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Question

A chain of length l and mass m lies of the surface of a smooth hemisphere of radius R>1 with one end tied to the top of the hemisphere. Taking base of the hemisphere as reference line, find the gravitational potential energy of the chain.
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Solution

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)


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