# If you have three masses in a line, determine the position of the middle masses given that the net force of the gravitational force is 0 [closed]

I have been working on this for some time now. So you have three masses m, M, and m3 (these values are known). The distance between m and M is d (this is known). m3 lies between m and M. What I am trying to figure out is the position of m3. I have defined the distance between m and m3 as r1 and the distance between m3 and M as r2. r1 = d - r2. So I have come up with this (using Newton's law of universal gravitation):

$$F = \frac{GMm_3}{r_2} - \frac{Gmm_3}{d-r_2}.$$

So I need to solve for r2 in order to find the position. Is this correct? How do I solve for r2?

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You're on the right track, but your expression for the gravitational force isn't quite right. Go back and look up the formula for gravitational force in your textbook, and try to find the difference from what you wrote. (Or if you wait a while, someone else will probably tell you. I'm a crotchety old man, so I make people track these things down for themselves.) – Ted Bunn Jul 25 '11 at 19:14
Once you've got the correct equation, you will need to set $F$ equal to zero and do some algebra to solve for $r_2$. Suggestion: do some sort of mathematical operation to both sides of the equation (doing the same thing to both sides, of course, so that the two sides remain equal), that will make it so that the unknown $r_2$ is no longer in the denominator. Once you've done that, see if you can figure out how to isolate $r_2$ on one side of the equation. – Ted Bunn Jul 25 '11 at 19:14
Ted, you are correct. I neglected to square the distance. Shouldn't the formula be: $$F = \frac{GMm_3}{(r_2)^2} - \frac{Gmm_3}{(d-r_2)^2}.$$ – justspamjustin Jul 25 '11 at 19:42
Almost there...you have a slight error in your formula. For a hint, check your denominators. Once you've figured that out, it'll just be some high school algebra, – MGZero Jul 25 '11 at 19:43
MGZero, was I correct in my above comment? – justspamjustin Jul 25 '11 at 19:45