Question

. Two large spherical planets of with masses of M and 2M are positioned a distance D apart (measured from the center to center distance) as shown in adjacent Fig. A small asteroid with mass m is located with its center exactly halfway between the two large planets. Find the magnitude and direction of the total gravitational force acting on the asteroid m.

Answer 1

According to the Newton's gravity law, the force acting on the mass $m$ from the mass $M$ is diracted toward the mass $M$ and given by the equation:

$$F_{M} = G\dfrac{Mm}{(D/2)^2}=4G\dfrac{Mm}{(D)^2}$$

The force acting on the mass m the mass 2M is diracted toward the mass 2M and given by the equation:

$$F_{2M} = G\dfrac{2Mm}{(D/2)^2}=8G\dfrac{Mm}{(D)^2}$$

As far as the forces are directed in the opposite directions, the resulting force will be:

$$F = F_{2M}-F_M = 4G\dfrac{Mm}{D^2}$$

Answer. $F = 4G\dfrac{Mm}{D^2}$, directed toward the $2M$.