Energy and Momentum of Rotating Systems
Energy and Momentum of Rotating Systems
Rotation carries kinetic energy ½Iω² and angular momentum L = Iω about a fixed axis, and for a rolling body the total is K = ½mv² + ½Iω². Rolling without slipping links the two motions through v = ωR. Angular impulse (torque times time) changes L, and when the net external torque about an axis or point is zero, L is conserved - the basis for skaters, collapsing stars and disks dropped onto turntables. Torque does work through W = τ Δθ, and the work-energy theorem extends to W_net = ΔK including rotational terms.
Traps: conserving angular momentum when an external torque (gravity, a rough pivot) actually acts; forgetting that a rolling object's energy includes both terms, so it reaches the bottom of a ramp later than a frictionless sliding block; believing static friction does work in rolling without slipping (the contact point is instantaneously at rest, so it does not); and using I about the wrong axis.
Worked example: a skater with moment of inertia I spins at ω. Pulling the arms in halves I, so ω doubles. Kinetic energy ½Iω² doubles, and the extra energy comes from the work the skater does pulling the arms inward.
Rule of thumb: pick the axis or point about which external torques vanish, then write I1 ω1 = I2 ω2.