Infinite number of masses, each of mass `m`, are placed along a straight line at distance of `r, 2r, 4r, 8r`, etc. from reference point `O`. Find the
(i) gravitational field intensity and
(ii) gravitational potential at point `O`.

Explanation

(i) The gravitational field intensity at point `O` will be
`I = Gm[(1)/(r^(2)) + (1)/((2r)^(2)) + (1)/((4r)^(2)) + (1)/((8r)^(2)) + ..]`
`= (Gm)/(r^(2)) [1 + (1)/(2^(2)) + (1)/(4^(2)) + (1)/(8^(2)) + ..]`
`= (Gm)/(r^(2)) [(1)/(2^(0)) + (1)/(2^(2)) + (1)/(2^(4)) + (1)/(2^(6)) + ...]`
`= (Gm)/(r^(2)) [(1)/(1 - (1)/(2^(2)))] = (Gm)/(r^(2)) [(1)/(1- (1)/(4))] = (4Gm)/(3r^(2))`
(ii) The gravitational potential at point `O` is
`V = - G [(1)/(r) + (1)/(2r) + (1)/(4r) + (1)/(8r) + ..]`
`= - (Gm)/(r) [1 + (1)/(2) + (1)/(4) + (1)/(8) + ..]`
`= - (Gm)/(r) [(1)/(2^(0)) + (1)/(2^(1)) + (1)/(2^(2)) + (1)/(2^(3)) + ..]`
`= - (Gm)/(r) [(1)/(1 - (1)/(2^(2)))] = - (2Gm)/(r)`