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 angular momentum vector 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:

$\^L_p$ = mv × 0 + mv × d

$= mvd ....(i)$

Angular momentum of the system about point Q :

$\^L_Q→ = mv × d + mv × 0$

$= mvd ....(ii)$

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 R :

$\^L_R = mv × (d - y) + mv × y$

$= mvd - mvy + mvy$

$= mvd ...(iii)$

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

$\^L_P = \^L_Q = \^L_R ...(iv)$

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