Understanding Shared Potential Energy in a Gravitational System

In summary, the gravitational potential energy of a system with two point masses, ##m_1## and ##m_2##, is defined as $$U_{grav} = -G \frac{m_1m_2}{r}$$ where ##r## is the separation between the two masses. When analyzing the motion of one component, such as ##m_1##, it is assumed that the other mass is significantly larger and essentially stationary. If the masses were more similar, the problem would become more complex, but it is possible to choose coordinates in which both masses are in motion. The approximation in this analysis is that one mass is considered to be stationary.
  • #1
PFuser1232
479
20
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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  • #2
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.
 
  • #3
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?
 
  • #4
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.
 
  • #5
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?
 
  • #6
MohammedRady97 said:
So the approximation is that we consider one mass to be stationary, correct?
Yes.
 

1. What is shared potential energy?

Shared potential energy is the energy stored within a system due to the interactions between two or more objects or particles. This energy is shared between the objects and can be converted into different forms of energy, such as kinetic energy or thermal energy.

2. How is shared potential energy different from regular potential energy?

Regular potential energy is the energy stored within a single object or system. Shared potential energy, on the other hand, is the energy stored within a system of multiple objects or particles interacting with one another.

3. What are some examples of systems with shared potential energy?

Some examples of systems with shared potential energy include a pendulum, where the potential energy is shared between the bob and the Earth, or a molecule, where the potential energy is shared between the atoms.

4. How is shared potential energy related to the concept of conservation of energy?

Shared potential energy follows the principle of conservation of energy, which states that energy cannot be created or destroyed, only transferred or converted into different forms. In a system with shared potential energy, the total amount of energy remains constant, but it can be transferred between objects or converted into other forms.

5. Can shared potential energy be negative?

Yes, shared potential energy can be negative. This can occur when the objects in the system have opposite charges, such as in an electric field, or when the objects are moving in opposite directions, such as in a gravitational field. Negative shared potential energy indicates that the objects are attracted to each other and the energy can be released if the objects move closer together.

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