### 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.

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.