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A thin uniform disc (see figure) of mas...

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

A

`(2GM)/(7R)(4sqrt(2)-5)`

B

`-(2GM)/(7R)(4sqrt(2)-5)`

C

`(GM)/(4R)`

D

`(2GM)/(5R)(sqrt(2)-1)`

Text Solution

Verified by Experts

The correct Answer is:
A
Area `=2pixdx`
`dm=(Mxx2pixdx)/(pi((4R)^(2))-pi(3R)^(2))`
`=(2M)/(7R^(2))xdx`
Potential `dV_(p)=(-Gdm)/(sqrt((4R)^(2)+x^(2)))=(-G)/(sqrt(16R^(2)+x^(2)))xx(2M)/(7R^(2))xdx`
`dV_(p)=(-2GM)/(7R^(2))=(pidx)/(sqrt(16R^(2)+x^(2)))`
Put `16R^(2)+x^(2)=t^(2)implies2xdx=2tdt`
`x=3R, t=5R`
`x=4R, t=4sqrt(2)R`
`V_(p)=(2GM)/(7R^(2))int_(5R)^(4sqrt(2)R)tdt`
`implies V_(p)=(-2GM)/(7R^(2))[t]_(5R)^(4sqrt(2)R)`
`V_(p)=(-2GM)/(7R^(2))[4sqrt(2)R-5R]`
`W=(V_(oo)-V_(p))=0-[(-2GM)/(7R^(2))(4sqrt(2)R-5R)]`
`W=(2GM)/(7R^(2))[4sqrt(2)R-5R]`
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