HW #9 due Tuesday, November 16, 11:59 p.m. Recitation Quiz #9 tomorrow Last Time: Rotational Kinetic Energy + the “Race of the Shapes” Today: Introduction to Angular Momentum Concepts, Example Problems

Define: L = Iω, the Angular Momentum Suppose a net tangential force starts to act on a mass m moving in a circle (radius r). F r What happens ? The net torque changes the mass’ angular speed from its initial value, ω0, to a value ω in some time interval Δt. Define: L = Iω, the Angular Momentum SI: kg-m2/s

Torque and Angular Momentum So, the angular momentum is L = Iω , and is related to the torque by : Note : The “derivation” on the previous slide assumed that the moment of inertia did not change. In general, this is not required. More generally, if we assume both the moment of inertia and the angular speed can change :

Example 100 N 2.0 kg 0.5 m A solid disc, with a mass of 2.0 kg and a radius of 0.5 m, is initially rotating at ω0 = 1.0 rad/s. A tangential force of 100 N is applied to the edge of the disc. What is its angular momentum after 3.0 s ?

Conservation of Angular Momentum We have seen that the relationship between torque and angular momentum is : If the net torque is zero, i.e., Στ = 0 : If there is no net torque, the angular momentum cannot change !

Rotational vs. Linear Motion [angular momentum] [momentum] A net torque causes an angular acceleration, which means a change in the angular speed A net force causes an acceleration, which means a change in the velocity If there is no net torque, the angular momentum is conserved. If there is no net force, the momentum is conserved.

Conceptual Question A constant net non-zero net torque is exerted on an object. Assume the object’s geometry and mass do not change. Which of the following quantities CANNOT be constant for this object? [More than one answer may be correct.] (a) angular acceleration (b) angular velocity (speed) (c) moment of inertia (d) center of mass (e) angular momentum

Conservation of Angular Momentum Example: Figure Skater http://www.youtube.com/watch?v=J6aSOpy2Z9g Watch video from 3:40 to 3:55. Why does the figure skater rotate faster when she draws her arms/legs in ? [Assume no torques in the system]

Why ?? Model the Figure Skater as a Dumbbell Q : What happens if R becomes smaller (when arms/legs drawn in) ? A : The moment of inertia I decreases. R m Q : What happens to the angular momentum ? m R A : Since we assume no torques, L is conserved. Moment of Inertia: Thus: If I decreases,  must increase !!

Real-Life Demo

Conceptual Question (Quick Quiz 8.6) If the ice at the polar caps continues to melt, the water that is trapped in the polar caps will re-distribute, such that there is more water closer to the equator. If this occurs, would the length of one day (i.e., the time for the Earth to rotate once about its axis) … increase decrease remain the same

Example A (horizontally-oriented) disc with moment of inertia I1 is rotating with angular speed ω1 about a vertical frictionless axle. A second (horizontally-oriented) disc with moment of inertia I2, which is initially not rotating, drops onto the first disc. The two discs eventually reach the same angular speed ω2 . What is the ratio ω2 / ω1 ?

Example A star with an initial radius of 1.0  108 m and a rotational period of 30 days suddenly collapses (due to gravitational forces) to a radius of 1.0  104 m. Assuming the star is spherical before/after the collapse … Find the rotational period after collapse. Find the work done by gravity during the collapse if the star’s mass is 2.0  1030 kg. What would be the speed of an object located at the star’s equator after the collapse?

Next Class 8.7 : More on angular momentum …