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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`
By defination, the work required to take a unit mass from `P` to infinity =`-V_(P)` where `V_(P)` is the gravitational potential at `P` due to the disc. To find `V_(P)`, we divide the disc into small elements, each of thickness dr. consider one such element at a distance `r` from the centre of the disc, as shown in the figure.
mass of the elements `dm=(M(2pirdr))/(pi(4R)^(2)-pi(3R)^(2))`
`=(2Mrdr)/(7R^(2))`

`V_(P)=-int_(3R)^(4P)(Gdm)/(sqrt(r^(2)+16R^(2)))=(2MG)/(7R^(2))int_(3R)^(4R)(rdr)/((r^(2)+16R^(2))^(1//2))`
Putting `r^(2)+16R^(2)=x^(2)`, we get `2rdr=2xdx` or `rdr=xdx`.
when `r=3R,x=sqrt(9R^(2)+16R^(2))=5R`
when `r=4R,x=sqrt(16R^(2)+16R^(2))=4sqrt(2)R` ,brgt `:. V_(P)=-(2MG)/(7R)(4sqrt(2)-5)` which is choice `(A)`
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