Understanding Shared Potential Energy in a Gravitational System

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In a gravitational system with two point masses, the gravitational potential energy is defined as U_grav = -G(m_1m_2/r). The discussion highlights a common confusion regarding why potential energy isn't shared between the two masses in a ratio. When analyzing the motion of one mass, it is typically assumed that the other mass is so large that it remains effectively stationary. This approach works well for scenarios like objects in Earth's gravitational field or planets orbiting the sun. However, if the masses are similar, the analysis becomes more complex, requiring a different approach that accounts for both bodies in motion.
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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##?
 
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MohammedRady97 said:
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.
 
Nugatory said:
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?
 
MohammedRady97 said:
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.
 
Nugatory said:
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?
 
MohammedRady97 said:
So the approximation is that we consider one mass to be stationary, correct?
Yes.
 

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