Newton's Law of Gravitation F = Gm₁m₂/r²: Formula & Worked Examples

Every object in the universe with mass attracts every other object with mass. This isn't a metaphor — it's a precise, quantitative law. Newton's law of universal gravitation, published in 1687 alongside his famous three laws of motion, gave humanity the first mathematical description of the force that holds planets in orbit, keeps you on the ground, and makes apples fall from trees.

Newton's Law of Universal Gravitation

The gravitational force between two objects with masses m₁ and m₂ separated by distance r is:

Here, G is the universal gravitational constant: G ≈ 6.674 × 10⁻¹¹ N·m²/kg². Several things jump out immediately from this equation.

First, the force is always attractive — there is no gravitational repulsion. Unlike electric forces, which can push or pull depending on charge signs, gravity only pulls.

Second, the force follows an inverse-square law: doubling the distance between two objects reduces the gravitational force to one quarter. Triple the distance and the force drops to one ninth. This rapid falloff with distance means that while gravity is theoretically infinite in range, it becomes negligible at large distances. The Sun's gravity, though 27 times stronger at its surface than Earth's, decreases enough over 150 million km that Earth orbits at a manageable speed rather than spiraling inward.

Third, the force scales with the product of both masses. Earth pulls on you with the same force you pull on Earth — Newton's third law applied to gravity. But because Earth's mass is ~10²⁴ times yours, Earth's resulting acceleration (a = F/m) is utterly negligible while yours is 9.8 m/s².

Weight vs. Mass: The Critical Distinction

Mass is a fundamental property of an object — a measure of its inertia and the quantity of matter it contains. It is the same everywhere in the universe. Weight is the gravitational force exerted on an object by a nearby massive body (usually a planet). Weight depends on both the object's mass and the local gravitational field strength:

On Earth's surface, g ≈ 9.8 m/s². On the Moon, g ≈ 1.6 m/s². An astronaut with mass 80 kg weighs 784 N on Earth and only 128 N on the Moon — but their mass is 80 kg in both places. This distinction matters enormously in physics: when you apply Newton's second law (F = ma), the m is always mass, not weight.

Why Do All Objects Fall at the Same Rate?

Galileo famously demonstrated (or at least argued convincingly) that objects of different masses fall at the same rate, dropping the famous cannonball-and-musket-ball thought experiment. Newton's law explains why.

The gravitational force on an object is proportional to its mass (F = mg). The acceleration produced by that force is also inversely proportional to mass (a = F/m). The mass cancels exactly: a = mg/m = g. Every object, regardless of mass, accelerates at the same rate under gravity — 9.8 m/s² downward near Earth's surface. A bowling ball and a feather would hit the ground simultaneously in a vacuum — as demonstrated famously on the Moon by Apollo 15 astronaut David Scott in 1971. This is exactly the same independence of mass that appears in projectile motion.

Orbital Mechanics: Gravity as a Centripetal Force

An orbit is what happens when an object falls toward a planet but moves sideways fast enough that the planet's surface curves away beneath it at the same rate it falls. The gravitational force provides the centripetal force required for circular orbital motion:

This tells you the orbital speed needed for a circular orbit at radius r. At Earth's surface (ignoring atmosphere), this works out to about 7.9 km/s — roughly 28,000 km/h. The International Space Station orbits at about 400 km altitude and 7.66 km/s. GPS satellites orbit much higher at ~20,200 km and move more slowly at ~3.9 km/s. In every case, the energy analysis shows a beautiful balance: kinetic energy and gravitational potential energy sum to a constant total — the orbit is a perpetual energy exchange.

From Newton to Einstein

Newton's law of gravitation is extraordinarily accurate for everyday scales and speeds. It predicts planetary orbits, tidal forces, and satellite trajectories with exceptional precision. It breaks down only in extreme conditions: near very massive, compact objects (neutron stars, black holes) or at very high speeds. In those regimes, Einstein's general relativity — which describes gravity not as a force but as the curvature of spacetime — takes over. But for everything from a falling apple to a spacecraft trajectory, Newton's law is the tool of choice, and understanding it deeply is foundational to the physics fundamentals every student needs

Newton's Law of Universal Gravitation

Newton's Law of Universal Gravitation states that every two masses in the universe attract each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between them:

where G = 6.674 × 10⁻¹¹ N·m²·kg⁻² is the gravitational constant, m₁ and m₂ are the masses (kg), and r is the centre-to-centre distance (m).

