# What is the Gravitational potential energy of two masses?

I'm having a bit of trouble understanding the gravitational potential energy. Suppose I have two masses $$m$$ and $$M$$ rotating around one another. Theres the gravitational force between them $$F=-G\frac{mM}{r^2}$$ and there's the potential energy $$U=-G\frac{mM}{r}$$.

Is the potential energy here the energy of both masses? of only one of them and the other one has the exact same energy? What is the total energy and is it easier to express with effective energy/center of mass system?

What is the energy of each mass and what is the energy of the entire system?

Potential energy is said to be of the $$m-M$$ system since it is defined as the work done against the gravitational force to take the masses from $$+\infty$$ to a distance $$r$$. Since the masses tend to move towards each other gravitationally, that work is negative for an external being to move them close to each other slowly. The force between the masses is mutual by Newton's third law.

• so can I look at the energy in the system of only one of the masses? and the energy of each mass is $U$ and together they have $2U$? (I think I'm wrong saying that) Commented Jul 11, 2020 at 19:51
• As for potential energy, Nope! As for their mass energies$(E=mc^2)$, yes! Commented Jul 11, 2020 at 19:53
• Understood, what about the effective energy and COM energy? Commented Jul 11, 2020 at 19:54
• Energy and Mass are equivalent by the famous $E=mc^2$ equation. Both the masses have their mass energy. Commented Jul 11, 2020 at 19:56
• To answer your main question, the potential energy of the $m-M$ system is $U=-\frac{GMm}{r}$, not of only $m$ or $M$. Commented Jul 11, 2020 at 19:58

The potential energy is always defined of system.

For example when we say when you raise a block of mass m by hieght h it's potential energy increase. What we mean by that is that combined potential energy of earth and block system. Suppose if there was no earth and you raise the block by hieght h then the change in potential energy is 0.

As the energy is stored between mutual gravitational field and in the above case that is absent.

• so in the total energy even though we should include the kinteic energy (including rotational) energy of earth we say it's negligible? Commented Jul 11, 2020 at 20:07
• I am only talking about gravitational potential energy. For an observer outside the earth , the earth possess rotational and kinetic energy. Commented Jul 11, 2020 at 20:09
• so in a frame inside earth and outside earth the energy is different (in one earth does not move)? Commented Jul 11, 2020 at 20:12
• Yes. Kinetic energy is a frame dependent quantity as velocity is frame dependent. Commented Jul 11, 2020 at 20:13
• Could you also answer the second part and explain how is the effective potential energy $U_{eff}=\frac{L^2}{2mr^2}-\frac{GmM}{r}$ used here and why does it only include $m$ and not $M$ (and what is $L$ in the angular momentum here?) Commented Jul 11, 2020 at 20:13