Newton's Law of Universal Gravitation

In 1687, Isaac Newton proposed something audacious: the same force that makes an apple fall from a tree also holds the Moon in its orbit around the Earth and the Earth in its orbit around the Sun. It is a single rule that governs falling objects, planetary motion, ocean tides, and the slow gathering of gas clouds into stars. That rule is the law of universal gravitation.

The equation

Newton's law states that the gravitational force between two objects is proportional to the product of their masses and inversely proportional to the square of the distance between them:

Each symbol carries a precise meaning:

• F — the gravitational force, measured in newtons (N). It acts along the line joining the two objects, always attractive.
• G — the gravitational constant, 6.674 × 10⁻¹¹ N·m²·kg⁻². It sets the overall strength of gravity and is the same everywhere in the universe.
• m₁, m₂ — the masses of the two objects, in kilograms.
• r — the distance between the centres of the two masses, in metres.

Why the inverse square?

The r² in the denominator is the heart of the law. Imagine gravity spreading out from a mass like light from a bulb. As you move away, that influence is diluted over the surface of an ever-larger sphere. The surface area of a sphere grows as the square of its radius, so the influence at any point falls off as one over the radius squared.

The practical consequence is dramatic. Double the distance between two objects and the force drops not to half but to a quarter. Triple the distance and it falls to a ninth. This steep fall-off is why you feel the Earth's pull firmly at its surface yet astronauts a few hundred kilometres up experience only slightly less, while a probe drifting between the planets feels almost nothing at all.

A worked example

Let's calculate the gravitational attraction between the Earth and the Moon. We need three numbers: the mass of the Earth (5.972 × 10²⁴ kg), the mass of the Moon (7.342 × 10²² kg), and the distance between them (about 3.84 × 10⁸ m).

Substituting into the formula gives a force of roughly 1.98 × 10²⁰ N. That is an almost unimaginably large number — about two hundred billion billion newtons — yet it is exactly the gentle, continuous tug that bends the Moon's path into an orbit rather than letting it fly off in a straight line.

From this law to everyday gravity

When you stay near the Earth's surface, the distance r barely changes, so the whole messy expression collapses into something simple. All the constants — G, the Earth's mass, and the Earth's radius squared — combine into a single number we call g, the acceleration due to gravity, about 9.81 m·s⁻². That is why, close to the ground, we can use the far simpler weight formula F = m·g instead. The universal law hasn't gone away; it has just been packaged into a convenient constant.

What the law does not tell us

Newton's law is extraordinarily accurate for everyday situations and for most of astronomy, but it describes what gravity does without explaining why. It also breaks down in extreme conditions — near a black hole, or when precision matters at the level of Mercury's orbit. For those cases, Einstein's general relativity replaces the idea of a force with the curvature of space and time. For nearly everything you will calculate, though, Newton's elegant inverse-square law is all you need.

Key takeaways

• Gravitational force grows with the product of the two masses and shrinks with the square of the distance.
• The gravitational constant G is universal and very small, which is why gravity is the weakest of the fundamental forces despite shaping the cosmos.
• Distance is measured centre to centre.
• Near a planet's surface, the law simplifies to F = m·g.