Centre of Mass

$x_{cm} = \frac{\sum m_i x_i}{\sum m_i} \quad [M^0 L^1 T^0] \quad (\text{m})$

Torque

$\tau = rF\sin\theta \quad [M^1 L^2 T^{-2}] \quad (\text{N·m})$

Angular Momentum

$L = I\omega = mvr\sin\theta \quad [M^1 L^2 T^{-1}] \quad (\text{kg·m}^2\text{/s})$

Moment of Inertia

$I = \sum m_i r_i^2 = \int r^2 \, dm \quad [M^1 L^2 T^0] \quad (\text{kg·m}^2)$

Radius of Gyration

$K = \sqrt{\frac{I}{M}}, \quad I = MK^2 \quad [M^0 L^1 T^0] \quad (\text{m})$

Parallel Axis Theorem (any body)

$I = I_{cm} + Md^2$

Perpendicular Axis Theorem (flat/2D bodies only)

$I_z = I_x + I_y$

Rolling Without Slipping

$v_{cm} = \omega R, \quad a_{cm} = \alpha R$ $KE_{total} = \frac{1}{2}mv_{cm}^2\!\left(1 + \frac{K^2}{R^2}\right)$ $a_{incline} = \frac{g\sin\theta}{1 + K^2/R^2}$

Standard Bodies (about central axis unless stated)

Conservation of Angular Momentum

$I_1\omega_1 = I_2\omega_2 \quad (\text{when } \tau_{net} = 0)$