What Is Gravitational Force? The Complete Physics Guide

Gravitational force is one of the four fundamental forces of nature, alongside electromagnetism, the strong nuclear force, and the weak nuclear force. It is the invisible attractive force that every object possessing mass exerts upon every other object with mass in the universe. Unlike nuclear forces that operate only at subatomic scales, gravity has infinite range — it acts between two atoms and between two galaxies billions of light-years apart, though the force diminishes rapidly with distance. Gravity is the architect of the cosmos: it formed galaxies, stars, and planets from diffuse clouds of gas and dust; it keeps the Moon in orbit around Earth; it causes the ocean tides; and it keeps your feet firmly planted on the ground.

The modern quantitative understanding of gravity began with Sir Isaac Newton in 1687 when he published his monumental work Philosophiæ Naturalis Principia Mathematica — arguably the most important scientific book ever written. Newton's genius was recognizing that the same force causing an apple to fall from a tree was also responsible for keeping the Moon in its orbit around Earth. This unification of terrestrial and celestial mechanics was revolutionary. Before Newton, it was widely believed that the heavens operated by entirely different rules than Earth. Newton demolished that distinction with a single elegant equation that described gravitational attraction between any two masses anywhere in the universe.

Newton's Law of Universal Gravitation — Deep Dive

Newton's law states that every particle of matter attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The mathematical form is:

In this equation, \(F\) is the magnitude of the gravitational force in Newtons (N). The symbol \(G\) is the universal gravitational constant, \(6.674 \times 10^{-11}\) N·m²/kg². The quantities \(m_1\) and \(m_2\) are the masses of the two objects in kilograms (kg), and \(r\) is the center-to-center distance between them in meters (m). The force is always attractive — gravity never repels — and acts symmetrically: object 1 pulls object 2 with the same magnitude of force that object 2 pulls object 1 (Newton's Third Law).

The Inverse-Square Law Explained

The "\(1/r^2\)" dependence — the "inverse-square law" — has a beautiful geometric explanation. Imagine the gravitational field as lines radiating outward from a mass in all directions. As you move further away, these lines must spread over the surface of an ever-larger sphere. Since the surface area of a sphere grows as \(4\pi r^2\), the density of field lines (and thus the field strength and force) decreases as \(1/r^2\). Double the distance? The force drops to one-quarter. Triple the distance? The force drops to one-ninth. This is why the Sun's gravity dominates the solar system despite being 150 million km from Earth.

Key Properties of Gravitational Force

• Universal: Acts between all objects with mass — from quarks to galaxy superclusters.
• Always Attractive: Unlike electric or magnetic forces, gravity is exclusively attractive — never repulsive.
• Infinite Range: Acts across all distances, though it weakens as \(1/r^2\).
• Mass-Dependent: Proportional to both masses; doubling either mass doubles the force.
• Conservative Force: Work done against gravity is path-independent; gravitational potential energy is well-defined.
• Superposition Principle: The total gravitational force on an object is the vector sum of individual forces from all other masses.
• Weakest Fundamental Force: Per unit mass, gravity is vastly weaker than electromagnetic or nuclear forces, yet it dominates at cosmic scales because it is always additive and acts over infinite range.

The Gravitational Constant G — History and Measurement

The gravitational constant \(G\) has a precisely measured value (CODATA 2018) of:

Newton derived the law of gravitation in 1687 but could not measure \(G\) directly — he could only determine relative gravitational effects. The first measurement of \(G\) came over a century later, in 1798, when the English scientist Henry Cavendish used a highly sensitive torsion balance apparatus. He suspended two small lead balls (each about 0.73 kg) from a horizontal rod, then brought two large lead spheres (each about 158 kg) close to them. The tiny gravitational attraction caused the suspension wire to twist, and Cavendish measured this deflection with great precision. His experiment, originally designed to "weigh the Earth" (determine Earth's density), effectively gave the first measurement of \(G\).

Despite being discovered over 200 years ago, \(G\) remains the least precisely measured of all fundamental physical constants. Its relative uncertainty of approximately \(2.2 \times 10^{-5}\) is many orders of magnitude larger than constants like the speed of light or the electron charge. This is because gravitational forces between laboratory objects are extraordinarily tiny and very difficult to isolate from vibrations, air currents, and electromagnetic interference. The extreme smallness of \(G\) (roughly \(6.67 \times 10^{-11}\)) explains why you don't feel gravitational attraction toward other people standing nearby — the force between two 70 kg people 1 meter apart is a mere \(3.27 \times 10^{-7}\) N, far below human perception.

