Rotational dynamics and angular momentum

Newton's second law has a rotational twin. For rotation about a fixed axis, τ = Iα: torque plays the part of force, moment of inertia the part of mass, and angular acceleration the part of acceleration.

The whole chapter lines up like this:

Work done by a torque is W = ∫τ dθ (just τθ for a steady torque), and the work–energy theorem holds: the work done equals the gain in ½Iω².

Example: a steady torque of 2 N m acts on a wheel with I = 0.5 kg m², starting from rest. α = 2/0.5 = 4 rad/s². After 5 s, ω = 20 rad/s, and the torque is then delivering P = τω = 2 × 20 = 40 W.

For a body symmetric about its axis, L = Iω points along the axis, and dL/dt = τ. So if there is no external torque, Iω stays the same: if I goes down, ω goes up.

You can see this anywhere. A skater spinning with arms out pulls them in: I falls, so she spins faster. A diver curls up tight to turn several somersaults, then opens out to slow down before entering the water. Example: if I falls from 3.0 to 1.2 kg m², a spin of 2 rev/s becomes 2 × 3.0/1.2 = 5 rev/s.

Is energy conserved too? No. K = ½Iω² = L²/2I, and with L fixed, a smaller I means more kinetic energy (2.5 times more in the example). The extra energy comes from the work the skater's own muscles do in pulling the arms in.