Summary

This chapter built up WBCHSE Unit 6's account of gravitation in the order the syllabus lists it. Kepler's three laws (Section 7.2) described planetary motion purely geometrically -- elliptical orbits with the Sun at one focus, equal areas swept in equal times, and $T^2 \propto a^3$ -- before Newton's universal law of gravitation (Section 7.3), $F = GMm/r^2$, supplied the underlying physical cause, with the universal constant $G = 6.674\times 10^{-11}\ \text{N}\,\text{m}^2/\text{kg}^2$ (Section 7.4) measured by Cavendish's torsion balance. Evaluated at the Earth's own surface, this law gives the familiar $g = GM/R^2$ (Section 7.5), which then varies in three distinct ways: with altitude ($g_h = g/(1+h/R)^2$), with depth ($g_d = g(1-d/R)$, reaching zero at the Earth's centre), and due to the Earth's own rotation ($g' = g - \omega^2R\cos^2\lambda$, least at the equator, unchanged at the poles) -- Sections 7.6-7.7. Gravitational potential energy, $U(r) = -GMm/r$ (Section 7.8), generalised the familiar $U=mgh$ to any separation, and gravitational potential, $V(r) = -GM/r$ (Section 7.9), stripped away the test mass to describe the field on its own -- both always negative, reflecting gravity's purely attractive nature. Escape velocity, $v_e = \sqrt{2GM/R} \approx 11.2\ \text{km/s}$ from Earth (Section 7.10), and orbital velocity, $v_o = \sqrt{GM/r}$ with $v_e = \sqrt{2}\,v_o$ close to the surface (Section 7.11), governed satellite motion, with the corresponding period $T = 2\pi\sqrt{r^3/GM}$ recovering Kepler's third law from Newtonian mechanics. The total orbital energy, $E = -GMm/2r$ (Section 7.12), always negative, marked a satellite as gravitationally bound. Finally, the geostationary satellite (Section 7.13) -- equatorial, eastward, with a pe …