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.

#### Solution

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:

`vecL_p = mv xx 0 + mv xx d`

= mvd ....(i)

Angular momentum of the system about point Q:

`vecL_Q = mv xx d + mv xx 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:

`vecL_R = mvxx(d-y) + mv xx y`

= mvd - mvy + mvy

=mvd ...(iii)

Comparing equation i, ii and iii we get

`vecL_p = vecL_Q = vecL_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