A particle of mass m, moving in a circular path of radius R with a constant speed v2 is located at point (2 R, 0) at time t=0 and a man starts moving with a velocity v1 along the positive y-axis from origin at time t=0. Calculate the linear momentum of the particle w.r.t. man as a function of time.

Question

A particle of mass m, moving in a circular path of radius R with a constant speed v2 is located at point (2 R, 0) at time t=0 and a man starts moving with a velocity v1 along the positive y-axis from origin at time t=0. Calculate the linear momentum of the particle w.r.t. man as a function of time. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Angular speed of particle about centre of the circle

ω = \frac{v _{2}}{R}, θ = ω t = \frac{v _{2}}{R} t

v_{p} = (- v_{2} sin θ \hat{i} + v_{2} cos θ \hat{j})

or

v_{p} = (- v_{2} sin \frac{v _{2}}{R} t \hat{i} + v_{2} cos \frac{v _{2}}{R} t \hat{j})

and

v_{m} = v_{1} \hat{j}

∴

Linear momentum of particle w.r.t. man as a function of time is

L_{p m} = m (v_{p} - v_{m})

= m [(- v_{2} sin \frac{v _{2}}{R} t) \hat{i} + (v_{2} cos \frac{v _{2}}{R} t - v_{1}) \hat{j}]