Net gravity problem ... how to solve?

Question

So far, I have:

$$\dfrac{G \times(2\cdot10^{-3})^2}{\sqrt{??}} $$ ???

What's next?

Answer 1

First of all, the equation you're using is incorrect. Please refer to the correct version.

Second, you have realized that the gravitational force of $m_1$ on $m_5$ and $m_4$ on $m_5$ cancel each other out, because the mass of $m_1$ and $m_4$ are the same, and so are their distance from $m_5$. This observation speeds things up.

Lastly, to lead you in the correct direction, the magnitude of the gravitational force of $m_2$ on $m_5$ is $$ G\cdot\frac{m_2 m_5}{d^2}, $$ where $d$ is the distance from $m_2$ to $m_5$, and the direction is to the north east.

What about the gravitational force of $m_3$ on $m_5$? What is its magnitude and direction? What is the net effect of these two gravitational forces? What about the net effect of all four gravitational forces on $m_5$?

If you can figure out these issues step by step, you will have learnt something.