# What is the force of gravity on two particles? [closed]

Suppose that I have a system of two particles. They have the same mass $$m$$. At the origin O is something with a very large mass $$M$$ which is attracting the two particles. The first particle has distance 1 to O. The second particle has distance 2 to O. Now, what is the total gravitational force on the system?

If I first calculate the two forces separately, I see that the force on particle 1 is $$\frac{C}{1^2}$$, where $$C=GMm$$ is a constant. The force on particle 2 is $$\frac{C}{2^2}$$. Therefore, the total force is $$\frac54 C$$.

However, we could also calculate the force on the entire system using the center of mass. The center of mass is at distance $$\frac32$$ from $$O$$. The total mass of the system is $$2m$$. Therefore, the total force is $$\frac{2C}{(\frac32)^2}=\frac89 C$$.

So is the total force equal to $$\frac54 C$$ or $$\frac89 C$$?

Update: the origin and the two particles are on a straight line. Also, to view the two particles as one ‘system’, we can say that the two particles are attached to the ends of a massless rope with a length larger than 1. Finally, the force between the particles is negligible.

• "total gravitational force on the system" What does that even mean? You should calculate the gravitational forces between each pair of objects. 2 days ago
• What should I clarify for the question to reopen? 16 hours ago

• The center of gravity of the system would be $\left(x_1 m_1 g(x_1) + x_2 m_2 g(x_2)\right)/\left(m_1 g(x_1) + m_2 g(x_2)\right) = \left(1 \frac{1}{1^2} + 2 \frac{1}{2^2}\right)/\left(\frac{1}{1^2} + \frac{1}{2^2}\right) = \frac{6}{5}$. And the force of gravity on an object of mass $2m$ located there would be $\frac{25}{18}C$. But I'm not sure what that tells you. I think the only answer is just the net (external) gravitational force on the system of smaller masses: $\frac{5}{4}C$. 2 days ago
• @Ben H, I think that your formula for the center of gravity is incorrect, because it should give $\frac54C$ for the total force 2 days ago