A point mass `m` is a distance `x` from the centre of mass `M` and radius `R` on its axis. Find the gravitational force between the two. What will this force be if `x gt gt R` and `x lt lt R`? For what value of `x` is the force maximum?

Explanation

Gravitational field due to ring at `P`
`E=(GMx)/((R^(2)+x^(2))^(3//2))`
Force on `m`
`F=mE=(GMmx)/((R^(2)+x^(2))^(3//2))`
If `x lt lt R,F=(GMmx)/(R^(3))`
`x gt gt R,F=(GMmx)/((R^(2)+x^(2))^(3//2))=(GMmx)/(x^(3)(1+(R^(3))/(x^(2)))^(3//2))`
`=(GMm)/(x^(2))`
`F=(GMmx)/((R^(2)+x^(2))^(3//2))`
For `F` to be maximum,
`(dF)/(dx)=0implies (R^(2)+x^(2))^(3//2).1-x.3/2(R^(2)+x^(2))^(1//2).2x=0`
`(R^(2)+x^(2))-3x^(2)=0`
`x+-R/(sqrt(2))`
`F_(max)=(GMm(R//sqrt(2)))/((R^(2)+x^(2))^(3//2))=(GMm(R//sqrt(2)))/((3R^(2))/2.(sqrt(3)R)/(sqrt(2)))=(2FMm)/(3sqrt(3)R^(2))`