Two bodies of masses m and M are placed at distance d apart. The gravitational potential (V) at the position where the gravitational field due to them is zero V is

Options

A. `- (G)/(d) (M_(1) + M_(2) + 2 sqrt(M_(1)) sqrt(M_(2)) ) `
B. `- (G)/(d) (M_(1) + M_(2) - 2 sqrt(M_(1)) sqrt(M_(2)) ) `
C. `- (G)/(d) (2M_(1) + M_(2) + 2 sqrt(M_(1)) sqrt(M_(2)) ) `
D. `- (G)/(2d) (M_(1) + M_(2) + 2 sqrt(M_(1)) sqrt(M_(2)) ) `

Answer

A

Explanation

Let the gravitational field be zero at a point distant x from `M_(1)`.
`(GM_(1))/(x^(2)) = (GM_(2))/((d - x)^(2)) , (x)/(d - x) = sqrt((M_(1))/(M_(2)))`
`x sqrt(M_(2)) = sqrt(M_(1)) d - x sqrt(M_(1))`
`x [ sqrt(M_(1)) + sqrt(M_(2)) ] = sqrt(M_(1)) `d
`x = (d sqrt(M_(1)))/(sqrt(M_(1)) + sqrt(M_(2)) ) , d - x = (d sqrt(M_(2)))/(sqrt(M_(1)) + sqrt(M_(2)))`
Potential at this point due to both the masses will be `- (GM_(1))/(x) - (GM_(2))/((d-x))`
` = - G [ (M_(1) ( sqrt(M_(1)) + sqrt(M_(2))))/(d sqrt(M_(1))) + (M_(2)( sqrt(M_(1)) + sqrt(M_(2))))/(d sqrt(M_(2))) ] `
` = - (G)/(d) ( sqrt(M_(1)) + sqrt(M_(2)))^(2)`
`= - (G)/(d) (M_(1) + M_(2) + 2 sqrt(M_(1)) sqrt(M_(2)) ) `