Inside This Unit: The Full Breakdown
This unit analyzes rotation: torque, moment of inertia as I = ∫r²dm, and the rotational form of Newton’s second law τ = Iα, plus rolling without slipping.
Why it matters
Rotational dynamics parallels translational dynamics and introduces integration for moment of inertia, a signature Physics C calculation.
Key concepts
- Torque τ = rF sinθ produces angular acceleration via τ = Iα.
- Moment of inertia I = ∫r² dm depends on mass distribution.
- Angular kinematics mirror linear kinematics with θ, ω, α.
- Rolling without slipping requires v_cm = Rω and a_cm = Rα.
Torque and Moment of Inertia
Torque is the rotational analog of force, computed from the lever arm. Moment of inertia, found by integrating r² over the mass (I = ∫r² dm) or summing mr², measures rotational inertia. The parallel axis theorem (I = I_cm + Md²) shifts the axis.
Rotational Dynamics and Rolling
The rotational second law Στ = Iα gives angular acceleration. Linear and angular quantities connect through v = rω and a_t = rα. Rolling without slipping links translation and rotation via v_cm = Rω, essential for ramps and pulleys with mass.
AP exam tip
Set up I = ∫r² dm with the correct mass element dm (e.g., λ dx for a rod) — showing the integral setup earns points even before evaluating.
Connections to other units
- Unit 2: τ = Iα is the rotational version of ΣF = ma.
- Unit 6: Torque and inertia feed rotational energy and angular momentum.
