The base of a hollow right cone of semi vertical angle 30° is fixed to a horizontal plane. Two particle each of mass m are connected by a light inextensible string hich asses through a small hole in the vertex of the cone. One particle A hangs at rest inside the cone. The other particle B moves on the outer smooth surface of the one at a distance  l from vertex in a horizontal circle with centre at A. Neglecting friction, now answer the following.

A is at rest on the axis at the same height as B, so its distance from the vertex is

VA=lcos30∘=23​​l.

For A: vertical string ⇒ $T=mg$.

For B on the smooth outer cone (semi-vertical angle $30^\circ$): forces are weight $mg$ (down), tension $T$ along the generator, and normal $N$ normal to the surface.

Vertical balance (no vertical acceleration):

Tcos30∘+Ncos60∘=mg⇒N=2mg−mg3​=mg(2−3​).

Radial (toward the axis provides centripetal force; radius $r=l\sin30^\circ=l/2$):

$$\Rightarrow mg\left(\tfrac12\right)+mg(2-\sqrt3)\left(\tfrac{\sqrt3}{2}\right) = \frac{mv^2}{r} \Rightarrow \frac{v^2}{r}=g(\sqrt3-1).$$