The Universal Law of Gravitation
Newton compared an apple with the Moon and found an inverse-square law.
Lesson at a glanceShow the key points
- Kepler's laws: elliptical orbits with the Sun at a focus; equal areas in equal times; T² ∝ a³.
- The Moon's centripetal acceleration (≈ 0.0027 m/s²) is 1/3600 of g, and it is 60 Earth radii away: acceleration falls as 1/r².
- Every pair of particles attracts with F = Gm₁m₂/r² (11.1), along the line joining them.
- G = 6.67 × 10⁻¹¹ N·m²/kg² is the same everywhere; the same laws govern the sky and the Earth.
- A uniform sphere attracts outside bodies as if all its mass were at its centre.
For most of history the heavens and the Earth were thought to run on different rules. Newton’s boldest move was to insist they don’t: the force that drops an apple also bends the Moon’s path.
Before Newton
Astronomers had tracked the planets for centuries. Aryabhata, in fifth-century India, argued that the Earth spins on its axis. A thousand years later Tycho Brahe’s precise naked-eye measurements let Johannes Kepler distil planetary motion into three rules:
- Each planet moves on an ellipse with the Sun at one focus.
- The line from the Sun to a planet sweeps out equal areas in equal times.
- The square of a planet’s period is proportional to the cube of the semi-major axis of its ellipse.
These described the motion beautifully but didn’t say why. That needed a force law.
The Moon is falling
The Moon circles the Earth at about m once every 27.3 days, so it is accelerating toward the Earth at
An apple falls with 9.8 m/s², 3600 times more. The apple is about 6400 km from the Earth’s centre and the Moon about 60 times further, and 60² = 3600. So the acceleration produced by the Earth falls off as the inverse square of distance.
The law
Acceleration ∝ 1/r² means the force on a body of mass m is ∝ m/r². By the third law the body pulls the Earth equally hard, so the force must also be proportional to the Earth’s mass. Newton then made the leap that every pair of bodies in the universe attracts this way:
The pull is 2.40 × 10^-7 N, about the weight of 24.5 micrograms: less than a speck of dust. Double the distance and it drops to a quarter; triple it and it drops to a ninth.
A 3.0 kg ball and a 5.0 kg ball sit with their centres 30 cm apart on a frictionless surface. Find the gravitational force between them and each ball’s initial acceleration.
- 1Kepler: ellipses with the Sun at a focus, equal areas in equal times, T² ∝ a³.
- 2The Moon's acceleration is g/3600 at 60 Earth radii: an inverse-square law.
- 3, attractive, with the same G for every pair of bodies.
- 4A uniform sphere attracts outside bodies as if its mass were at its centre.
Check yourself
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