Angular Momentum and Angular Velocity

Now specialise angular momentum to the case that gives it its most familiar, everyday meaning: a point mass $m$ that is part of a rigid body, executing genuine circular motion about a fixed axis at distance $r$ from that axis.

Because the point mass's linear momentum is, at every instant, tangential to its circular path, it is always exactly perpendicular to the position vector $\vec r$ (measured from a point on the axis) — so the angle between them is always $\theta=90°$, and the general formula $L=rp\sin\theta$ simplifies to

Using the relation between linear and angular speed in circular motion, $v=r\omega$, this becomes

Direction. Since $\vec L$ is perpendicular to both $\vec r$ and $\vec p$ (both of which lie in the plane of the circular motion), $\vec L$ is necessarily directed along the axis of rotation itself — exactly the same direction as $\vec\omega$.

Generalising to a full rigid body. Recognising $mr^2$ as the moment of inertia of this point mass, and summing over every point mass making up an entire rigid body — all of which, crucially, share the same angular velocity $\omega$ even though they sit at different distances $r_i$ from the axis — gives

$\vec L=\left(\sum_i m_ir_i^2\right)\vec\omega=I\vec\omega.$ …