Animated Solution for Physics - System of Particles and Rotational Motion: A particle of mass $m$ moves along line $PC$ with velocity $v$ as shown. What is the angular momentum of the particle about $O$?

Our mission is to determine its angular momentum about the origin $O$. At first glance, you might wonder: "If it's moving in a straight line, how can it have angular momentum?"

Well, angular momentum isn't just for spinning wheels or orbiting planets. It's a measure of the "turning effect" of a particle's linear momentum with respect to a specific reference point. As long as the particle's path doesn't pass directly through the origin, it possesses angular momentum relative to it.

The fundamental definition of angular momentum $\vec{L}$ about a point is given by the cross product of the position vector $\vec{r}$ and the linear momentum $\vec{p}$:

Since linear momentum is simply mass times velocity ($\vec{p}=m\vec{v}$), we can rewrite this as:

To find the magnitude of this cross product, we multiply the magnitudes of the vectors and the sine of the angle between them. Let $\theta$ be the angle between the position vector $\vec{r}$ and the velocity vector $\vec{v}$. The magnitude becomes:

Drop a perpendicular from the origin $O$ to the line of motion $PC$. Let the length of this perpendicular be $l$.

This creates a beautiful right-angled triangle formed by the origin $O$, the particle's position $M$, and the foot of the perpendicular $Q$. In this triangle, the hypotenuse is the distance $r$. By basic trigonometry, the side opposite to the angle $\theta$ is exactly the perpendicular distance $l$. Therefore, we can write:

Rearranging this gives us a powerful geometric substitution:

We group the terms to reveal the magic:

Replacing $r\sin \theta$ with $l$, we get our final, elegant result:

Notice the profound implication here: As the particle moves along the straight line, its distance $r$ from the origin changes, and the angle $\theta$ changes. However, the product $r\sin \theta$ is always equal to the perpendicular distance $l$, which is a constant!

Thus, the angular momentum of a particle moving in a straight line with constant velocity is absolutely constant. It depends only on its mass, its speed, and the perpendicular distance from the reference point to its line of motion.