Angular Momentum

Angular momentum is the rotational counterpart of linear momentum, and — importantly — it is a well-defined quantity for any moving particle, not only for particles undergoing circular or rotational motion. For a particle of mass $m$ with linear momentum $\vec p=m\vec v$, located at position $\vec r$ relative to a chosen point (or axis), the angular momentum of the particle about that point is defined as the moment of its linear momentum:

Its magnitude is $L=rp\sin\theta$, where $\theta$ is the angle between $\vec r$ and $\vec p$, and (as with torque) it can equivalently be written $L=(r\sin\theta)p=r_\perp p$ or $L=r(p\sin\theta)=rp_\perp$.

Worked idea — a particle moving in a straight line has constant angular momentum. Consider a particle of mass $m$ moving with constant velocity $\vec v$ (so its path is a straight line), and fix an origin $O$ at a perpendicular distance $d$ from that path. At any instant, if the particle is at a point $Q$ with position vector $\vec r=\vec{OQ}$, making angle $\theta$ with the momentum $\vec p=m\vec v$, then $L=rp\sin\theta$; but $r\sin\theta$ is exactly the fixed perpendicular distance $d$ from the origin to the particle's straight-line path, so

with no dependence on the angle $\theta$ (or on the particle's changing distance $r$ from $O$) at all — as the particle moves further along its straight path, $r$ increases but $\theta$ correspondingly shrinks, and the product $r\sin\theta$ stays exactly fixed at $d$. This shows conclusively that angular momentum is not a quantity confined to rotational motion — a particle in ordinary straight-line motion has a perfectly well-defined, and generally nonzero, angular momentum about any point not lying on its path, and that angular momentum stays constant throughout the straight-line motion (as long as no external torque acts to change $\vec p$). …