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. The normal reaction at 'B' just after the bead is released to move is : `(3 sqrt3 mg)/(3)`
B. The tangential acceleration of the bead just after it is released to move is : g/2
C. The normal reaction at 'B' just after the head is released to move is : `(3mg)/(2)`
D. Just after the bead is released to move the normal acceleration and tangential acceleration are numerically equal .

Answer

A, B

Explanation

Force developed in the spring due to elongation is :
`F_(s) = kx = k (2 R cos 30^(@) - R)`
=`kR (sqrt3 -1) = ((sqrt3 + 1) mg)/(R ) xx (sqrt( 3 -1)) = 2 mg `

Normal reaction N = `F_(s) cos 30^(@) + mg cos 30^(@)`
= ` 3 mg cos 30^(@) = (3sqrt3 mg)/(2)`
Tangential force `F_(T) = F_(s) sin 30^(@) - mg sin 30^(@)`
= ` mg sin 30^(@) = ma_(T)`
`a_(T) = (g)/(2)`