Gravitational Potential Energy
  • As mentioned before, Gravitational Potential Energy is the energy of a mass due to its position within a gravitational field.
  • Whilst on a small(er) scale, GPE can be determined by $E_p = mgh$ (See 1.2), on a planetary scale we need to find a proper "zero" point (that is where the GPE of an object is zero.)
  • All gravitational fields are infinite (See 1.1), hence the force acting on an object only drops to zero if you at the furthest point from the centre of the gravitational.
    • Subsequently, the GPE of an object at that point would be zero.
    • However what happens if you move from that point toward the centre of the gravitational field. What happens is that you gain kinetic energy in exchange for GPE, GPE decreases as KE increases. Starting from the point of infinity you would then have (from zero) an increasingly negative value of gravitational potential energy as you approach the centre remember that as you get closer to the centre and the value is smaller as a bigger negative value is a smaller value.

FUN FACTS: HOW TO DERIVE THE FORMULA FOR GPE

  • First: Remember that $W = Fd$ is for a CONSTANT force.
    • For a non-constant force, we would need to integrate Newton's law of Universal Gravity. (Why? Shut up, that's why)
  • The limits would be infinite and the r since it is moving from infinite to r.
(1)
\begin{align} Work = \int\limits_\infty^r \frac{GMm}{x^2} dx \end{align}
(2)
\begin{align} Work = -GMm[x^{-1}]_\infty^r \end{align}
(3)
\begin{align} Work = -\frac{GMm}{r} \end{align}