Where Is the Gravity Crossover Point Between Earth and the Moon?

[AhmadG]

Homework Statement

The problem asks to find the crossover point between the Earth and the moon. It will be the point where the net force due to gravity from both sides will equate to 0.
Me=mass of earth
Mm=mass of the moon
d=distance between Earth and moon

Homework Equations

Gmm/r^2

The Attempt at a Solution

Me/x^2=Mm/(d-x)^2

I got to the answer, but it is a negative number, and it is of the same degree as the answer in the back of the book, meaning I'm doing something wrong. I can't seem to work it out. I want to know whether (d-x) has to be negative, because the forces are in the opposite direction. I will post back with a proper detailed attempt at the solution once I get more time, I have to go to class. In the meantime, if someone could set up the initial equation with the correct negative signs, that would be great, I'll learn what I was doing wrong. I'm a strong physics student overall, this is just ticking me off lol. I'm bad with algebra :P.

 

[member 553083]

The correct equation should look like this: Me/x^2 = Mm/(d-x)^2This equation is used to calculate the crossover point between the Earth and the moon, where the net force due to gravity from both sides equals 0. The equation can be rearranged to solve for x, the distance from the Earth to the crossover point:x^2 = Me/(Mm/(d-x)^2)Solving for x yields:x = ±√(MeMm/d^2) where the negative sign corresponds to the crossover point being on the side of the moon, and the positive sign corresponds to the crossover point being on the side of the Earth.