Universal Law of Gravitation

Gravity is the most familiar force in everyday life — it keeps you on the ground, holds the Moon in orbit, and governs the motion of planets around the Sun. Yet until Isaac Newton published his landmark work in 1687, no one had expressed gravity as a single, universal mathematical law. Newton's Universal Law of Gravitation states that every object with mass attracts every other object with mass, across any distance, no matter how large. This was a revolutionary idea: the same force pulling an apple downward is identical in nature to the force keeping the Earth in orbit around the Sun. The strength of this gravitational attraction depends on two things: the masses of the objects involved and the distance between them. More mass means a stronger pull, while greater separation means a weaker pull — and crucially, the force weakens with the square of the distance. This 'inverse-square' relationship means that if you double the distance between two objects, the gravitational force between them drops to one quarter of its original value. If you triple the distance, it falls to one ninth. This pattern appears throughout nature and is one of the most important relationships in all of physics. Gravitation is not just a local phenomenon — it truly is universal. It acts between a proton and an electron inside an atom, between you and your desk, and between galaxies billions of light-years apart. Of course, gravitational forces between small everyday objects are extraordinarily tiny and undetectable without sensitive instruments. It is only when at least one object has an enormous mass — like a planet or a star — that gravity becomes the dominant force shaping the universe around us.

Key takeaways
  • Gravity Acts Between All Masses
  • Force Is Proportional to Mass
  • The Inverse-Square Law
  • Newton's Third Law and Mutual Attraction
  • The Gravitational Constant G

Gravity Acts Between All Masses

Every particle of matter in the universe exerts a gravitational pull on every other particle. There are no exceptions and no shields — unlike electric forces, which can be cancelled by opposite charges, gravity is always attractive and always present. Two books sitting on a shelf attract each other gravitationally, but the force is so minuscule that no everyday effect is noticeable.

Force Is Proportional to Mass

The gravitational force between two objects increases in direct proportion to each of their masses. If you double the mass of one object, the force doubles. If you double both masses, the force quadruples. Think of mass as the 'gravitational charge' of an object — the more mass, the more powerfully it interacts with other masses. This is why the Earth, with its enormous mass, exerts a strong pull on everything near it.

The Inverse-Square Law

The force decreases with the square of the distance separating the two objects' centres. Mathematically, F ∝ 1/r². Imagine the gravitational influence of a mass spreading out in all directions like the surface of an expanding sphere. As the radius doubles, the same influence is spread over four times the surface area, so the force per unit area (and thus the force on a distant object) drops to a quarter. This geometric reasoning explains why the inverse-square law is so natural.

Newton's Third Law and Mutual Attraction

Gravitational forces always come in pairs. The Earth pulls you downward with a force equal to your weight. Simultaneously, you pull the Earth upward with an identical force in magnitude. According to Newton's Third Law, forces are equal and opposite. You do not noticeably accelerate the Earth because of its enormous mass (a = F/m), but the Earth accelerates you significantly. Both forces are real, and both are described by the same equation.

The Gravitational Constant G

The proportionality in Newton's law requires a universal constant G to make the equation exact. G = 6.674 × 10⁻¹¹ N m² kg⁻² is an extraordinarily small number, which is precisely why gravity between everyday objects is imperceptible. G was first measured accurately by Henry Cavendish in 1798 using a delicate torsion balance experiment. Cavendish measured the tiny twist caused by the attraction between lead spheres, effectively 'weighing the Earth' by determining its density.

The Key Equation

F=Gm1m2r2F = \frac{G m_1 m_2}{r^2}
  • F — Gravitational force (Newton (N)): The attractive force each mass exerts on the other, directed along the line joining their centres. Both objects experience forces of equal magnitude.
  • G — Universal gravitational constant (N m² kg⁻²): A fixed constant of nature equal to 6.674 × 10⁻¹¹ N m² kg⁻². Its very small value explains why gravity is only noticeable when at least one mass is very large.
  • m_1 — Mass of first object (kilogram (kg)): The mass of the first body. Increasing this mass proportionally increases the gravitational force.
  • m_2 — Mass of second object (kilogram (kg)): The mass of the second body. The force depends equally on both masses — neither is more 'responsible' for the attraction than the other.
  • r — Separation distance (metre (m)): The distance between the centres of mass of the two objects. For spherical bodies, this is the centre-to-centre distance. The force varies as 1/r², making distance extremely influential.

Worked Example

Problem. The Earth has a mass of 5.97 × 10²⁴ kg and the Moon has a mass of 7.34 × 10²² kg. The average distance between their centres is 3.84 × 10⁸ m. Calculate the gravitational force between the Earth and the Moon.

F≈1.98×1020 NF \approx 1.98 \times 10^{20} \text{ N}

Answer. The gravitational force between the Earth and the Moon is approximately 1.98 × 10²⁰ N. This enormous force is what continuously deflects the Moon from a straight-line path and keeps it in orbit.

Figure
Diagram
m₁m₂rF on m₁F on m₂Universal Law of GravitationF = Gm₁m₂ / r²Forces are equal, opposite, and directed along the line joining the centresLarger massSmaller mass
Origins · 1 minIsaac Newton and how this was discovered
IN
Isaac Newton
1643–1727 · English

Newton formulated the Universal Law of Gravitation in his monumental work Philosophiæ Naturalis Principia Mathematica (1687). He demonstrated that a single mathematical law — F = Gm₁m₂/r² — could account for both the fall of objects on Earth and the orbital motion of planets, unifying terrestrial and celestial mechanics for the first time. His law remained the definitive description of gravity for over 200 years, until Einstein's General Theory of Relativity provided a deeper geometric interpretation. The famous story of Newton and the falling apple is not entirely myth. Newton himself recounted to friends late in his life that observing a falling apple prompted him to think about whether the force pulling the apple also extended to the Moon. The apple tree at Woolsthorpe Manor — Newton's birthplace — still stands today.

  • Weight and Gravitational Field Strength — Weight is the gravitational force exerted on a mass by a planet. Understanding the Universal Law of Gravitation is essential for calculating surface gravitational field strength g = GM/r² and hence the weight of any object.
  • Newton's Second Law (F = ma) — To find the acceleration produced by gravity — for instance, the free-fall acceleration near Earth's surface — you must combine the gravitational force law with F = ma. These two laws together are the foundation of classical mechanics.
  • Circular Motion and Orbits — Gravitational force provides the centripetal force that keeps planets, moons, and satellites in approximately circular orbits. Combining F = Gm₁m₂/r² with the centripetal force equation allows calculation of orbital speeds and periods.
  • Newton's First Law and Inertia — Newton's First Law establishes that objects continue in straight-line motion unless a net force acts. Gravity is precisely the force that curves a projectile's path or bends a planet's trajectory away from a straight line.