My question is about two equations regarding uniform spheres that I've run into:
- $\quad V=\frac{GM}{r},$
and
- $\quad U = \frac{3}{5}\frac{GM^2}{r}.$
1) On one hand, $V$ is unknown to me, and is described (in Solved Problems in Geophysics) as "the gravitational potential of a sphere of mass M." I also found it online called "the potential due to a uniform sphere."
2) On the other hand, $U$ is what I've seen before and I know it by the descriptions "sphere gravitational potential energy" or "gravitational binding energy."
My understanding is that $U$ is the amount of energy required to build the sphere piece by piece from infinity. I also recognize $GMm/r$ as the gravitational potential between two masses.
Can someone explain the difference between these concepts? How can $GM/r$ be the "gravitational potential of a sphere"? Isn't that what $U$ is?
Best Answer
There is a mistake in one of your formulas, $U=\frac{3 G M^2}{5 R}$ with $R$ equal to the sphere radius is the energy required to blow every tiny shred of the sphere apart so that its pieces no longer interact gravitationally, as you said, while $V$ as given above with $r$ equal to distance from the sphere center describes how the sphere interacts with other (celestial) bodies, i.e test particles moving in the sphere's gravitational field feel $V$.
To elaborate: the gravitational field around a point mass and around an object that's spherically symmetric is the same outside of the object due to symmetry considerations, which is why $V$ agrees with the formula for the gravitational potential between 2 masses.