Gravitation

Formulas for Gravitation

Newton's Law of Universal Gravitation

$F = \frac{Gm_1m_2}{r^2}$ G is the universal gravitational constant, r is the distance between the centers of masses m1 and m2.

Acceleration due to gravity on a planet's surface

$g = \frac{GM}{R^2}$ M is the mass of the planet, R is its radius.

Variation of g with altitude h

$g_h = g\left(1 + \frac{h}{R}\right)^{-2}$ For h << R, this can be approximated as g_h \approx g(1 - 2h/R).

Variation of g with depth d

$g_d = g\left(1 - \frac{d}{R}\right)$ This assumes the planet has a uniform density.

Gravitational Potential Energy

$U = -\frac{GMm}{r}$ The zero of potential energy is taken at infinite separation (r = ∞).

Gravitational Potential

$V = -\frac{GM}{r}$ V is the work done to bring a unit mass from infinity to a point.

Escape Velocity

$v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR}$ This is the minimum speed to escape a planet's gravity from its surface.

Orbital Velocity of a satellite

$v_o = \sqrt{\frac{GM}{r}}$ r is the radius of the circular orbit from the planet's center (r = R+h).

Time Period of a satellite

$T^2 = \frac{4\pi^2}{GM}r^3$ This is a form of Kepler's Third Law for circular orbits.

Total Energy of an orbiting satellite

$E = -\frac{GMm}{2r}$ For a satellite in a stable orbit, Total Energy = - (Kinetic Energy).

Kepler's Third Law

$T^2 \propto a^3$ 'a' is the semi-major axis of the elliptical orbit.