A Bead of mass `m` is attached to one end of a spring of natural length `'R'` and spring cosntant `'k=((sqrt3+1)mg)/R'`. The other end of the spring is fixed at point `'A'` on a smooth vertical ring of radius `'R'` as shown

Options

A. `mg//2`
B. `sqrt(3)mg`
C. `3sqrt(3)mg`
D. `(3sqrt(3)mg)/(2)`

Answer

D

Explanation

Extension in the spring is
`x=AB-R=2R cos 30^(@)-R=(sqrt(3)-1)R`
Spring force: `F=kx=((sqrt(3)+1)mg)/(R ) xx (sqrt(3)-1)R=2mg`
From the fig, we have
`N=(F+mg)cos 30^(@)= (3sqrt(3)mg)/(2)`