System of Particles & Rotatory Motion — Formula Sheet

X_{cm} = \frac{\sum m_i x_i}{\sum m_i}

\vec{r}_{cm} = \frac{m_1\vec{r}_1 + m_2\vec{r}_2}{m_1 + m_2}

\theta = \omega t

\omega = \omega_0 + \alpha t

\theta = \omega_0 t + \frac{1}{2} \alpha t^2

\omega^2 = \omega_0^2 + 2\alpha\theta

v = r\omega

I = \sum m_i r_i^2

I = MK^2

I = I_{cm} + Md^2

I_z = I_x + I_y

Torque = r × F; magnitude is rF sinθ.

For pure rolling without slipping, v = Rω and a = Rα.

For continuous bodies, use integration

X_cm = (1/M) ∫x dm.

Angular momentum (L) is the rotational counterpart of linear momentum, given by L = Iω.