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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 vector angular momentum of the two particle system is the same whatever be the point about which the angular momentum is taken.

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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:
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=dy
Angular momentum of the system about point R:
LR=mv×(dy)+mv×y=mvdmvy+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.


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