A thin uniform annular disc (see figure) of mass M has outer radius 4R and inner radius 3R. If the work required to take a unit mass from point P on its axis to infinity is 2GMnR(4√2−5), then n = (n is an integer)
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Solution
Surface mass density σ=Mπ(4R)2−π(3R)2=M7πR2
Consider an elemental ring of radius dr at a distance r from the centre of the disc.
Then, mass of the ring
dm=σ2πrdr
Gravitational potential due to the ring
Up=−∫4R3RGdm(1)r′
where r′ is the distance from the ring to the point P