A chain having mass m and length l rests on the surface of a smooth...

Consider a small element of the chain of length $dx$ subtending an angle $d\theta$ at the centre of the sphere, located at an angle $\theta$ from the vertical.

The length of the element is $dx = R d\theta$.

The mass of the element is $dm = \dfrac{m}{l} dx = \dfrac{m}{l} R d\theta$.

The height of this element from the centre of the sphere is $h = R \cos\theta$.

The gravitational potential energy of this element is $dU = dm g h = \left(\dfrac{m}{l} R d\theta\right) g (R \cos\theta) = \dfrac{m R^2 g}{l} \cos\theta d\theta$.

The total gravitational potential energy of the chain is obtained by integrating $dU$ from $\theta = 0$ to $\theta = \dfrac{l}{R}$.

$U = \int_{0}^{l/R} \dfrac{m R^2 g}{l} \cos\theta d\theta$

$U = \dfrac{m R^2 g}{l} [\sin\theta]_{0}^{l/R}$

$U = \dfrac{m R^2 g}{l} \sin$\left(\dfrac{l}{R}\right)