Q.7:- 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:-

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

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.