Finding Equilibrium: Solving for Net Gravitational Force on a Mass

[moaath]

??how to get there

I tried to solve this question its from old exam and the answer is posted down but I didn't know how to get there please explain for me how we solve these kind of question??

two particles with masses M and 4*M are separated by distance D what is the distance from the mass M for which the net gravitational force on a mass m is zero.

the answer is (D/3)

 

[hokie1]

What you want to do is to place an object of mass m between M and 4M so that the same force is applied by the two masses,but in opposite directions.

F=G m1 m2/r^2 is the force formula

notice that m=m1 and G cancel out
Place the mass a distance x from M, then the mass is a distance (D-x) from 4M

This leaves:
M/x^2 = 4M/(D-x)^2

Cancel the Ms, multiply and expand

3x^2+2Dx-D^2 = 0
use quadratic formula to get the solution for x.