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Question

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(425), then n = (n is an integer)

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Solution

Surface mass density σ=Mπ(4R)2π(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

=4R3RGdm(1)(16R2+r2)1/2
=4R3R2πGσrdr(16R2+r2)1/2

After solving Up=2GM7R2(r2+16R2

After applying limits,
Up=2GM7R(425)

Work done by external agent =UUP
=2GM(425)7R

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