Centre of Mass

$x_{cm} = \frac{\sum m_i x_i}{\sum m_i} \quad \text{Dimensions: } [L] \quad \text{Unit: m}$

Moment of Inertia

$I = \sum m_i r_i^2 \quad [M\,L^2\,T^0] \quad \text{kg m}^2$

Theorems

$I = I_{cm} + Md^2 \quad \text{(Parallel — any body)} \qquad I_z = I_x + I_y \quad \text{(Perpendicular — flat bodies only)}$

Torque and Angular Momentum

$\tau = r F\sin\theta \quad [M\,L^2\,T^{-2}] \quad \text{N m} \qquad L = I\omega \quad [M\,L^2\,T^{-1}] \quad \text{kg m}^2\text{/s}$

$\tau = \frac{dL}{dt} = I\alpha \qquad P = \tau\omega \quad [M\,L^2\,T^{-3}] \quad \text{W}$

Rolling Motion

$v_{cm} = \omega R \qquad KE = \frac{1}{2}mv^2\!\left(1 + \frac{K^2}{R^2}\right) \qquad a = \frac{g\sin\theta}{1 + K^2/R^2}$

$K^2/R^2$ Values: Solid sphere $\frac{2}{5}$, Disc $\frac{1}{2}$, Hollow sphere $\frac{2}{3}$, Ring $1$.

Conservation of Angular Momentum

$I_1\omega_1 = I_2\omega_2 \quad \text{when } \tau_{net} = 0 \qquad \Delta KE = \frac{L^2}{2}\!\left(\frac{1}{I_2} - \frac{1}{I_1}\right)$