# Shared Potential Energy

Given two point masses, ##m_1## and ##m_2##, we define the gravitational potential energy of this system as:

$$U_{grav} = -G \frac{m_1m_2}{r}$$

Where ##r## is the separation between ##m_1## and ##m_2##.

When we analyze the motion of a single component, say ##m_1## in this system, we usually say things like:

The potential energy of ##m_1## is:

$$U_{grav} = -G \frac{m_1m_2}{r}$$

This is where my intuition fails. As dumb as this may sound, why isn't potential energy shared in some ratio between ##m_1## and ##m_2##?

Nugatory
Mentor
This is where my intuition fails. As dumb as this may sound, why isn't potential energy shared in some ratio between ##m_1## and ##m_2##?
When we're analyzing the problem in terms of the motion of only one of the two bodies, we are making an assumption that mass of the other body is so great that it is effectively not moving at all. That works just fine for objects moving around in Earth's gravitational field (where you probably first saw this treatment of potential energy), planets orbiting the sun, and the like.

When we're analyzing the problem in terms of the motion of only one of the two bodies, we are making an assumption that mass of the other body is so great that it is effectively not moving at all. That works just fine for objects moving around in Earth's gravitational field (where you probably first saw this treatment of potential energy), planets orbiting the sun, and the like.
What if the masses of the two bodies were similar? How would our analysis differ in that case?

Nugatory
Mentor
What if the masses of the two bodies were similar? How would our analysis differ in that case?
The problem becomes appreciably harder, but you can choose coordinates in which the center of mass of the two bodies is at rest and both objects are in motion and you'll get sensible results.

When we're analyzing the problem in terms of the motion of only one of the two bodies, we are making an assumption that mass of the other body is so great that it is effectively not moving at all. That works just fine for objects moving around in Earth's gravitational field (where you probably first saw this treatment of potential energy), planets orbiting the sun, and the like.
So the approximation is that we consider one mass to be stationary, correct?

Nugatory
Mentor
So the approximation is that we consider one mass to be stationary, correct?
Yes.