Newton's Law and g

F = \frac{G m_1 m_2}{r^2} \quad [M^1 L^1 $T^{-2}$]\ \text{(N)}

G = 6.674 \times 10^{-11}\ \text{N m}^2\ \text{kg}^{-2} \quad [$M^{-1}$ L^3 $T^{-2}$]

g = \frac{GM}{R^2} \approx 9.8\ \text{m/s}^2 \quad [M^0 L^1 $T^{-2}$]

Variation of g

$g'_\text{alt} = \frac{gR^2}{(R+h)^2} \quad \text{(exact)}; \qquad g'_\text{alt} \approx g\!\left(1 - \frac{2h}{R}\right) \quad (h \ll R)$

g'_\text{depth} = g\!\left(1 - \frac{d}{R}\right) \quad [M^0 L^1 $T^{-2}$]

$g_\text{eff,lat} = g - R\omega^2\cos^2\lambda$

Gravitational PE and Potential

U = -\frac{GMm}{r} \quad [M^1 L^2 $T^{-2}$]\ \text{(J)}

V = -\frac{GM}{r} \quad [M^0 L^2 $T^{-2}$]\ \text{(J/kg)}

Escape and Orbital Velocities

v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR} \approx 11.2\ \text{km/s} \quad [M^0 L^1 $T^{-1}$]\ \text{(m/s)}

v_0 = \sqrt{\frac{GM}{r}} \approx 7.9\ \text{km/s (near surface)} \quad [M^0 L^1 $T^{-1}$]\ \text{(m/s)}

$\boxed{v_e = \sqrt{2}\, v_0}$

Kepler's Third Law

$T^2 = \frac{4\pi^2}{GM}\, r^3 \quad [M^0 L^0 T^2] \propto r^3$

Satellite Energy

$KE = \frac{GMm}{2r},\quad PE = -\frac{GMm}{r},\quad E = -\frac{GMm}{2r}$

$KE : PE : E = 1 : {-2} : {-1}; \quad |PE| = 2KE$

Key Constants