A thin uniform annular disc (see figure) of mass $M$ has outer radius $4 R$ and inner radius $3 R$ . The work required to take a unit mass from point $P$ on its axis to infinity is

Question

A thin uniform annular disc (see figure) of mass M has outer radius 4 R and inner radius 3 R . The work required to take a unit mass from point P on its axis to infinity is

TestHub · Accepted Answer

W = Δ U ​ = U f ​ − U i ​ = U ∞ ​ − U p ​ = − U P ​ = − m V P ​ = − V P ​ ( as m = 1 ) ​ Potential at point P will be obtained by integration as given below Let d M be the mass of small ring as shown d M d V P ​ ∴ W ​ = π ( 4 R ) 2 − π ( 3 R ) 2 M ​ ( 2 π r ) d r = 7 R 2 2 M r d r ​ = − 16 R 2 + r 2 ​ G ⋅ d M ​ = − 7 R 2 2 GM ​ ∫ 3 R 4 R ​ 16 R 2 + r 2 ​ r ​ ⋅ d r = − 7 R 2 GM ​ ( 4 2 ​ − 5 ) = + 7 R 2 GM ​ ( 4 2 ​ − 5 ) ​ ∴ correct option is (a).

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