The form of gravitational potential energyby henpen Tags: celestial mechanics, classical mechanics, potential energy 

#1
Mar813, 02:34 PM

P: 50

http://www.scholarpedia.org/article/...tial_Mechanics In this source, the gravitational potential energy is given as [itex] \frac{MmG}{r}\frac{mmG}{r}[/itex], seeming to imply that the [itex]\frac{MmG}{r}[/itex] result only applies to a body, mass [itex]m[/itex], in a gravitational potential, not a twobody sytem. Why is this, or is it an error?
I would have thought, given that the force is [itex]\frac{mMG}{r^2}[/itex], the potential energy is [itex]\int (\frac{mMG}{r^2}) =\frac{mMG}{r}[/itex]: certainly not the result in the source. 



#2
Mar813, 05:06 PM

P: 5,634

I suspect the radius r in the Force and energy equations in Scholarpedia are slightly different.
Often because M is so much greater than m, I have elsewhere see the approximation M + m about equal to M...which matches your proposed integration result. The equation here http://en.wikipedia.org/wiki/Escape_...scape_velocity for escape velocity shows  GMm/r [as you suspected] for the potential energy BUT that 'r' is from the surface of earth...and my old physics book has the same GMm/r when the sun is considered to be at rest.... and I just watched Leonard Susskind derive the same potential energy in lecture 2 of his Cosmology series. In other formulations, the center of mass of a system of an orbiting sun and planet about each other [sun is not stationary] is based on mr =mR.....where such a center of mass is stationary.... So you are on the right track, I think, but exactly what was assumed and approximated in 'Scholarpedia' is not clear to me. 



#3
Mar913, 12:03 AM

P: 177





#4
Mar913, 04:07 AM

P: 50

The form of gravitational potential energyBy the way, in the model where the sun isn't stationary (i.e.[itex]F=\frac{G(M+m)m}{r^2}[/itex]), is it a simplification to assume that the sun is stationary, and exerts a force of [itex]\frac{G(M+m)m}{r^2}[/itex] on the planet: does this pedagogical simplification actually reproduce the same results as reality, or must one use the frame of reference where the sun and Earth are moving? 



#5
Mar913, 05:37 PM

P: 177





#6
Mar1313, 03:44 PM

P: 50




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