question_answer

A bead of mass m is attached to one end of a spring of natural length R and spring constant\[k=\frac{(\sqrt{3}+1)mg}{R}\] The other end of the spring is fixed at a point A on a smooth vertical ring of radius R as shown in the figure. The normal reaction at B just after it is released to move is

A) \[mg/2\]           

B) \[\sqrt{3\,}mg\]

C) \[3\sqrt{3\,}mg\]           

D) \[\frac{3\sqrt{3}mg}{2}\]

Correct Answer: D

Solution :

[d] Extension in the spring is \[x=AB-R=2R\cos 30{}^\circ -R=(\sqrt{3}-1)R\] Spring force: \[F=kx=\frac{(\sqrt{3}+1)mg}{R}\times (\sqrt{3}-1)R=2mg\] From the figure we have \[N=(F+mg)cos30{}^\circ =\frac{3\sqrt{3}mg}{2}\]