2

I'm confused over the concept of gravitational potential energy inside a sphere. I understand that the gravitational potential energy inside the sphere is supposed to be a constant $U = \frac{GMm}{R}$ where $R$ is the radius of the sphere.

What I am confused about is how this applies in conservation of energy equations. For instance, the classic "Drill a hole through the planet". In that instance, GPE at the surface is converted to KE at the core and then back to GPE as it continues through. But, if the object still has gravitational potential energy, this equation doesn't hold. What am I missing concept wise?

Ali
  • 6,052
  • 6
  • 36
  • 50
user41720
  • 21

1 Answers1

6

Your equation is incorrect. The gravitational potential is $$\phi(r)=-GM\frac{3a^2-r^2}{2a^3}$$ when you're inside a uniform sphere of radius $a$ with total mass $M$. This is a quadratic potential in $r$, which is why it gives rise to harmonic exchange of energy when you oscillate between the planet surface and the core.

DumpsterDoofus
  • 10,699