
Physics 1D03 - Lecture 30

6. Angular momentum of a particlez f r v L O x y m
This is the real definition of L.
L is a vector.
Like torque, it depends on the
choice of origin (or “pivot”).
If the particle motion is all in the x-y
plane, L is parallel to the z axis..
Physics 1D03 - Lecture 30

7. Angular momentum of a particle (2-D):
|L| = mrvt
= mvr sin θ r v m
For a particle travelling in a circle (constant |r|, θ=90), vt = rw, so:
L = mrvt = mr2w = Iw
Physics 1D03 - Lecture 30

8. Quiz
As a car travels forwards, the angular momentum vector L of one of its wheels points:
forwards
backwards
C) up
D) down
E) left
F) right
Physics 1D03 - Lecture 30

9. Quiz
A physicist is spinning at the center of a frictionless turntable, holding a heavy physics book in each hand with his arms outstretched. As he brings his arms in, what happens to the angular momentum?
increases
decreases
remains constant
What happens to the angular velocity?
Physics 1D03 - Lecture 30

10. Example:
A student sits on a rotating chair, holding two weights each of mass 3.0kg. When his arms are extended to 1.0m from the axis of rotation his angular speed is 0.75 rad/s. The students then pulls the weights horizontally inward to 0.3m from the axis of rotation. Given that I = 3.0 kg m2 for the student and chair, what is the new angular speed of the student ?
Physics 1D03 - Lecture 30

11. Example
Angular momentum provides a neat approach to Atwood’s Machine. We will find the accelerations of the masses using “external torque = rate of change of L”. m1 m2 v w O R
Physics 1D03 - Lecture 30

12. w m2 v m1 p1 Atwoods Machine, frictionless (at pivot), massive pulley
For m1 : L1 = |r1 x p1|= Rp1 m1 m2 v w R O p1
so L1 = m1vR
L2 = m2vR
Lpulley= Iw = Iv/R
Thus L = (m1 + m2 + I/R2)v R
so dL/dt = (m1 + m2 + I/R2)a R p1 r R
Torque, t = m1gR - m2gR
= (m1 - m2 )gR
Write t = dL/dt, and complete the calculation to solve for a.
Note that we only consider the external torques on the entire system.
Physics 1D03 - Lecture 30

13. Solution
Physics 1D03 - Lecture 30

14. Summary Particle: Any collection of particles: L = I w.
Rotating rigid body:
L = I w.
Newton’s 2nd Law for rotation:
Angular momentum is conserved if there is no external torque.
Physics 1D03 - Lecture 30