A smooth semicircular wire-track of radius R is fixed in a vertical plane. One end of a massless spring of natural length `3R//4` is attached to the lowest point O of the wire-track. A small ring of mass m, which can slide on the track, is attached to the other end of the spring. The ring is held staionary at point P such that the spring makes an angle of `60^@` with the vertical. The spring constant `K=mg//R`. Consider the instant when the ring is released, and (i) draw the free body diagram of the ring, (ii) determine the tangential acceleration of the ring and the normal reaction.

Explanation

The forces acting on the ring are
(a) weight of the ring
(b) force in the spring
(c ) Normal reaction passing through center `C`
Extension of spring `x=R-(3R)/(4)=(R )/(4)`
Since the ring starts from rest,
`a_(c)=0`
Along radial direction:
`N+kx sin 30^(@)-mg cos 60^(@)=ma_(c)=0`
`N+(mg)/(R )xx(R )/(4)xx(1)/(2)-mgxx(1)/(2)=0`
`N=(3mg)/(8)`
Along tangential direction : `kx cos 30^(@)+mg cos 30^(@)=ma_(t)`
`(mg)/(R )xx(R )/(4)xxsqrt(3)/(3)=ma_(t)`
`a_(t)=(Rsqrt(3)g)/(8)`