Universal Gravitation
Newton's law of universal gravitation states that every mass in the universe attracts every other mass with a force proportional to the product of their masses and inversely proportional to the square of the distance between them:
where G is the gravitational constant (G ≈ 6.674×10⁻¹¹ N·m²/kg²), m₁ and m₂ are the two masses, and r is the distance between their centers.
The key word is universal. Newton's great insight was not that the Earth pulls objects downward — everyone knew that — but that the same force that pulls an apple off a tree also pulls the Moon around the Earth and the Earth around the Sun. The mathematical form is identical; only the masses and distances vary.
The Moon Falling
Feynman describes Newton's reasoning vividly: the Moon is falling. At every instant, the Moon would travel in a straight line if there were no force acting on it. Instead, it continuously curves toward Earth. How much does it fall? About 1/20 inch per second. This matches exactly what you'd predict from the apple's fall if you scale by the square of the distance ratio: the Moon is 60 Earth radii away, so gravity is 60² = 3600 times weaker, and the fall per unit time is correspondingly smaller. The quantitative match was the confirmation of the inverse-square law.
Kepler and Newton
Newton's law explains why Kepler's three empirical laws hold. Kepler deduced the laws from Tycho Brahe's measurements without knowing why they were true. Newton showed they follow mathematically from F = Gmm'/r²:
- Orbits are ellipses because the inverse-square force produces conic sections.
- Equal areas in equal times (the area swept by the radius vector is constant) because angular momentum is conserved — gravity exerts no torque.
- The orbital period T scales as the semi-major axis a to the 3/2 power (T² ∝ a³) because the force law determines how speed scales with orbit size.
Measuring G
The gravitational constant G could not be measured by astronomical observation alone — those measurements only give the product GM for the attracting body, not G and M separately. Henry Cavendish resolved this in 1798 with a torsion balance: two small lead balls on a rod suspended by a thin fiber, attracted to two larger lead balls. The tiny gravitational deflection of the fiber was measured optically. Cavendish thereby "weighed the Earth" — measured G and from it computed Earth's mass.
Where It Fails
Newton's law is accurate to high precision for planetary motion and ordinary scales. It fails in two regimes: very strong gravitational fields (near massive compact objects like neutron stars or black holes) and in the prediction that gravity propagates instantaneously — which violates special relativity. Einstein's general relativity corrects both failures.
Connections
- keplers-laws — the observational pattern that Newton's law explains
- galilean-inertia — inertia and gravitational mass are equal; the equivalence principle
- general-relativity-gravity — Einstein's extension; gravity as geometry of spacetime
- gravitational-constant — G and how it was measured