Angular Momentum

Use the angular-momentum method when you want to describe how strongly an object's motion is tied to turning about a chosen point or axis, and especially when a conservation argument is cleaner than chasing every force. It applies to a single particle, a rigid body spinning about a fixed axis, or a system that changes shape while external torque stays negligible.

Step 1: Choose The Reference Point

Angular momentum is always measured about some point or axis, and the same moving object can have different values relative to different points. Define that reference before anything else. This is the detail students miss most often, which is why many problems start by telling you exactly which origin or axis to use.

Step 2: Pick The Right Model

For a single particle, the general definition is

where $\vec{r}$ points from the reference point to the particle and $\vec{p} = m\vec{v}$ is linear momentum. In magnitude form,

For a rigid body rotating about a fixed axis, use instead

where $I$ is moment of inertia and $\omega$ is angular speed. This second form is valid only under the fixed-axis rigid-body condition; it is not the general definition.

Step 3: Check The Angle

From $L = rp\sin\theta$, four things raise angular momentum: more mass, more speed, more distance from the reference point, and a larger angle between $\vec{r}$ and $\vec{p}$, up to $90^\circ$. Only the part of the momentum perpendicular to the radius contributes. The shortcut $L = mvr$ works only when the motion is perpendicular to the radius; otherwise keep the $\sin\theta$ factor. If the motion points directly toward or away from the reference point, $\sin\theta = 0$ and the angular momentum about that point is zero.

Step 4: Test Conservation Carefully

Torque is what changes angular momentum:

If the net external torque is zero about the same point or axis, then $d\vec{L}/dt = 0$ and angular momentum is conserved. Do not assume this too early: conservation holds only when that torque condition is genuinely satisfied.

Full Worked Example: A Ball Moving Tangentially

A $0.20\ \mathrm{kg}$ ball moves at $6.0\ \mathrm{m/s}$ along a path tangent to a circle of radius $0.50\ \mathrm{m}$ centered on a chosen origin. Find the magnitude of its angular momentum about that origin.

Following the steps: the reference point is the origin given. The model is the particle definition $L = rp\sin\theta$. For the angle, the velocity is tangent to the circle, hence perpendicular to the radius, so $\theta = 90^\circ$ and $\sin\theta = 1$. That collapses the magnitude to

Substituting,

So the magnitude is $0.60\ \mathrm{kg \cdot m^2/s}$. If the ball moved along the same tangent line but you chose a different origin, the answer could change. That dependence on the reference point, not the arithmetic, is the real lesson.

Where Each Step Trips People Up

Step 1 (reference point): Forgetting to name it. Angular momentum is always about some point or axis; without one, the statement is incomplete. Self-check: can you point to the origin in your diagram?

Step 2 (model): Using $L = I\omega$ when the body is not a rigid body on a fixed axis. Self-check: is the axis fixed and the body rigid?

Step 3 (angle): Using $L = mvr$ in every problem. It works only when momentum is perpendicular to the radius. Self-check: is $\theta$ really $90^\circ$? If not, keep $\sin\theta$.

Step 4 (conservation): Assuming conservation without checking torque. Angular momentum is conserved only if the net external torque is zero about the same point or axis. Self-check: is there any external torque about my reference point?

One more habit: angular momentum is a vector. Many introductory problems need only the magnitude, but direction still matters in the full definition.

Where The Method Is Used

Angular momentum shows up in orbit problems, spinning wheels, gyroscopes, rotating machinery, and figure-skater examples where pulling mass inward changes the spin rate. It is especially powerful when conservation is easier to apply than a force-by-force analysis, as in a system that changes shape internally while external torque stays negligible.

To see the change in spin from the conservation side, vary the worked example: cut the speed from $6.0\ \mathrm{m/s}$ to $3.0\ \mathrm{m/s}$, then instead keep $6.0\ \mathrm{m/s}$ and double the radius to $1.0\ \mathrm{m}$. Comparing those two cases shows quickly what controls angular momentum and by how much.