A bead of mass `m` is attached to one end of a spring of natural length `R` and spring is fixed at point A on a smooth vertical ring of radius R as shown in figure. The normal reaction at B just after it is released to move is

Options

A. 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)`