Gravitational Potential Energy

Gravitation is a conservative force, because the work done going ``around'' the attractor (perpendicular to the force) is zero, and the work done varying $r$ is the same going out as in, so the work done is independent of path (see figure above). So:

$$U(r) = -\int_{r_0}^r \vec{\bf F}\cdot d\vec{\bf r}$$ (15)

$$= -\int_{r_0}^r -\frac{GMm}{r^2} dr$$ (16)

$$= -(\frac{GMm}{r}-\frac{GMm}{r_0})$$ (17)

$$= -\frac{GMm}{r} + \frac{GMm}{r_0}$$ (18)

where $r_0$ is the radius of an arbitrary point where we define the potential energy to be zero. By convention, unless there is a good reason to choose otherwise, we require the zero of the potential energy to be at $r_0 = \infty$. Thus:

$$U(r) = -\frac{GMm}{r}$$ (19)