# Resource: Moments-of-inertia

Expressions for moments-of-inertia $I$ of various shapes about various axes-of-rotation.

Moments-of-inertia
RodAxis through centre$$I=\frac{1}{12}mL^2$$Proof 27
Axis at end$$I=\frac13 mL^2$$
PlateAxis through centre$$I=\frac1{12} m(w^2+l^2)$$Proof 28
Axis along edge$$I=\frac13 mw^2$$
Holed cylinderAxis through centre$$I=\frac12 m(r_1^2+r_2^2)$$Proof 29
Solid cylinder$$I=\frac12 mr^2$$
Hollow (thin-walled) cylinder$$I=mr^2$$
Solid ballAxis through centre$$I=\frac25 mr^2$$Proof 30
Hollow (thin-walled) ball$$I=\frac23 mr^2$$

For other axis placements, the parallel-axis theorem can be used ($d$ is distance from a parallel axis through the centre-of-mass ($\text{com}$) to the axis-of-rotation) (see proof 26):

$$I=I_\text{com}+md^2$$

## Proof 26: Parallel-axis theorem

When rotating about the centre-of-mass, the moment-of-inertia $I_\text{com}$ is often easier to find. When the axis-of-rotation is somewhere else, we’ll below derive a formula that…

## Proof 27: Rod’s moment-of-inertia

A rod rotating perpendicularly about its centre-of-mass With a single point having the moment-of-inertia $I_\text{point}=r^2m$ ($m$ and $r$ are mass and distance from the axis-of-rotation),…