Chapter 7 · Introduction

Chapter 7 — Gravitation

One force holds you to the ground, keeps the Moon in its orbit and steers the planets round the Sun. This chapter finds its law and uses it to weigh the Earth, launch satellites and escape into space.

An apple falls from a branch. The Moon, $384\,000\ \text{km}$ away, sails on round the Earth and never comes down. Mars wanders across the night sky, now forwards, now backwards. For most of history these looked like three quite different things — earthly objects fall, heavenly ones go round. The great discovery of the seventeenth century was that they are the same thing: the Moon is falling too, all the time, but it is moving sideways so fast that it keeps missing the Earth.

This chapter follows that idea from the sky to the launch pad. You will start with Kepler's three laws, distilled from years of careful observation of the planets, and see how Newton explained all of them with one law of universal gravitation and one constant, $G$. Then you will bring the law down to Earth: where the value $g=9.8\ \text{m/s}^2$ comes from, how it weakens up a mountain and down a mine, and how it lets you calculate the mass of the Earth. Finally you will replace $mgh$ with a potential energy that works at any distance, and use it to find how fast a rocket must go to escape the Earth, how fast a satellite must move to stay in orbit, and how much energy it holds there.

Kepler's laws→$F=\dfrac{Gm_1m_2}{r^2}$→$T^2\propto r^3$→$g=\dfrac{GM}{R^2}$→$g$ with height and depth→$U=-\dfrac{GMm}{r}$→Escape speed→Satellites
🎯 By the end of this chapter you can…
  • state Kepler's three laws, use $r_p=a(1-e)$, $r_a=a(1+e)$, $v_pr_p=v_ar_a$ and $T^2\propto a^3$, and explain the law of areas as conservation of angular momentum;
  • use Newton's law of gravitation $F=\dfrac{Gm_1m_2}{r^2}$ with superposition, and the shell results for spheres;
  • derive $T^2=\dfrac{4\pi^2}{GM}r^3$ for a circular orbit and use it to find the masses of the Sun, the Earth and the planets;
  • derive $g=\dfrac{GM}{R^2}$ and its variation with height, $\dfrac{g}{(1+h/R)^2}\approx g\left(1-\dfrac{2h}{R}\right)$, and with depth, $g\left(1-\dfrac dR\right)$;
  • use the gravitational potential energy $U=-\dfrac{GMm}{r}$ and the potential $V=-\dfrac{GM}{r}$, and show that they reduce to $mgh$ near the surface;
  • calculate escape speed $\sqrt{2gR}$, orbital speed $\sqrt{GM/r}$, period, and the energies $K=-E$, $U=2E$ of a satellite;
  • tackle Board, JEE Main, JEE Advanced and NEET-style problems on gravitation (with geostationary orbits as a JEE Advanced extension).
Teaser — The Falling Moon

The Moon circles the Earth once every $27.3$ days at $3.84\times10^8\ \text{m}$. Without gravity it would carry straight on along the dashed tangent. Choose a time interval: the red gap shows how far the Moon has fallen towards the Earth, compared with the straight line, in that time. Before you look at the readouts, guess: in one second, does the Moon fall more or less than a millimetre? An apple falls $4.9\ \text{m}$ in its first second.

72 h
straight-line distance—
fall towards the Earth—
Moon's fall in 1 s—
apple's fall ÷ Moon's—

Sections in this chapter

7.1

Kepler's Laws and the Universal Law of Gravitation

Elliptical orbits, the law of areas, $T^2\propto a^3$, Newton's inverse-square law, the constant $G$ and weighing the Sun.

$F=\dfrac{Gm_1m_2}{r^2}$
Open section →7.2

Acceleration due to Gravity and Its Variation

Where $g=9.8\ \text{m/s}^2$ comes from, the mass of the Earth, and how $g$ changes with height and with depth.

$g(h)\approx g\left(1-\tfrac{2h}{R_E}\right)$
Open section →7.3

Gravitational Potential Energy, Escape Speed and Satellites

$U=-GMm/r$, gravitational potential, escape speed, orbital speed and period, and the energy of a satellite.

$v_e=\sqrt{2gR}$
Open section →
✅ Before you start
You will need uniform circular motion and centripetal acceleration $a_c=\dfrac{v^2}{r}=\omega^2r$, with period and angular speed (Class 11 Physics, Chapter 3, Section 3.4); Newton's laws, especially the second and third, and the idea of a net force supplying the centripetal force (Chapter 4); and work, kinetic and potential energy, conservation of mechanical energy and $U=mgh$ (Chapter 5, Sections 5.1–5.2). Section 7.1 mentions angular momentum about a point (Chapter 6). From mathematics you need powers and roots ($T\propto a^{3/2}$), the area under a graph as an integral (Chapter 2), and significant figures and standard form (Chapter 1).
← 6.3 Moment of Inertia and Rotation about a Fixed Axis7.1 Kepler's Laws and the Universal Law of Gravitation →