Angular momentum

Angular momentum is to rotation what momentum is to translation: it measures “how much” a body is rotating about a point, taking into account both its momentum and how far it passes from the point.

Principle — Angular momentum

The angular momentum of a body about a point O is $\ev{\vec{L} = \vec{r} \times \vec{p} = \vec{r} \times (m\,\vec{v})}$

The cross product $\vec{r} \times \vec{p}$ tells us that three things matter: the momentum $\vec{p}$, the distance $r$ from point O, and the angle between the two. A body moving straight towards point O (or straight away from it) has zero angular momentum about O, because $\vec{r}$ and $\vec{p}$ are parallel. Angular momentum is maximum when the motion is perpendicular to the radius, i.e. when the body “goes around” the point.

In two dimensions, with $\vec{r} = (x, y)$ and $\vec{p} = (p_x, p_y)$, the cross product has just one component (perpendicular to the plane):

$L = x\,p_y - y\,p_x$

Like momentum and kinetic energy, angular momentum depends on the choice of pole O and reference frame — and it is precisely this dependence that makes the decomposition in the next note natural.

Links

Topics: Centre of mass Concepts: Angular momentum · Momentum

Related exercises: The rotating diver · Velocity and momentum of the CM · Finding the masses from the CM