Gravitational Field Strength — Concept and Formula

Rather than always thinking of gravity as an interaction between two specific objects, physicists find it more powerful to describe gravity in terms of a gravitational field. A massive object creates a gravitational field in the space surrounding it — a region where any other mass will experience a force. The gravitational field strength \(\vec{g}\) at a point in space is defined as the gravitational force per unit mass that a test mass placed at that point would experience:

The units of gravitational field strength are Newtons per kilogram (N/kg), which is dimensionally identical to meters per second squared (m/s²). This is not a coincidence — the gravitational field strength numerically equals the acceleration due to gravity at that point. On Earth's surface, \(g \approx 9.807\) m/s², meaning a freely falling object accelerates at 9.807 m/s² downward (in the absence of air resistance).

Variation of g With Altitude

Note that even at the altitude of the International Space Station (400 km), gravitational field strength is still 88.6% of the surface value. Astronauts experience "weightlessness" — more correctly called microgravity — not because gravity has disappeared, but because they are in continuous free fall. They and the station fall toward Earth at the same rate, so there is no contact force between them and the floor — creating the sensation of weightlessness.

Orbital Mechanics — How Gravity Creates Orbits

An orbit is a continuous free fall around a massive body. When an object is placed in orbit, the gravitational force acts as the centripetal force that continuously deflects its path into a curve. Crucially, the object never "falls to the ground" because it is also moving forward fast enough that Earth's surface curves away beneath it at the same rate. This is the key insight: orbital motion is horizontal free fall.

This derivation shows that orbital velocity depends only on the mass of the central body and the orbital radius — not on the mass of the orbiting object. The International Space Station (450,000 kg) and a 1 kg tennis ball at the same orbit have identical orbital velocities. For low Earth orbit (~400 km altitude), this works out to about 7.66 km/s, or roughly 27,600 km/h — fast enough to circle the globe in about 90 minutes.

Kepler's Three Laws from Newtonian Gravity

Johannes Kepler (1571–1630) discovered his three laws of planetary motion empirically from astronomical data. Newton later showed that all three laws can be derived mathematically from the gravitational force law — a triumph of theoretical physics:

• Kepler's First Law: Planetary orbits are ellipses with the Sun at one focus. (Newton: derived from the inverse-square force law.)
• Kepler's Second Law: A line from the Sun to a planet sweeps equal areas in equal times. (Newton: this is conservation of angular momentum.)
• Kepler's Third Law: \(T^2 \propto r^3\) — the square of the orbital period is proportional to the cube of the semi-major axis. (Newton: derived as \(T^2 = 4\pi^2 r^3 / GM\).)

Escape Velocity — Breaking Free from Gravity

Escape velocity is the minimum speed an object must achieve to escape a gravitational field without any further propulsion. It is derived by equating kinetic energy to the magnitude of gravitational potential energy:

The escape velocity at Earth's surface is approximately 11.19 km/s (about 40,320 km/h or 25,054 mph). Crucially, this is independent of the mass of the escaping object — a golf ball and a rocket need the same minimum speed to escape Earth's gravity (ignoring air resistance). The relationship between escape velocity and orbital velocity is elegantly simple:

The Moon's low escape velocity (2.38 km/s) directly explains why it has virtually no atmosphere: gas molecules at typical lunar temperatures easily reach this speed through thermal motion and escape into space over geologic time. Earth's higher escape velocity (11.19 km/s) retains most atmospheric gases, though light hydrogen and helium do slowly leak away from the upper atmosphere.

Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy stored in the gravitational interaction between two masses. Near Earth's surface, the familiar formula \(U = mgh\) is used, where \(h\) is the height above an arbitrary reference level. However, this is only an approximation valid for small height changes. The exact formula, valid at any distance from a massive body, is:

The negative sign is physically crucial. We define the reference level as \(U = 0\) when two objects are infinitely far apart (\(r \to \infty\)). As the objects move closer, \(U\) becomes more negative — energy is released (converted to kinetic energy). This is why comets speed up as they fall toward the Sun. To separate two gravitationally bound objects, you must input energy equal to \(|U|\) — the "binding energy."

For a satellite in a circular orbit of radius \(r\), the total mechanical energy (kinetic + potential) is:

Gravity on Other Planets — Why Your Weight Changes

Your mass (the amount of matter in your body, measured in kg) remains constant throughout the universe. But your weight (the gravitational force pulling you toward a planet, measured in Newtons) changes depending on the planet's surface gravity. Surface gravity depends on both the planet's mass and radius:

This formula reveals something surprising: a planet can be much more massive than Earth yet have similar surface gravity if it is also much larger. Saturn has 95 times Earth's mass but a surface gravity of only 10.44 m/s² (barely more than Earth's 9.807 m/s²), because its radius is about 9.5 times Earth's. Conversely, a neutron star — with 1.4 times the Sun's mass packed into a sphere only 10 km across — has an estimated surface gravity of about \(10^{12}\) m/s², one trillion times Earth's.

