Newton’s Law of Universal Gravitation

Newton’s law of universal gravitation states that every particle attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers: F = Gm1m2 ÷ r2, where G = 6.674 × 10−11 N·m2/kg2. In brief: gravity is always attractive and acts along the line joining the centers of the masses. The forces on the two bodies are equal and opposite (Newton’s third law). Doubling the distance reduces the force to a quarter. The same law explains why objects fall, why the Moon orbits the Earth and why planets orbit the Sun.

The force that pulls an apple to the ground also holds the Moon in orbit. This guide states the law and the meaning of G, explains the inverse-square law and the shell theorem, discusses the relation to weight, compares forces in a table, works through numerical examples including the Earth-Moon force and corrects common mistakes.

What does the law say?

For two point masses m1 and m2 a distance r apart, each experiences an attractive force of magnitude

F = G m1 m2 ÷ r2,

directed toward the other mass. The universal gravitational constant G = 6.674 × 10−11 N·m2/kg2 is a tiny number: gravity is the weakest of the fundamental forces. The force between two 1 kg masses 1 m apart is only 6.7 × 10−11 N. We notice gravity only when at least one mass is as large as a planet. The law applies to spherical bodies by taking r as the distance between their centers (by the shell theorem, a uniform sphere acts as if all of its mass were at its center). It is the foundation of celestial mechanics, though for strong fields, or very high accuracy (such as the precession of Mercury’s orbit), Einstein’s general relativity is needed.

m₁m₂FFr

What does the inverse-square law imply?

The force falls as 1/r2: at twice the distance, the force is one quarter; at three times the distance, one ninth. A force that spreads out in three dimensions in this way is a consequence of geometry: the force is “diluted” over the area of a sphere, which grows as r2. The force is proportional to each mass: doubling either mass doubles the force. Since F = ma, the acceleration of a body of mass m due to a large mass M is a = GM ÷ r2, independent of m, so all bodies fall with the same acceleration (apart from air resistance).

Gravitational force on a person against distance from the Earth center R 2R 3R 4R 0 100 200 300 400 500 600 700 Distance r from the center of the Earth Force on a 70 kg person (N) 687 N (weight) 172 N 76 N
Gravitational force of the Earth on a 70 kg person, F = 687 N × (R/r)². At twice the distance the force is one quarter (172 N), and at three times the distance it is one ninth (76 N): the inverse-square law.

How does gravitation relate to weight?

The weight of an object at the surface of the Earth (mass ME, radius RE) is mg = GMEm ÷ RE2, so g = GME ÷ RE2 = 9.82 m/s2. This fact was used by Henry Cavendish in 1798 to determine G with a torsion balance, and thereby “weigh the Earth”: ME = gRE2 ÷ G = 5.97 × 1024 kg. At a height h above the surface, g(h) = GME ÷ (RE + h)2, which decreases slowly: only by about 12 percent at 400 km, the altitude of the International Space Station.

Gravitational forces compared

Examples of gravitational attraction between pairs of bodies
Bodies Separation Force (N)
Two people (70 kg and 60 kg)1 m2.8 × 10−7
Person (70 kg) and the EarthEarth’s radius, 6.371 × 106 mabout 687 (the weight)
Earth and Moon3.84 × 108 m2.0 × 1020
Sun and Earth1.50 × 1011 m3.5 × 1022

Worked example: calculating gravitational force

  1. Two people. F = G m1m2 ÷ r2 = 6.674 × 10−11 × 70 × 60 ÷ 1.02 = 2.8 × 10−7 N, about a ten-millionth of the weight of a grain of sand.
  2. Earth and Moon. ME = 5.97 × 1024 kg, MM = 7.35 × 1022 kg, r = 3.84 × 108 m. F = 6.674 × 10−11 × 5.97 × 1024 × 7.35 × 1022 ÷ (3.84 × 108)2 = 2.93 × 1037 ÷ 1.475 × 1017 = 1.99 × 1020 N.
  3. Changing the distance. If the Moon were twice as far away, the force would be 1.99 × 1020 ÷ 4 = 5.0 × 1019 N.
  4. g at the space station. At 400 km above the surface, r = 6.771 × 106 m: g = GME ÷ r2 = 3.986 × 1014 ÷ 4.585 × 1013 = 8.69 m/s2, which is 88.5 percent of 9.82 m/s2. Astronauts feel weightless because they are in free fall, not because gravity is absent.
  5. Mass of the Earth. From g = 9.82 m/s2 and R = 6.371 × 106 m: M = gR2 ÷ G = 9.82 × 4.059 × 1013 ÷ 6.674 × 10−11 = 5.97 × 1024 kg.

How was the law discovered?

Johannes Kepler had found three laws describing the motion of the planets (1609 and 1619). Isaac Newton showed that an inverse-square attraction explains them, and compared the acceleration of the Moon with the fall of an apple, publishing the law in the Principia (1687). Robert Hooke and Edmond Halley had suggested an inverse-square force earlier. Henry Cavendish measured G in 1797 to 1798, and the 2015 detection of gravitational waves, predicted by general relativity, was a new test of gravity. The value of G is known with the lowest precision of the fundamental constants, about 22 parts per million.

Which mistakes are common?

Mistake 1: Using the surface distance instead of the distance between centers

For an object at height h, r = R + h.

Mistake 2: Thinking that the larger mass exerts the larger force

Both bodies feel forces of equal size (third law). The accelerations differ.

Mistake 3: Believing there is no gravity in space

Gravity decreases with distance but never becomes zero.

Mistake 4: Confusing G with g

G is a universal constant. g is the local gravitational field strength (acceleration).