Newton's Law of Universal Gravitation Explained

In 1687, Isaac Newton published his Principia Mathematica — arguably the most important scientific work ever written. Among its results was a single equation that unified the fall of an apple with the orbit of the Moon, the tides of the ocean, and the paths of the planets: Newton's law of universal gravitation. Every mass in the universe attracts every other mass. The force between them follows an inverse-square law. This one insight transformed astronomy, enabled space travel, and established the template for all subsequent physics.

The Gravitational Constant G

G = 6.674 × 10⁻¹¹ N·m²/kg² is one of the fundamental constants of nature. It was first measured by Henry Cavendish in 1798 using a torsion balance — two small lead balls attracted to two larger ones, with the tiny deflection revealing G. Cavendish called his experiment "weighing the Earth": knowing G and measuring g at Earth's surface, he could calculate Earth's mass.

G is extraordinarily small — gravity is the weakest of the four fundamental forces by a factor of ~10³⁶ compared to electromagnetism. Yet it dominates at cosmic scales because it is always attractive (unlike electric forces, which can cancel), and because large masses accumulate in planets, stars, and galaxies.

The Inverse-Square Law

The F ∝ 1/r² dependence of gravity — the inverse-square law — is one of the most important patterns in physics. It also governs the electric force (Coulomb's Law), light intensity, sound intensity, and radiation from a point source. The inverse-square relationship arises geometrically: the surface area of a sphere is 4πr², so any quantity spreading equally in all directions from a point source is diluted in proportion to r².

Deriving g from Newton's Law

Newton's law allows us to calculate the gravitational field strength g at Earth's surface from first principles. Setting the gravitational force equal to mg:

Using M_Earth = 5.97 × 10²⁴ kg, R_Earth = 6.37 × 10⁶ m, G = 6.674 × 10⁻¹¹:

This agrees with the measured value — confirming that the same gravitational law governing the apple falling from a tree also governs the acceleration of objects at Earth's surface. This unification is one of the great triumphs of classical physics.

Orbital Mechanics: Gravity as Centripetal Force

For a satellite of mass m in circular orbit at radius r around Earth (mass M), gravity provides the centripetal force:

The orbital speed depends only on radius — not on the satellite's mass. Lower orbit → faster speed (ISS at 400 km: ~7.66 km/s; geostationary at 35,786 km: ~3.07 km/s).

The orbital period T follows from v = 2πr/T:

This is Kepler's third law — the orbital period squared is proportional to the orbital radius cubed — derived from Newton's laws. Newton's theory explained all three of Kepler's empirical laws of planetary motion, providing the mathematical foundation that Kepler had described observationally.

Escape Velocity

The escape velocity is the minimum speed needed to escape a gravitational field completely — to reach infinity with zero remaining kinetic energy. Setting kinetic energy equal to the magnitude of gravitational potential energy:

For Earth's surface: v_escape = √(2 × 6.674 × 10⁻¹¹ × 5.97 × 10²⁴ / 6.37 × 10⁶) = 11.2 km/s ≈ 40,200 km/h.

Note that escape velocity is √2 × orbital velocity at the same radius. A spacecraft needs 41% more speed to escape Earth's gravity entirely than to orbit at the surface — explaining why low Earth orbit is a significant milestone, but escaping to the Moon requires substantially more energy.

Gravitational Potential Energy

The gravitational potential energy of mass m at distance r from mass M is:

The negative sign is fundamental: gravitational potential energy is always negative (taking zero at infinity). Objects bound in orbit have negative total energy (KE + PE < 0). To escape, total energy must reach zero or above. The work required to move a mass from Earth's surface (r = R) to infinity:

Black Holes: Escape Velocity Exceeds c

If a mass M is compressed into a sufficiently small radius r_s (the Schwarzschild radius), the escape velocity reaches c — the speed of light. Nothing, not even light, can escape. This is a black hole:

For the Sun (M = 2 × 10³⁰ kg): r_s ≈ 3 km. The Sun would need to be compressed from its actual radius of 696,000 km into a sphere 3 km across to form a black hole. Earth's Schwarzschild radius is about 9 mm.

Newton's formula gives the right Schwarzschild radius through a fortunate coincidence — the full calculation requires general relativity — but it illustrates how Newton's law of gravitation naturally leads to the concept of objects from which light cannot escape.

Worked Examples

Example 1: Force between Earth and Moon

M_Earth = 5.97 × 10²⁴ kg, M_Moon = 7.35 × 10²² kg, r = 3.84 × 10⁸ m.

Example 2: Gravitational field at altitude

Find g at 600 km above Earth's surface (r = R_E + 600,000 = 6.97 × 10⁶ m).

Frequently Asked Questions