Mass Moment of Inertia

Moment of inertia $(I)$, also known as mass moment of inertia, measures a rigid body’s resistance to angular acceleration about a given axis.
$I={\int }_m{r}^{2}dm$
Units: $\text{kg}\cdot {\text{m}}^2$

Alternatively, for a point mass, mass moment of inertia is equal to:
$I=m{r}^2$

Mass moment of inertia $I$ is related to angular acceleration by:
$M=I\alpha$

The moment of inertia about any axis can be calculated given you know the $\bar{I}$ about a parallel axis passing through the centre of mass.

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Radius of Gyration

Instead of quoting $I$ directly, many problems (including textbook wheel/drum problems) give a radius of gyration $k$:

$I=m{k}^2$

$k$ is the distance from the axis at which the entire mass could be concentrated to give the same $I$. It is not a physical dimension of the body.

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Moment of Inertia for Common Shapes

Shape	Axis	$\bar{I}$
Slender rod, length $l$	$\perp$, through centre	$\frac{1}{12}m{l}^2$
Slender rod, length $l$	$\perp$, through end	$\frac{1}{3}m{l}^2$
Thin ring/hoop, radius $r$	Through centre, $\perp$ to plane	$m{r}^2$
Circular disk, radius $r$	Through centre, $\perp$ to plane	$\frac{1}{2}m{r}^2$
Rectangular plate, sides $a,b$	Through centre, $\perp$ to plane	$\frac{1}{12}m(a^2+b^2)$
Solid sphere, radius $r$	Through centre	$\frac{2}{5}m{r}^2$

Composite bodies

For a body built from several simple shapes, find $I$ (about the same axis) for each part separately. Apply the parallel-axis theorem to each part individually as needed and then sum them. Subtract for holes/cut-outs.

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