Initially p is at rest relative to the wire at a distance a

Question

A particle P of mass m can slide along a smooth rigid straight wire. The wire has one of its points fixed at the origin O, and is made to rotate in the (x, y)-plane with angular speed . By using the vector equation of motion of P in polar co-ordinates, show that r, the distance of P from O, satisfies the equation

r^·· - π²r = 0,

and find a second equation involving N, where Nθ^{^} is the force the wire exerts on P. [Ignore gravity in this question.] Initially, P is at rest (relative to the wire) at a distance a from O. Find r as a function of t in the subsequent motion, and deduce the corresponding formula for N.