Worked Example 1: Gravitational Force Between Earth and Moon

m_Earth = 5.97 × 10²⁴ kg, m_Moon = 7.34 × 10²² kg, r = 3.84 × 10⁸ m.

This enormous force keeps the Moon in orbit, producing a centripetal acceleration of 2.73 × 10⁻³ m/s² — about 1/3600 of g at Earth's surface.

Worked Example 2: Surface Gravity

Derive g at Earth's surface from Newton's Law. m_Earth = 5.97 × 10²⁴ kg, R_Earth = 6.371 × 10⁶ m.

The slight difference from 9.8 m/s² reflects Earth's non-uniform density and rotation.

Gravitational Field Strength

Gravitational field strength g at distance r from mass M is:

This is the acceleration due to gravity at that point. At Earth's surface: 9.8 m/s². At 400 km altitude (ISS): g = GM/(R+h)² = 9.8 × (6371/(6771))² = 9.8 × 0.886 = 8.68 m/s². The ISS is still under ~89% of surface gravity — astronauts feel weightless because they are in free fall, not because gravity is absent.

Orbital Mechanics from F = Gm₁m₂/r²

For a circular orbit, gravitational force provides centripetal force:

Orbital speed decreases with distance — further orbits are slower. For Earth: a 400 km orbit requires v = √(GM/(R+h)) = 7.67 km/s. A geostationary orbit (35,786 km altitude) requires only 3.07 km/s. Kepler's Third Law follows: T² ∝ r³.

Frequently Asked Questions

Tidal Forces and the Roche Limit

Gravity doesn't just attract — it creates differential forces across extended objects. The Moon's gravity is stronger on the near side of Earth than the far side (because of the inverse-square law). This differential — the tidal force — stretches Earth into a slightly prolate shape and is responsible for ocean tides. The tidal acceleration across an object of diameter d at distance r from mass M is approximately: a_tidal ≈ 2GMd/r³. The Roche limit is the minimum distance at which a self-gravitating body (held together by its own gravity) can survive in the tidal field of a larger body. Saturn's rings exist within Saturn's Roche limit — any moon there would be shredded by tidal forces.

Escape Velocity from F = Gm₁m₂/r²

The minimum speed to escape a planet's gravity well (ignoring atmosphere) follows from energy conservation. At the surface, the object has kinetic energy KE = ½mv² and gravitational PE = −GMm/R. At infinity, both are zero. Setting total energy = 0:

For Earth: v_esc = √(2 × 6.674 × 10⁻¹¹ × 5.97 × 10²⁴ / 6.371 × 10⁶) = 11,185 m/s ≈ 11.2 km/s. For the Moon (weaker gravity, smaller mass): 2.38 km/s. For the Sun: 617.5 km/s. Black holes have escape velocity > c — which is why nothing, not even light, can escape from within the event horizon. Use the escape velocity calculator to compute this for any body.

Common Mistakes with Gravitational Force Problems

Using diameter instead of radius. r in F = Gm₁m₂/r² is the centre-to-centre distance — for objects near Earth's surface, this is Earth's radius (~6,371 km), not diameter. Using diameter gives a force four times too small.

Confusing G and g. G = 6.674 × 10⁻¹¹ N·m²·kg⁻² is the universal gravitational constant (appears in F = Gm₁m₂/r²). g = 9.8 m/s² is the local acceleration due to gravity at Earth's surface (derived from G via g = GM/R²). G is universal; g is local and varies with location.

Forgetting the inverse-square relationship. Double the distance and the gravitational force drops to ¼, not ½. This is easy to miss in calculations — always square the distance ratio when comparing forces at different distances.

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