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Here is the question:

A planet with mass $m$ and a second with mass $M$ are separated by a distance $d$. A third planet with mass $m_3$ happens to be midway between $M$ and $m$. Where could the third planet be positioned (distance from the larger planet $M$ in meters) so that the net gravitational force is zero?

My confusion lies with how to solve for the position of the third planet. I am given this equation to find the force between two planets at a given distance $r$: $$F = {GMm}/{r^2}$$ With this I can then set the sum of the two forces to zero: $$0 = {GMm_3}/r_1^2 + {Gmm_3}/r_2^2$$ My confusion lies in the fact that both $r_1$ and $r_2$ are unknown. However, we do know that $r_1 + r_2 = d$. But I am confused with how to solve for either $r_1$ or $r_2$.

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Thank you for a very well-written homework question :-) – David Zaslavsky♦ Jul 29 '11 at 6:29
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@user768900 , it might be that your homework question as it has been set is a little unclear. When it says "give the distance from the larger planet M in meters", it can't expect an answer that's a number, like 1 million, or something, if none of the values m,M,m3,d are given as absolute values. So express the answer as a function of those variables. – EnergyNumbers Jul 29 '11 at 9:22
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This is the same question as physics.stackexchange.com/questions/12751/… just camouflaged as "planets" instead of masses. That original version was closed as "too localized". – Georg Jul 29 '11 at 9:39
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If I'm interpreting the question correctly, in addition to the sign issue pointed out by Marek, the main issue is that you don't know how to do the algebra to solve these two equations for two unknowns. I don't want to be rude, but I think it's worth saying that, if that's the case, you're going to have continual troubles in any physics course. I'd recommend a systematic review of algebra before trying to take a physics course. I tried to supply some general algebraic hints in the previous iteration of this question. Did those make any sense at all? – Ted Bunn Jul 29 '11 at 14:43
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It is indeed puzzling that the previous question, which basically the same one, got closed and this one didn't. – Marek Jul 29 '11 at 15:00
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1 Answer

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Check the signs in your equation. Draw the three planets in a line with the central planet at the origin. What is the direction of the force each of the side planets exert on the middle one?

If you resolve the above problem you can approach solving the equations. Notice that you have as many equations as you have unknows. Therefore each equation lets you (at least in principle) express one unknown as the function of the others which reduces the problem to one with smaller number of unknows (although possibly with a more complicated equations).

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