Particle Motion in a Polar Groove: Solving for R, v, and θ as Functions of Time

[iitjee10]

Homework Statement

A particle follows a trajectory given as R = Aθ, where θ is the polar angle.in a horizontal plane. The trajectory is such that the walls are vertical and the particle moves in a groove made by them. The particle remains in contact with both the walls throughout its motion. The side walls are smooth but the horizontal surface on which it moves is rough with coefficient of friction = μ. The particle was initially at R=A and given an initial velocity v_o in the direction of the trajectory at that point. Find:

(a) R as a function of time
(b) velocity of particle v as a function of time
(c) θ as a function of time

Homework Equations

general polar coordinate equations of motion
Newtons laws
work energy theorem

The Attempt at a Solution

i tried to do the question by using polar coordinates. However I realized that the normal reaction was not in the direction of r(unit vector). and consequently, v(vec) was not in tangential direction. Then I tried to use work energy theorem as only friction was doing work. but that lead to an ugly integral. and further in that integral kmg.dr was quite ugly.

 

[jbsteele]

On the other hand, I tried to use newtons second law, but there also the normal reaction was not in direction of unit vector. and this became a dead end for me. Please help me in solving the question.