Q.Find the components along the $x$, $y$, $z$ axes of the angular momentum $\vec{l}$ of a particle, whose position vector is $\vec{r}$ with components $x$, $y$, $z$ and momentum is $\vec{p}$ with components $p_x$, $p_y$ and $p_z$. Show that if the particle moves only in the $x$-$y$ plane the angular momentum has only a $z$-component.

Angular Momentum: The Rotational Cousin of Momentum

You already know linear momentum — how hard it is to stop a moving object. A truck moving at 20 m/s has more momentum than a bicycle at the same speed. Now imagine something spinning: a bicycle wheel, a spinning top, or a figure skater pulling their arms in. There is a similar "quantity of motion" for rotation, and that is angular momentum.

The core intuition is simple: angular momentum measures how much rotation an object has, and how hard it is to change that rotation. Just as a heavy truck is hard to stop, a heavy flywheel spinning fast is hard to stop spinning.

The Two Faces of Angular Momentum

Angular momentum appears in two forms, depending on what you are studying.

For a single particle moving in a straight line or a curve, angular momentum is defined relative to a chosen point (usually the centre of rotation). It is the cross product of the position vector and the linear momentum:

Here $\mathbf{r}$ is the vector from the reference point to the particle, and $\mathbf{p} = m\mathbf{v}$ is its linear momentum. The magnitude is $L = r p \sin\theta$, where $\theta$ is the angle between $\mathbf{r}$ and $\mathbf{p}$. This tells you: the farther the particle is from the point, and the faster it moves perpendicular to that line, the greater its angular momentum.

For a rigid body rotating about a fixed axis, the formula simplifies beautifully. Every particle in the body contributes, and when you add them all up, you get:

where $I$ is the moment of inertia (the rotational analogue of mass) and $\omega$ is the angular velocity (how fast it spins). This is the direct parallel of $p = mv$.

Why Angular Momentum Matters

The real power of angular momentum is its conservation. In the absence of an external torque (the rotational analogue of force), angular momentum stays constant. This is why a figure skater spins faster when she pulls her arms in — her moment of inertia $I$ decreases, so $\omega$ must increase to keep $L$ constant.

This principle explains everything from why a bicycle stays upright to why neutron stars spin at incredible speeds after a supernova collapse.

Connecting the Two Definitions

The particle definition $\mathbf{L} = \mathbf{r} \times \mathbf{p}$ is the fundamental one. The rigid-body formula $L = I\omega$ is derived from it by summing over all particles in the body. For a single particle moving in a circle of radius $r$ with speed $v$, you get $L = rmv = mr^2\omega = I\omega$, since $I = mr^2$ for that particle.

So the two definitions are not separate — they are the same idea at different levels of description.

A Quick Check on Direction …