Given,
Mass of chain is m
Length is l
Radius of sphere is R
⇒h=Rsinθ
⇒dl=Rdθ
The mass of dl length of the chain dm=mdl
Potential energy.
dU=(dm)gh
⇒dU=mdllgh=mghdll=mgh2sinθdθl
⇒dU=mgR2sinθdθl
On Integrating
U=∫x2θmgR2sinθdθl
U=−mgR2l[cosθ]π2θ=−mgR2l[cos(π2)−cosθ]
U=−mgR2lcosθ=mgR2lsin(lR)
Hence, potential energy is mgR2lsin(lR)