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.
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:
→LP=mv×0+mv×d=mvd …(i)
Angular momentum of the system about point Q:
→LQ=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:
→LR=mv×(d−y)+mv×y=mvd−mvy+mvy=mvd …(iii)
Comparing equations (i), (ii), and (iii), we get:
→LP=→LQ=→LR …(iv)
We infer from equation (iv) that the angular momentum of a system does not depend on the point about which it is taken.