Two bodies of masses `M_(1)` and `M_(2)`, are placed at a distance `R` apart. Then at the position where the gravitational field due to them is zero, the gravitational potential is

Options

A. `-G(sqrtM_(1))/R`
B. `-G(sqrtM_(2))/R`
C. `-(sqrtM_(1)+sqrtM_(2))^(2)G/R`
D. `-(sqrtM_(1)-sqrtM_(2))^(2)G/R`

Answer

C

Explanation

`(GM_(1))/(x^(2))=(GM_(2))/((R-x)^(2))`
`((M_(2))/(M_(1)))x^(2)=R^(2)+x^(2)-2Rx`
Let `(M_(2))/M=k`
`x^(k-1)+2Rx-R^(2)=0`
`x=-(2R+sqrt(4R^(2)+4(k-1)R^(2)))/(2(k-1))=(Rsqrt(M_(1)))/(sqrt(M_(1))+sqrtM_(2))`
`R-x=(Rsqrt(M_(2)))/(sqrt(M_(1))+sqrtM_(2))`
Gravitational potential at point `P` is
`-((GM_(1))/x+(GM_(2))/(R-x))`
`=-[(GM_(1)(sqrt(M_(1))+sqrt(M_(2))))/(Rsqrt(M_(1)))+(GM_(2)(sqrt(M_(1))+sqrt(M_(2))))/(Rsqrt(M_(2)))]`
`=-[(G(sqrt(M_(2))+sqrt(M_(1))))/R(sqrtM_(1)+sqrtM_(2))]`
`=-(G(sqrtM_(1)+sqrtM_(2))^(2))/R`