Using the planetary calculator above, a 70 kg person would weigh approximately: 258 N on Mercury (26 kg equivalent), 621 N on Venus (63 kg equiv.), 686 N on Earth (70 kg equiv.), 113 N on the Moon (11.5 kg equiv.), 260 N on Mars (26.5 kg equiv.), and 1,735 N on Jupiter (177 kg equiv.). Apollo astronauts, despite wearing spacesuits weighing 80 kg on Earth, found them weighing only about 13 kg on the Moon's surface — enabling their famous high jumps and bounding steps.

Einstein's General Relativity vs. Newton's Gravity

Newton's law of gravitation is astonishingly accurate for everyday and most astronomical purposes, yet it is fundamentally incomplete. Albert Einstein's General Theory of Relativity (1915) provides a deeper, more accurate description. In Einstein's framework, gravity is not a force at all — it is the curvature of four-dimensional spacetime caused by mass and energy. Objects naturally follow the straightest possible paths (called "geodesics") through curved spacetime, and what we observe as gravitational attraction is simply the result of this curved geometry.

General relativity predicts phenomena impossible to explain with Newton's gravity: the precession of Mercury's perihelion (off by 43 arcseconds per century in Newtonian theory — perfectly explained by GR), gravitational lensing of light around massive objects, gravitational time dilation (GPS satellites must account for this), gravitational waves (first directly detected by LIGO in 2015 from merging black holes 1.3 billion light-years away), and the behavior of black holes and neutron stars. For the purposes of our gravitational force calculator, Newtonian gravity is the standard approach and gives results accurate to well within measurement precision for virtually all educational and engineering applications.

Real-World Applications of Gravitational Force

• Satellite Communications & GPS: Communication satellites (GEO at ~35,786 km) and GPS satellites (~20,200 km) are precisely positioned using orbital mechanics equations derived from gravitational theory. GPS also requires relativistic corrections for both special and general relativistic effects on satellite clocks.
• Space Mission Design: NASA and ESA engineers use gravitational calculations to design gravity-assist (slingshot) maneuvers, where spacecraft accelerate by swinging around planets. The Voyager missions used this technique to reach the outer solar system and eventually interstellar space.
• Tidal Forces: The differential gravitational force (tidal force) across extended objects causes Earth's ocean tides (primarily from the Moon, secondarily from the Sun), deforms moons like Io (creating volcanism), and eventually can tear apart objects that venture too close to massive bodies (the "Roche limit").
• Geophysics & Geology: Sensitive gravimeters measure tiny variations in Earth's gravitational field to map subsurface rock density, locate mineral deposits, detect underground cavities, monitor volcanic activity, and track changes in ice sheets and groundwater.
• Structural Engineering: Every building, bridge, dam, and skyscraper is designed to withstand gravitational loads. The entire practice of structural engineering is rooted in accurate gravitational force calculations.
• Astrophysics: Gravitational lensing allows astronomers to detect and study dark matter, distant galaxies, and exoplanets. Gravitational wave astronomy has opened an entirely new observational window on the universe.
• Medical Imaging: The GRACE satellite mission measured Earth's gravitational field variations to detect changes in groundwater aquifers and ice sheet mass loss, providing crucial data on climate change.
• Particle Physics: Scientists test Newton's and Einstein's gravity at very small scales, searching for extra dimensions and quantum gravity effects that would modify the inverse-square law at sub-millimeter distances.

Gravitational Force in Academic Exams — What You Need to Know

Gravitational force is tested extensively across all major physics curricula. Here is what each major exam expects:

• SAT Physics Subject Test: Newton's law of gravitation, inverse-square law, satellite motion. Questions typically require substituting values into \(F = Gm_1m_2/r^2\) and reasoning about proportional changes.
• AP Physics 1: Gravitational force, field strength, Newton's Third Law applied to gravity, conceptual orbital mechanics.
• AP Physics C: Mechanics: Gravitational potential energy \(U = -GMm/r\), derivation of orbital mechanics formulas, Kepler's Laws, gravitational field as a vector field, the Shell Theorem.
• IB Physics HL/SL (Topic 6): Gravitational field strength, orbital speed, gravitational potential, escape velocity, and the relationship between g at surface and orbital height.
• A-Level Physics (UK): Full derivation of orbital mechanics from first principles, gravitational potential wells, satellite energy calculations, and geostationary orbits.
• University Physics (Halliday/Serway): Shell Theorem proof, tidal forces, orbital mechanics in non-circular orbits, introduction to general relativity.

Our calculators are designed to reinforce understanding at all these levels. Each result displays the formula applied with your values substituted in, helping you learn the mathematics rather than just obtain an answer.

Frequently Asked Questions About Gravitational Force