particle dynamics, circular motion?

Question

A particle of mass m can move freely on a horizontal frictionless surface. It is connected by a spring to a fixed point which can pivot freely; the spring is of natural length L and spring constant k. If the particle undergoes circular motion given by polar coordinates:-

R = P0 and theta = wt

where P0 and w are constants.

Q1. write down the equation of motion of the partical and find the equilibrium radial coordinate P0 in terms of m,k and w.

Q2. If the particle is made to undergo small radial perturbations to this equilibrium: R = P0 + z with theta = wt, derive an equation for the time dependence of z and hence show that if k >w^2, the particle unergoes radial oscillations and find their period.

I really don't know where to start with these questions, I understand the circular motion but don't know how the spring affects it.

Answer 1

Q1:
P0 is the distance from the center of rotation to the particle.
The extension of the spring will be P0-L
There is a radial force on the particle that will act against the spring such that
(P0-L)*k=m*P0*w^2
P0*(k+m*w^2)-L*k=0
P0=L*k/(k+m*w^2)

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Answer 2

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