A chain having mass m and length l rests on the surface of a smooth sphere of radius R > l . One end of the chain is fastened to the top of the sphere. Determine the gravitational potential energy of the chain, taking the reference level at the centre of the sphere.

Step-by-step solution

Consider a small element of the chain of length dx subtending an angle d at the centre of the sphere, located at an angle from the vertical. The length of the element is dx = R d . The mass of the element is dm = m l dx = m l R d . The height of this element from the centre of the sphere is h = R . The gravitational potential energy of this element is dU = dm g h = ( m l R d ) g (R ) = m R^2 g l d . The total gravitational potential energy of the chain is obtained by integrating dU from = 0 to = l R . U = ₀^ l/R m