A particle describes a horizontal circle of radius r in a funnel type vessel of frictionless surface with half one angle `theta` (as shown in figure). If mass of the particle is m, then in dynamical equilibrium the speed of the particle must be -

Explanation

The normal reaction N and weight mg are the only forces acting on the particle (inertial frame view),
the N is making an angle `((pi)/(2) - theta)` with the vertical.
The vertical component of N balances the weight mg and the horizontal component provides the centripetal force required for circular motion. Thus
N cos `((pi)/(2) - theta)` = mg
N sin `((pi)/(2) - theta) = (mv^(2))/(r )`
or N sin `theta` = mg.
N cos `theta = (mv^(2))/(r )`,
on dividing we get tan `theta = (rg)/(v^(2))`,
so v = `sqrt((rg)/(tan theta))`