Angular momentum and its conservation

Take one particle at distance r⊥ from the fixed axis: the part of its angular momentum along the axis is mr⊥²ω. Adding over the body gives L_z = Iω.

For a particle, l need not be along the axis: l and ω are not necessarily parallel, unlike p and v, which always are.

If the axis is a symmetry axis of the body, the perpendicular parts of the particles' angular momenta cancel in pairs, and L = Iω exactly along the axis.

For a fixed axis, dL_z/dt = τ along the axis, while the component of L perpendicular to the axis stays constant. With I constant this gives τ = Iα again.

Conservation: if the external torque about the axis is zero, Iω = constant. If I decreases, ω increases in the same ratio, and the other way round.

A person on a frictionless swivel chair spinning with arms folded slows down on stretching the arms out (I increases) and speeds up again on pulling them in.

Acrobats, divers, skaters and dancers doing a pirouette on the toes of one foot use this: pulling the arms and legs in reduces I and raises the spin rate.

Kinetic energy ½Iω² = L²/2I is not conserved when I changes: pulling the arms in raises it, and the extra energy comes from the work done by the person's muscles.