What is mathematical equation of the Gravitational force between three objects?

Question

This is mathematical equation of the Gravitational force between two objects: $$F_G=\frac{GM_1M_2}{r^2}$$

what is mathematical equation of the Gravitational force between three objects?

Answer 1

The gravitational force due to two objects, $M_1, M_2$ on another object, $M_3$, is the vector sum of the forces due to $M_1$ and $M_2$ individually on $M_3$, i.e. $$\vec{F}_{\text{net}} = \vec{F}_{1,3} + \vec{F}_{2,3}$$ where $|\vec F_{1,3}| = \frac{GM_1M_3}{r_{1,3}^2}$, and $|\vec F_{2,3}| = \frac{GM_2M_3}{r_{2,3}^2}$.

This is generally not (I repeat : NOT) the same as the sum of the magnitude of the forces, which is otherwise given by: $$\frac{GM_1M_3}{r_{1,3}^2} + \frac{GM_2M_3}{r_{2,3}^2}$$

A simple illustration is to put three equal masses on the vertices of an equilateral triangle, whose base is parallel to the ground: You will find that the net gravitational force on the top-most mass due to the other two masses is upwards.

If you want a more explicit formula, you will need information of the direction of the individual forces