Solve the following :
Question:

A chain of length 1 and mass $m$ lies on the surface of a smooth sphere of radius $R>1$ with one end tied to the top of the sphere. (a) Find the gravitational potential energy of the chain with reference level A chain of length 1 and mass $m$ lies on the surface of a smooth sphere of radius $R>1$ with one end tied to the top of the sphere. (a) Find the gravitational potential energy of the chain with reference level

Solution:

(a) G.P.E $=d m g R \cos \theta$

$=(\mathrm{mgR} \cos \theta \mathrm{d} \Theta) / \mathrm{l}[\mathrm{dm}=\mathrm{m} / \mathrm{l} \mathrm{R} \mathrm{d} \Theta]$

Total G.P.E $={ }_{0}^{\int 1 / r}\left(\mathrm{mgR}^{2}\right) / / \cos \theta \mathrm{d} \Theta[\Theta=\mathrm{l} / \mathrm{R}]$

G.P.E $=\left(m g R^{2}\right) / / \sin (I / R)$

(b) K.E. = change in P.E

K.E. $=\left(m g R^{2}\right) / l[\sin (l / R)+\sin \theta-\sin (\theta+l / R)]$

(c) From above equation

K.E. $=\left(m g R^{2}\right) / I[\sin (I / R)+\sin \theta-\sin (\Theta+l / R)]$

as $\theta=0$

$\frac{d v}{d t}=(g R / I)[1-\cos (l / R)]$