Torque

$\vec{\tau} = \vec{r} \times \vec{F}, \quad |\tau| = rF\sin\theta \quad [M^1L^2T^{-2}]$

• Maximum torque when $\theta = 90°$ (force perpendicular to position vector)
• Zero torque when force passes through the axis ($\theta = 0°$ or $180°$)
• Direction: right-hand rule (thumb along $\vec{\tau}$ when curling $\vec{r} \to \vec{F}$)
• Torque is the "turning effect" of a force

Angular Momentum

$\vec{L} = \vec{r} \times \vec{p} = I\vec{\omega} \quad [M^1L^2T^{-1}]$

• For a particle: $L = mvr\sin\theta$
• For a rigid body: $L = I\omega$ (about the rotation axis)

Newton's Second Law for Rotation

$\vec{\tau}_{net} = \frac{d\vec{L}}{dt} = I\vec{\alpha}$

Conservation of angular momentum: When $\tau_{net} = 0$: $L = I\omega = \text{constant}$

Connection to Linear Laws