Four equal masses (each of mass M) are placed at the corners of a square of side a. The escape velocity of a body from the centre O of the square is

Options

A. `4sqrt((GM)/(x))`
B. `sqrt((4sqrt(2)GM)/(x))`
C. `sqrt((8sqrt(2)GM)/(x))`
D. `sqrt((2GM)/(x))`

Answer

C

Explanation

`A_(O)=(1)/(2)(AC)=(sqrt(2)x)/(2)=(x)/(sqrt(2))`
Suppose that a mass m is kept at the centre O.
Potential at the centre O, due to all masses,
`v_(O)=4(-(GM)/(A_(O)))=-(4GM)/(x//sqrt(2))=-(4sqrt(2)GM)/(x)`
`therefore` P.E. of all the masses m `(v_(O))=-(4sqrt(2)GMm)/(x)`
`therefore B.E. = (+ 4 4sqrt(2)GMm)/(x)`
`therefore` If `v_(e)` is the escape velocity of mass 'm' then
`(1)/(2)mv_(e)^(2)=(4sqrt(2)GMm)/(x)`
`therefore v_(e)^(2)=(8sqrt(2)GM)/(x) " " therefore v_(e)=sqrt((8sqrt(2)GM)/(x))`