A Question Concerning Gravitational Potential Energy

[Silvius]

Hey guys,

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

U_{r} = -\frac{Gm_{1}m_{2}}{r}

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!

 

[dipole]

\vec{r} is measured from the origin, wherever you choose to place that. Each object in general has it's own \vec{r} coordinate, and thus will have a different potential. Note that the potential of each object depends on the location of the origin.

 

[rcgldr]

U_r is the total gravitational potential energy of the system. Wiki article:

http://en.wikipedia.org/wiki/Potential_energy#General_formula

 

[Silvius]

U_r 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!

 

[algebrat]

Hey guys,

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

U_{r} = -\frac{Gm_{1}m_{2}}{r}

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!

 

[Silvius]

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?