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

Let P be the point where the field is zero and a unit mass is placed at P.
By Newton's law of gravitationa,
`(Gmxx1)/(x^(2))=(G4m xx1)/((r-x)^(2))`
`therefore (x^(2))/((r-x)^(2))=(1)/(4)`
or `(x)/(r-x)=(1)/(2) " " therefore 2x=r-x` or `x=(r )/(3)`
`therefore` The gravitational potential at P is
`V_(P)=-(Gm)/(x)-(G(4m))/(r-x)`
`=-(Gm)/((r )/(3))-(G(4m))/((r-(r )/(3)))`
`=-(3Gm)/(r )-(3G(4m))/(2r)=-(9Gm)/(r )`