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.

Angular momentum of the system about point P:
${\overrightarrow{L}}_P=mv\times 0+mv\times d=mvd\dots \left(i\right)$
Angular momentum of the system about point Q:
${\overrightarrow{L}}_Q=mv\times d+mv\times 0=mvd\dots \left(ii\right)$
Consider a point R, which is at a distance y from point Q, i.e.,
$QR=y\therefore PR=d-y$
Angular momentum of the system about point R:
${\overrightarrow{L}}_R=mv\times (d-y)+mv\times y=mvd-mvy+mvy=mvd\dots \left(iii\right)$
Comparing equations (i), (ii), and (iii), we get:
${\overrightarrow{L}}_P={\overrightarrow{L}}_Q={\overrightarrow{L}}_R\dots \left(iv\right)$
We infer from equation (iv) that the angular momentum of a system does not depend on the point about which it is taken.