Unit 7: Torque and Rotational Motion

Executive Summary

Rotation mirrors translation: torque replaces force, rotational inertia replaces mass, and angular momentum replaces momentum — with conservation laws to match.

Kinematics and dynamics

Rotational kinematics use the same equation shapes with θ, ω, α. Torque $\tau = rF\sin\theta$ (lever arm × perpendicular force). Newton's second law for rotation: $\tau_{net} = I\alpha$.

Rotational inertia

$I$ depends on mass AND its distribution: point mass $mr^2$, solid cylinder $\frac{1}{2}mr^2$, hollow cylinder $mr^2$, solid sphere $\frac{2}{5}mr^2$. Mass farther from the axis means larger I and slower spin for a given torque.

Energy and rolling

$K_{rot} = \frac{1}{2}I\omega^2$. Rolling without slipping couples translation and rotation via $v = r\omega$; total kinetic energy is the sum of translational and rotational parts. A rolling object down an incline splits its energy, so it accelerates slower than a sliding block.

Angular momentum

$L = I\omega$ (and $L = mvr$ for a point mass). Conserved when net external torque is zero — the spinning ice skater pulling in arms spins faster.

Exam traps

Torque depends on the lever arm (perpendicular distance), not the raw distance. Rolling objects reach the bottom later than sliding ones. Angular momentum conservation applies only with zero net external torque. Static equilibrium requires BOTH net force zero and net torque zero about any axis.