Gravity Revisited

Astronomy Fundamentals Chapter 3 covered gravity from the sky's own side — Kepler's three empirical laws of planetary motion, discovered decades before anyone knew why they held. This chapter returns to the same real physics from mechanics' own side: the single force law Newton used to explain, mathematically, why Kepler's laws were true at all.

Newton's Law of Universal Gravitation

Newton's own 1687 Principia proposed that every object with mass attracts every other object with mass, with a force given by:

Here m₁ and m₂ are the two masses, r is the distance between their centres, and G is the gravitational constant — a fixed number that sets the real strength of gravity throughout the universe, with a measured value of 6.674×10&supminus;¹¹ m³·kg&supminus;¹·s&supminus;².

The Real Apple Story

Newton spent much of 1666 at his family home, Woolsthorpe Manor in Lincolnshire, after Cambridge University closed because of plague — the same real "annus mirabilis" period behind several of his major breakthroughs. The apple story genuinely traces back to Newton himself: William Stukeley, a personal friend and contemporary of Newton's, recorded that Newton told him directly how watching an apple fall from a tree in the Woolsthorpe orchard had set him thinking about gravity.

The tree itself still survives at Woolsthorpe Manor today, now maintained by England's National Trust; tree-ring dating (dendrochronology) confirms the orchard's apple tree is genuinely over 400 years old, having regrown from roots that survived after the original trunk blew down in a storm in 1820.

Worked Example: Gravitational Force Between Two People

Two people, each with a mass of 70 kg, stand 1 m apart. What gravitational force do they exert on each other?

This is an almost immeasurably tiny force — roughly the weight of a single grain of table salt — which is exactly why gravity between everyday objects is never noticeable, and why it took a genuinely delicate laboratory experiment to measure G directly at all.

The Real Story of Measuring G: Michell and Cavendish

Newton's law gives the shape of gravity's force, but not its real strength — G itself had to be measured experimentally, and not by Newton. The real credit is more layered than it's often given: English geologist and clergyman John Michell designed the delicate torsion-balance apparatus needed to measure it before 1783, but died in 1793 before completing the experiment. Henry Cavendish inherited Michell's own apparatus, rebuilt it, and completed the measurement in 1797–1798 — the first laboratory measurement of gravity between everyday-sized masses, using a horizontal rod suspended by a thin wire, with small lead spheres at each end drawn very slightly toward larger fixed lead balls nearby.

Explaining Kepler: Why the Planets Move the Way They Do

Astronomy Fundamentals Chapter 3 presented Kepler's three laws as empirical — discovered from Tycho Brahe's careful observational data, decades before anyone could explain why planets actually obeyed them. Newton's real achievement in the Principia was combining his own laws of motion (Chapters 2 and 5 of this course) with the inverse-square gravity law above to derive Kepler's three laws mathematically, from first principles, rather than simply describing what was observed.

An elliptical orbit (Kepler's First Law) is exactly the trajectory a body follows under an inverse-square attractive force; a planet sweeping out equal areas in equal times (Kepler's Second Law) is a direct consequence of conservation of angular momentum (this course's own Chapter 5) applied to a body orbiting under gravity's pull, which always points directly toward the Sun and so exerts zero torque about it; and the real relationship between orbital period and orbital radius (Kepler's Third Law) falls directly out of setting gravitational force equal to the centripetal force (Chapter 5) required for a circular orbit.

Escape Velocity, Derived from Energy

Chapter 3's own conservation of energy gives a clean way to find escape velocity — the minimum speed needed to break free of a body's gravity entirely, without further propulsion. An object "just barely escapes" when its kinetic energy exactly equals the gravitational potential energy binding it, with zero energy left over at infinite distance:

Equivalently, at a body's own surface, this can be written v = √(2gr), using the local surface gravity g directly — useful since g is often easier to look up than the body's full mass M.

Worked Example: Earth's Escape Velocity

Using Earth's real surface values (g ≈ 9.81 m/s², radius r ≈ 6.371×10&sup6; m):

This matches the real, measured value for Earth's escape velocity exactly — a rocket must reach roughly 11.2 km/s (about 40,270 km/h) to leave Earth's gravity behind entirely without further thrust.

Gravity: From Empirical Law to Explained Law

Hands-On Exercises

Next chapter: The Laws of Thermodynamics — where this course leaves motion and gravity behind and turns to heat, energy, and the real zeroth, first, second, and third laws that govern them.