Rotational Inertia and Newton's Second Law for Rotation

The translation table

Every rotational law is a translation of a linear one:

• $x\to\theta$, $v\to\omega$, $a\to\alpha$
• $F\to\tau$, $m\to I$, $p\to L$
• $F=ma\ \to\ \tau=I\alpha$
• $K=\tfrac12mv^2\ \to\ K_{\text{rot}}=\tfrac12I\omega^2$

Rotational inertia depends on where the mass is

The $r^2$ is the whole story: moving mass twice as far from the axis multiplies its contribution by four. That is why a hoop ($I=MR^2$) is far harder to spin up than a solid sphere ($I=\tfrac25MR^2$) of the same mass and radius.

For a point mass on a rotating rigid body, $v=\omega r$ — points farther out move faster even though $\omega$ is the same everywhere.