CHECKLIST

Angular momentum. Moment of inertia and more properties of angular motion.

Moment of inertia, I
For linear momentum, p = mv
Similarly, for angular momentum,
L = “something”x ω
That is, we seek a property of the body that measures ‘angular inertia’. This is defined as the moment of inertia, I.
Thus: L = I ω

One way to get a formula for I
Consider a body rotating with angular speed ω. Then consider the body is divided into many small masses, m1, m2, etc at distances r1, r2, etc, respectively, from a pivot.

Moment of inertia
is the angular speed

Moment of inertia

More on I
Note: we know that 1) if no external forces act on the system, linear momentum is conserved.
We now know that 2) if no external torques act on the system, angular momentum is conserved. This explains why, e.g., an ice skater rotates faster when she pulls in her arms.

Conservation of angular momentum
Since L = I ω, if no external torques act, L is constant.
So, when the skater moves her arms in, closer to her body,
I decreases and therefore ω increases.

Conservation of angular momentum
In (1) I = m1a2 +m2a2
In (2) I = m1b2 + m2b2
For L = I ω to remain constant, the skater rotates faster in (2).

Please don’t try this!
A crazy physics professor stands on a swivel chair with weights in his outstretched arms. A student causes him to rotate.
The professor brings in his arms and…

Comparison of equations for linear and angular motion

Example 1. Conservation of angular momentum
A horizontal disc rotates around a vertical axis at 90 rev/min. A small piece of chewing gum of 20 g falls vertically onto the disc and sticks to it at a distance of 5.0 cm from the axis. If the number of revs/min is reduced to 80, find the moment of inertia, I, of the disc.

Work done by torque
Note: We can also define work done by a constant torque in moving through an angle θ as: W = τθ
The power

Example 2
An electric motor supplies a power of 500 W to drive a flywheel of I = 2.0 kgm2 at a speed of 600 rev/min. How long will it take the flywheel to come to rest after the power is switched off, assuming the retarding torque, due to friction, is constant?