Gravitation — from Newton’s apple to Earth’s mass

The story we tell here is a genuine mystery from the history of science:

If I drop a stone, why does it fall? — this question is simple. What is "the mass of the Earth"? — this question seems impossible. You can't put the Earth on a scale!

Yet in 1798, an English scientist named Henry Cavendish, without ever leaving the Earth, computed its mass — with a small experiment on a wooden bench. How? That answer is three centuries of physics in a chain.

The chain of discovery

No big physics question is answered in one stroke. It gets answered through a chain of simpler measurements. For gravity, the chain is:

Date	Who	What they got	How
~240 BC	Eratosthenes	R_E ≈ 6370 km	Sun's shadow in two cities
~1600	Galileo	g ≈ 9.8 m/s²	Timing falling bodies (see his method)
1687	Newton	Law F ∝ Mm/r²	Math (Principia) — from Kepler's laws
1798	Cavendish	G ≈ 6.67 × 10⁻¹¹	Torsion balance with laboratory masses
after 1798	anyone	M_E ≈ 6 × 10²⁴ kg	Simple algebra: M_E = gR²/G

Crucial point: before 1798, Earth's mass was practically unknown. Newton had the law of gravity in Principia, but not the value of G. g and R were known but g = GM_E/R² doesn't separate G from M_E — only their product GM_E = gR² was accessible.

Cavendish, by measuring G independently (in the laboratory, without involving the Earth), broke that deadlock. Once G was known, computing Earth's mass was one multiplication and division away.

The four articles in this collection

Following the chain above:

1. Article 1 — How Newton reached 1/r² — from Kepler's laws and the Moon's orbit to the universal form of the gravity law. Using Kepler's third law + centripetal acceleration.

2. Article 2 — How Cavendish measured G — the torsion balance, sensitivity to 10⁻⁷ N, and why it was called "weighing the Earth". A laboratory that filled a room.

3. Article 3 — Earth's mass, the value of g, and its variation with altitude — once G is known, everything cascades: M_E, Earth's density, the numerical origin of g = 9.8, g on Everest, in the ISS, on the Moon.

Prerequisites

The level here is university introductory — so we assume you have:

• Algebra: ratios, roots, exponents, solving quadratics
• Circular motion: centripetal acceleration a_c = v²/r = 4π²r/T²
• Kepler's laws: especially the third (T² ∝ a³) — we review it in Article 1
• Basic calculus: power rule, polynomial derivatives, definite integrals. We differentiate 1/r² a few times
• Scientific notation (Ch1 §1.6) — numbers span 10⁻¹¹ to 10²⁴

Where calculus is not needed, we use plain algebra. Where derivatives or integrals give better insight (e.g., deriving potential energy U = -GMm/r from F = GMm/r²), we use them — but final formulas are always usable at the algebra level.

Suggested reading order

Read Articles 1 → 2 → 3 in order. Each rides on the previous:

• Article 1 gives the law (but not the number G).
• Article 2 gives the number G.
• Article 3 harvests the payoff: M_E, g, and its variation.

If you have only an hour: read Article 2. Cavendish's experiment is one of the most elegant in physics — from design philosophy to execution.

Why this story matters

Most "universal constants" (like c, h, k_B) are numbers pinned down by experiment. But G has the worst precision of any fundamental constant — to this day, only about 3 significant figures:

\[ G = 6.67430 \times 10^{-11}\ \mathrm{N \cdot m^2 / kg^2} \pm 0.00015 \times 10^{-11} \]

Why? Because gravity is weak. For two 1-kg balls 1 m apart, the force is 6.67 × 10⁻¹¹ N — about the weight of a single red blood cell. Every vibration, air current, and stray charge drowns out this tiny force. Cavendish solved it in 1798; the same intrinsic weakness of gravity still limits us today.

📚 Primary source: Henry Cavendish, Experiments to determine the Density of the Earth, Philosophical Transactions of the Royal Society, 1798. 📖 Open reference: OpenStax University Physics Vol 1 — Chapter 13: Gravitation. 📖 Feynman Lectures Vol I — Ch 7: The Theory of Gravitation.