Physics · Ch 8 — Gravitation

Summary

Newton’s universal law of gravitation: every particle attracts every other particle with a force $F = G \frac{m_1 m_2}{r^2}$ , where $G = 6.67 \times 10^{-11} \, \text{N m}^2 \text{kg}^{-2}$ .
Acceleration due to gravity on Earth’s surface: $g = \frac{GM}{R^2} \approx 9.8 \, \text{m/s}^2$ . It varies with altitude ( $g_h = g \left(1 - \frac{2h}{R}\right)$ for $h \ll R$ ) and depth ( $g_d = g \left(1 - \frac{d}{R}\right)$ ).
Gravitational potential energy of a two-body system: $U = -\frac{G m_1 m_2}{r}$ , with zero at infinite separation.
Escape velocity from a planet: $v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR}$ . For Earth, $v_e \approx 11.2 \, \text{km/s}$ .
Orbital velocity of a satellite in a circular orbit at radius $r$ : $v_o = \sqrt{\frac{GM}{r}}$ . For a low Earth orbit ( $r \approx R$ ), $v_o \approx 7.9 \, \text{km/s}$ .
Kepler’s three laws:
- First law: planets move in elliptical orbits with the Sun at one focus.
- Second law: areal velocity is constant — $\frac{dA}{dt} = \frac{L}{2m}$ (angular momentum conserved).
- Third law: $T^2 \propto a^3$ , i.e. $\frac{T^2}{a^3} = \frac{4\pi^2}{GM}$ for a body orbiting a central mass $M$ .
Principle of superposition: the net gravitational force on a particle due to several other particles is the vector sum of the individual forces, $\vec F_R = \sum \vec F_i$ . …