Moment of Inertia Reference Table Cheat Sheet

Key Facts

• For point masses, moment of inertia is $I = \sum m_i r_i^2$, where $r_i$ is the perpendicular distance from the axis of rotation.
• For a continuous object, moment of inertia is $I = \int r^2\,dm$, so mass farther from the axis has a larger effect.
• The parallel axis theorem is $I = I_{\mathrm{cm}} + Md^2$, where $d$ is the distance between the center-of-mass axis and the parallel new axis.
• The perpendicular axis theorem for a flat lamina in the $xy$-plane is $I_z = I_x + I_y$ when the axes meet at the same point.
• A thin hoop or ring about its central axis has $I = MR^2$.
• A solid disk or solid cylinder about its central symmetry axis has $I = \frac{1}{2}MR^2$.
• A solid sphere about a diameter has $I = \frac{2}{5}MR^2$, while a thin spherical shell has $I = \frac{2}{3}MR^2$.
• Rotational kinetic energy is $K_{\mathrm{rot}} = \frac{1}{2}I\omega^2$, and net torque obeys $\tau_{\mathrm{net}} = I\alpha$ for rotation about a fixed axis.

Vocabulary

Moment of inertia

A measure of an object's resistance to angular acceleration about a chosen rotation axis.

Rotation axis

The line about which an object rotates, and the line from which perpendicular distances $r$ are measured.

Center of mass

The average position of an object's mass, often used as the reference axis for standard moment of inertia formulas.

Parallel axis theorem

A rule stating that $I = I_{\mathrm{cm}} + Md^2$ for an axis parallel to a center-of-mass axis.

Angular acceleration

The rate of change of angular velocity, represented by $\alpha$ and usually measured in $\mathrm{rad/s^2}$.

Rotational kinetic energy

The energy of rotation given by $K_{\mathrm{rot}} = \frac{1}{2}I\omega^2$.