Question

Two particles, each of mass $m$ and speed $v$, travel in opposite directions along parallel lines separated by a distance $d$. Show that the vector angular momentum of the two particle system is the same whatever be the point about which the angular momentum is taken.

Answer 1

Let at a certain instant two particles be at points P and Q, as shown in the following figure.

Consider a point R, which is at a distance $y$ from point Q, i.e.,

$QR=y$
∴ $PR=d-y$

Angular momentum of the system about point P:

$L_P=mv\times 0+mv\times d=mvd......\left(i\right)$

Angular momentum of the system about point Q:

$L_Q=mv\times d+mv\times 0=mvd....\left(ii\right)$

Angular momentum of the system about point R:

$L_R=mv\times (d-y)+mv\times y=mvd....\left(iii\right)$

Comparing equations (i), (ii), and (iii), we get:

$L_P=L_Q=L_R......\left(iv\right)$

We infer from equation $\left(iv\right)$ that the angular momentum of a system does not depend on the point about which it is taken.