Newton's Law of Gravitation

$\boxed{F = \frac{GMm}{r^2}}$

$G = 6.67 \times 10^{-11}$ N m² kg⁻²

Always attractive, acts along line joining centres.

$g = \frac{GM}{r^2}$

At surface: $g = GM/R^2$. $g$ is also free-fall acceleration.

Gravity provides centripetal force: $\frac{GMm}{r^2} = \frac{mv^2}{r}$

$v = \sqrt{\frac{GM}{r}}$

$T = 2\pi\sqrt{\frac{r^3}{GM}}$

Kepler's Third Law: $T^2 \propto r^3$

$E_p = -\frac{GMm}{r}$

Negative: work needed to separate masses to infinity.

Total orbital energy: $E = KE + PE = \frac{1}{2}mv^2 - \frac{GMm}{r} = -\frac{GMm}{2r}$

• $F = GMm/r^2$; $g = GM/r^2$
• Orbits: $v = \sqrt{GM/r}$; $T^2 \propto r^3$
• $E_p = -GMm/r$; total orbital energy = $-GMm/(2r)$