Angular Momentum of a Particle

Angular momentum of a particle characterizes the rotational aspect of particle motion relative to a selected point or axis.

Angular momentum about a point

For particle $M$ relative to point $O$:

$$\vec L_O=\vec r\times m\vec v,$$

where $\vec r=\overrightarrow{OM}$. The direction of $\vec L_O$ follows from the right-hand rule.

Magnitude

The magnitude is $L_O=mvr\sin\theta=mv h$, where $h$ is the perpendicular distance from point $O$ to the velocity line.

Angular momentum about an axis

Angular momentum about an axis is the projection of $\vec L_O$ onto that axis. For the $z$ axis, $L_z=(\vec r\times m\vec v)\cdot\vec e_z$.

Angular-momentum theorem

For a fixed point $O$ in an inertial frame:

$$\frac{d\vec L_O}{dt}=\vec M_O,$$

where $\vec M_O=\vec r\times\sum\vec F$ is the moment of the resultant force about $O$.

Conservation

If the resultant external moment about the point is zero, then $\vec L_O=const$. Likewise, if the sum of moments about a fixed axis is zero, the corresponding component of angular momentum is conserved.

Example

A 2 kg particle moves at 4 m/s perpendicular to a 0.5 m position vector. Its angular momentum magnitude is $L_O=mvr=2\cdot4\cdot0.5=4$ kg·m²/s.

Common mistakes

• confusing angular momentum with moment of force;
• reversing the cross-product order;
• using the full distance $r$ instead of the perpendicular lever arm $h$ when the vectors are not perpendicular;
• applying conservation without checking the external moment.