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.
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:
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.
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