Rotation, Torque and Angular Momentum

Rotation fills two units of AP Physics C: Mechanics, Torque and Rotational Dynamics and Energy and Momentum of Rotating Systems. Every linear idea from the first half of the course has a rotational twin, and the exam tests whether you can switch between them.

The rotational twins

Torque

τ = r × F, with magnitude τ = rF sin θ

θ is the angle between the position vector from the pivot and the force. A force pointing straight at the pivot has zero torque however large it is.

Moment of inertia by integration

I = ∫ r² dm

For a uniform rod of mass M and length L about one end, write dm = (M/L) dx:

I = ∫₀ᴸ x² (M/L) dx = ML²/3

The parallel axis theorem moves an axis away from the center of mass:

I = I_cm + Md²

About its center the rod has I_cm = ML²/12; moving the axis L/2 to the end gives ML²/12 + M(L/2)² = ML²/3, the same answer.

Rolling without slipping

The contact point is momentarily at rest, which links the two motions: v = ωR. The kinetic energy has two parts:

K = ½Mv² + ½Iω²

Worked example

A solid disk (I = ½MR²) rolls without slipping from rest down a ramp of height h. Find its speed at the bottom.

1. Energy: Mgh = ½Mv² + ½(½MR²)(v/R)².
2. Simplify: Mgh = ½Mv² + ¼Mv² = ¾Mv².
3. Result: v = √(4gh/3), independent of mass and radius.

A hoop (I = MR²) gets v = √(gh), so a disk beats a hoop down the same ramp. The object with the smaller I/MR² wins.

Angular momentum

With no external torque, L = Iω is conserved. A skater who pulls in the arms reduces I, so ω increases. Kinetic energy is not conserved in that case: it increases, supplied by the work of the skater's muscles.

The free response approach

Start from Στ = Iα or from conservation, set up any integral with dm written explicitly, and state the no-slip condition when you use v = ωR.