what is potential energy of a chain kept on hemispherical surface...

Question

i want ans in both case......

1...when chain is kept on convex surface means outside the bowl

2...when chain is kept on concave surface means inside the bowl

Answer 1

I'm going to assume that you mean gravitational potential energy, not tension or torsion. I will also assume that the chain has mass distribution m(x), 0 ≤ x ≤ L, where L = length of the chain, and that the chain is distributed symmetrically on a great circle running through the peak of the hemisphere. Say the hemisphere has radius R, where πR > L. Then the gravitational potential energy of the chain is

V = gR ∫ dφ sin φ m(R(φ - φ₀))

where the integral is over the interval [π(1 - L/πR), L/R] and φ₀ = π(1-L/πR). Draw a diagram with the chain placed symmetrically on top of a circle of radius R and this will make sense. You are just counting the gravitational contributions of each element of mass along the chain and adding them up.