Torque and Angular Momentum
- Torque: $\vec{\tau} = \vec{r} \times \vec{F} = I\vec{\alpha}$
- Angular Momentum: $\vec{L} = \vec{r} \times \vec{p} = I\vec{\omega}$
- Law of Conservation of Angular Momentum: If net external torque $\tau_{\text{ext}} = 0$, then $I_1\omega_1 = I_2\omega_2 = \text{constant}$.
- Kinetic Energy of Rolling without Slipping:
$$K_{\text{total}} = K_{\text{trans}} + K_{\text{rot}} = \frac{1}{2}M v_{\text{cm}}^2 \left(1 + \frac{k^2}{R^2}\right)$$
