Angular Momentum (L = Iω)

Angular momentum, $L = I\omega$, is the rotational counterpart of linear momentum $p = mv$, and like its linear cousin its importance rests on a conservation law: with no external torque acting, $L$ does not change. That is a genuinely deep statement — by Noether's theorem it is the consequence of the fact that the laws of physics look the same in every direction — and it is the reason angular momentum is worth tracking rather than recomputing.

Sit on a rotating stool with weights held out at arm's length, then pull them in. Nothing exerts a torque about the vertical axis, so $L$ is fixed; but $I$ has dropped sharply, because the weights are now much closer to the axis and $I$ goes as the square of that distance. Since $I\omega$ must stay constant, $\omega$ has to rise. That is the figure skater's spin, and it is not a trick of technique — it is arithmetic. If $I$ falls to a third, $\omega$ triples.

The same law works at every scale. A star's core collapsing at the end of its life shrinks from something the size of the Earth to a neutron star perhaps 20 km across; $I$ collapses with it and the rotation rate climbs to hundreds of revolutions per second, which is what a millisecond pulsar is. In the other direction, the Moon's tidal drag exerts a small torque on the Earth, so terrestrial angular momentum is not conserved and the day is lengthening by about 1.8 milliseconds per century — the angular momentum lost is transferred to the Moon's orbit, which is why it recedes about 3.8 cm a year.

The thing conserved is $L$, not $\omega$, and not the kinetic energy. The skater who triples her rotation rate does not spin at the same energy — rotational kinetic energy is $\tfrac{1}{2}I\omega^2$, and if $I$ falls to a third while $\omega$ triples, the energy triples. Where did it come from? From her muscles: pulling the weights inward against the outward pull required real work, and that work is exactly the extra energy. People often assume conservation of angular momentum means everything is conserved, and it does not. Two further points. $L$ is a vector along the axis of rotation, which is why a spinning bicycle wheel resists being tilted and why gyroscopes precess rather than fall over — the torque changes the direction of $L$, not its magnitude. And "no external torque" is a condition to check, not to assume; a system with friction at a bearing is losing angular momentum to whatever the bearing is bolted to.