# Does the moment of inertia affect angular momentum?

Sep 14, 2015

Yes.
Angular momentum $L$ is directly proportional to angular velocity $\omega$ , with moment of inertia $I$ being the constant of proportionality.

#### Explanation:

Note that the Law of Conservation of Angular Momentum states that total angular momentum is always conserved.
Since $\vec{L} = I \vec{\omega}$ , it implies that for a rotating object, as the moment of inertia decreases, the angular velocity increases, and vice versa.

For example, suppose that an ice skater rotates at one spot about a fixed axis with arms outstretched, an with constant angular velocity $\omega$ .

Suppose now that he moves his arms towards his centre and places them on his chest. Automatically his angular velocity will increase and he will begin to rotate faster in order to satisfy the principle of conservation of linear momentum. This is so since by pulling his arms inwards he decreases his moment of inertia since by definition
$I = \frac{1}{M} {\sum}_{j} {m}_{j} {r}_{\bot j}^{2} = \frac{1}{M} {\int}_{M} {r}^{2} \mathrm{dm}$ ,
where r is the position vector from the axis of rotation to the mass element dm.