Two bodies of masses m and 4m are placed at a distance r. The gravitational potential at a point on the line joining them where the gravitational field is zero is:

Options

A. zero
B. `-(4Gm)/r`
C. `-(6Gm)/r`
D. `-(9Gm)/r`

Answer

D

Explanation

At `P`:
Gravitational Field due to `m,E_(1)=(Gm)/(x^(2))`, along `A`
Gravitational Field due to `4m, E_(2)=(G.4m)/((r-x)^(2))`, along `B`
`E_(p)=0 implies E_(1)=E_(2)`
`(Gm)/(x^(2))=(4Gm)/((r-x)^(2)) implies ((r-x)/x)^(2)=4`
`(r-x)/x=2 implies r-x=2x implies x=r//3`
Gravitational, potential at `P`
Due to `m,V_(1)=-(Gm)/x=-(Gm)/(r//3)=-(3Gm)/r`
Due to `4m,V_(2)=-(G.4m)/(r-x)=-(4Gm)/(2r//3)=-(6Gm)/r`
`V_(p)=V_(1)+V_(2)=-(9Gm)/r`