Question

prove that if two particles are moving along parallel directions with equal and opposite momentum, their total angular momentum about any point remains constant

Answer 1

Let at a certain instant two particles be at points P and Q, as shown in the following figure.(given below)

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,

QR = y

PR = d – y

Angular momentum of the system about point R:

LR = mv × (d – y) + mv × y

mvd – mvy + mvy

= mvd  ….(iii)

From (i), (ii), and (iii), we get:

LP = LQ = LR    …(iv)

We infer from equation (iv) that the angular momentum of the particles Is constant.