A Question Concerning Gravitational Potential Energy

In summary, if two bodies are attracted to each other and are in deep space, the Earth-sized body will be accelerated towards the sun-sized body. The Earth-sized body will have a lesser amount of potential energy than the sun-sized body, so the kinetic energy of the Earth-sized body will be half that of the sun-sized body.
  • #1
Silvius
13
0
Hey guys,

I'm currently trying to get my head around the concept of gravitational potential energy;

U[itex]_{r}[/itex] = -[itex]\frac{Gm_{1}m_{2}}{r}[/itex]

My question concerns whether this relates to both bodies in the system individually, or together.

That is, if I have two masses separated by a distance r, does the above formula describe the gravitational potential energy possessed by one of those bodies, or does it describe the total potential energy of the interaction between the two bodies? Or, another way of looking at it - is the total energy of the system U or 2U?

Thanks!
 
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  • #2
[itex]\vec{r}[/itex] is measured from the origin, wherever you choose to place that. Each object in general has it's own [itex]\vec{r}[/itex] coordinate, and thus will have a different potential. Note that the potential of each object depends on the location of the origin.
 
  • #4
rcgldr said:
Ur is the total gravitational potential energy of the system.

Hmm okay. I guess it's just a bit confusing since people/questions often seem to refer simply to one of the bodies having the potential energy given by that equation (e.g., "a satellite at radius R has potential energy U = -Gm1m2/r").

Is this just an issue of expression, or am I missing something deeper here...?

Thanks!
 
  • #5
Silvius said:
Hey guys,

I'm currently trying to get my head around the concept of gravitational potential energy;

U[itex]_{r}[/itex] = -[itex]\frac{Gm_{1}m_{2}}{r}[/itex]

My question concerns whether this relates to both bodies in the system individually, or together.

That is, if I have two masses separated by a distance r, does the above formula describe the gravitational potential energy possessed by one of those bodies, or does it describe the total potential energy of the interaction between the two bodies? Or, another way of looking at it - is the total energy of the system U or 2U?

Thanks!...

... often seem to refer simply to one of the bodies having the potential energy given by that equation (e.g., "a satellite at radius R has potential energy U = -Gm1m2/r").

U as it is given is the energy for the system of both objects.

When they mention the potential energy of one object, they are splitting the symmetry in a sense, like U=m1*P(r). I made up the symbol P=-Gm2/r for gravitational potential, which is definitely different from gravitational potential energy.

I don't think I found an eloquent way to say it, it's complicated, there are more situations to be confused by, study more, ask us more questions!
 
  • #6
algebrat said:
U as it is given is the energy for the system of both objects.

When they mention the potential energy of one object, they are splitting the symmetry in a sense, like U=m1*P(r). I made up the symbol P=-Gm2/r for gravitational potential, which is definitely different from gravitational potential energy.

I don't think I found an eloquent way to say it, it's complicated, there are more situations to be confused by, study more, ask us more questions!

Thanks so much for the help =)

Bearing the above in mind, how would I approach the following type of problem;

"Two Earth sized/shaped bodies are separated by a distance R in deep space. The bodies are attracted to each other, and hence accelerate towards each other. How fast will each body be when they collide?" (Obviously this isn't a specific problem).

If one body were Earth sized and the other were, say, the size of the sun, the problem would be much simpler; I could simply assume that the sun wouldn't move much - being so much bigger than the Earth - and hence my expression for the potential energy would only really apply to the Earth. Thus, I would find the difference in potential energy of the system at R and then at twice the radius (when the bodies are touching), and I would be able to say that that difference in potential energy has been converted entirely into the kinetic energy of the Earth-sized body.

But in this case, I cannot make such an assumption. Would I then find the difference in potential energy of the system at each of the two points, and halve that to find the kinetic energy of each of the bodies? Am I thinking about this the right way?
 

1. What is gravitational potential energy?

Gravitational potential energy is the energy stored in an object due to its position in a gravitational field. It is the amount of work that is required to move the object from its current position to a reference position where the gravitational potential energy is defined as zero.

2. How is gravitational potential energy calculated?

Gravitational potential energy can be calculated using the formula U = mgh, where U is the potential energy, m is the mass of the object, g is the acceleration due to gravity, and h is the height of the object relative to the reference position.

3. What factors affect gravitational potential energy?

The factors that affect gravitational potential energy include the mass of the object, the strength of the gravitational field, and the distance between the object and the center of the gravitational field.

4. How does gravitational potential energy relate to kinetic energy?

Gravitational potential energy is converted into kinetic energy when an object moves from a higher position to a lower position in a gravitational field. This is known as the law of conservation of energy, which states that energy cannot be created or destroyed, only transferred from one form to another.

5. What are some real-world applications of gravitational potential energy?

Gravitational potential energy is used in many everyday applications, such as hydroelectric power plants, roller coasters, and launching rockets into space. It is also an important concept in understanding the behavior of celestial bodies in our solar system and beyond.

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