How Do You Calculate Gravitational Field Strength Between Two Planets?

[thereddevils]

Homework Statement

Write an expression for the gravitational field strength E at a point distance r from a planet of mass M. (In my diagram, there are two planets with mass, m and distance 2a apart)

Homework Equations

The Attempt at a Solution

The resultant gravitational field strength is towards the left. Using the parallelogram method to solve for the resultant vector and by the cosine rule,

$$|E|^2=(\frac{GM}{R^2})^2+(\frac{GM}{R^2})^2-2(\frac{GM}{R^2})(\frac{GM}{R^2})\cos 120$$

$$|E|=\frac{GM\sqrt{3}}{(r^2+a^2)^2}$$

But the answer given is

$$E=\frac{2GMr}{(a^2+r^2)^{\frac{3}{2}}}$$

 

[Mindscrape]

I don't know what a parallelogram method is. You know from symmetry that the y components will cancel, so all you have to do is look at the two x components, which will both add to give

2cosØ(GM)/(a^2+r^2)^(1/2), and cosØ=r/(r^2+a^2)

 

[thereddevils]

I don't know what a parallelogram method is. You know from symmetry that the y components will cancel, so all you have to do is look at the two x components, which will both add to give

2cosØ(GM/R^2)/(a^2+r^2)^(1/2), and cosØ=r/(r^2+a^2)

Thanks for your help, Mindscrape!