The conservation law

Angular momentum $\vec{L}$ is the rotational analogue of momentum $\vec{p}$. For a rotating body $L = I\omega$ holds, and like $\vec{p}$ it obeys a conservation law: it is conserved when the system is subject to no net external moments.

Conservation of angular momentum

If the sum of the moments of the external forces about a pole $P$ is zero, then the total angular momentum about $P$ is conserved: $\ev{\sum\vec{M}_\text{ext} = \vec{0} \;\;\Longrightarrow\;\; \vec{L}_\text{tot}^A = \vec{L}_\text{tot}^B}$

The intuition is the same as for momentum: without an external “rotational push”, the system’s overall rotation stays unchanged. But there is a spectacular difference: whereas in translation $\vec{p} = m\vec{v}$ has a fixed mass, in the rotational case $L = I\omega$ contains $I$, which the system can change by redistributing its own mass. If $I$ decreases, $\omega$ must increase to keep $L$ constant — and that is exactly what a skater does when pulling in their arms.

A first example: the disc with the rope

Consider a disc rotating about a pivot while a rope unwinds from its rim, dragging a package that progressively moves away from the centre. Choosing the pivot as the pole, both the pivot’s reaction and the (radial) tension have zero arm or a radial direction: they produce no moment. Hence the total angular momentum is conserved:

$\left(\tfrac{1}{2}m_d R^2 + m_p r_A^2\right)\omega_A = \left(\tfrac{1}{2}m_d R^2 + m_p r_B^2\right)\omega_B$

As the package moves further away ($r_B > r_A$), the system’s moment of inertia grows and the angular velocity $\omega_B$ decreases: the system slows down. It is the same mechanism as the skater, in reverse.

Rotational analogue of $\vec{p}$

Angular momentum is roughly what momentum is for translation. It is conserved if $\sum M_\text{ext} = 0$, and it is the pivot around which every problem in this section turns.

▶ Anti-Gravity Wheel and angular momentum — Veritasium

Collegamenti

Argomenti: Dinamica rotazionale Concetti: Conservazione del momento angolare · Momento angolare Competenze: Conservazione del momento angolare

Esercizi collegati: Il tuffatore che ruota · Cometa al perielio e all’afelio · Problema — Momento torcente rettangolare e L(t)