PX155 - C3 - gravitational potential energy

d W = \vec{F} \cdot d \vec{l} = - \frac{G M m}{r^{2}} \underset{―}{\hat{r}} \cdot d \vec{l} = - \frac{G M m}{r^{2}} d r

- work done:

W (A \to B) = - G M m \int_{A}^{B} \frac{1}{r^{2}} d r = - G M m [\frac{1}{r (B)} - \frac{1}{r (A)}]

- only depends on distance $r$ of $m$ from $M$ , therefore, gravity is conservative

Δ U = - W (A \to B)

- so, if $A$ is at $r = \infty$ , $Δ U = U$ :

U (B) = - G M m [\frac{1}{r (B)}]

- in general:

U (r) = - \frac{G M m}{r}

- WARNING: sometimes see "gravitational potential" to mean $\frac{U}{m}$ or $ϕ$