A chain of length l and mass m lies of the surface of a smooth hemisphere of radius `Rgtl` with one end tied to the top of the hemisphere. Taking base of the hemisphere as reference line, find the gravitational potential energy of the chain.

Explanation

Consider a small element of chain of width `dtheta` at an angle `theta` from the vertical
The mass of the element `dm = (m/l)Rdtheta`
The gravitational potential energy of the element `du = (dm)gy`
The gravitational potential energy of total chain
`U = int _(0)^(1/R)(dm)gy = int_(0)^(1/R)((m)/(l)Rd theta)g(R cos theta)`
` U = (mgR^(2))/l[sin theta]_(0)^(1/R) = (mgR^(2))/l sin (l/R)`