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Find the point between Mars and the Sun at which an object can be placed so that the net gravitational force exerted by Mars and the Sun on this object is zero.

We are given a chart with information on the mass, etc. of planetary objects. I know to use law of universal gravitation but how do I manipulate the equations and what am I solving for?

At some arbitrary point inbetween Mars and the Sun the force in the direction of the sun is:

M_{sun} m G/r^2 -
M_{mars} m G/(R_{mars} - r)^2

Here r is the distance from the object to the sun and R_{mars} is the distance from the Sun to Mars, and m is the mass of the object. If you equate this to zero you get:

M_{sun}/r^2 = M_{mars}/(R_{mars} - r)^2

Take the reciprocal of both sides, and you have a quadratic equation.

You must then solve for r...

I am having trouble getting the right equation. After I take the reciprocal he numerator and denominator should be flipped, correct? Than do I multiply both sides by the masses?

That's right:

M_{sun}/r^2 = M_{mars}/(R_{mars} - r)^2

becomes:

r^2/M_{sun} = (R_{mars} - r)^2 /M_{mars}

and then you get:

r^2 =(M_{sun}/M_{mars})*(R_{mars} - r)^2

Take the squate root of bith sides:

r=sqrt[M_{sun}/M_{mars}]})*(R_{mars}-r)

This is a linear equation for r.

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