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A chain of length l and mass m lies on t...

A chain of length l and mass m lies on the surface of a smooth sphere of radius R>l with one end tied to the top of the sphere.

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The mass is distributed in chain uniformly along its length.
Choose a small element of chain of width `dtheta` at an angle `theta` from the vertical.
The mass of the element, `dm=(m/lRd theta)`
The gravitational potential energy of the element `dU=(dm)gy`
Thus, the gravitational potential energy of whole chain
`U=int(dm)gy`
`=underset0overset((l//R))int(m/lRd theta)g(Rcostheta)`
`=(mR^2g)/(l)underset0overset((l/R))intcosthetad theta`
`=(mgR^2)/(l)|sintheta|_0^(l//R)=(mgR^2)/(l)sin(l/R)`
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