Gravitational Constant (G)

When Newton proposed that every object in the universe attracts every other object, he needed a number to make that idea quantitative — a number that converts masses and distances into an actual force. That number is the gravitational constant, G. It appears in Newton's Universal Law of Gravitation and sets the overall strength of gravity throughout the universe. Without G, the law would tell us that gravity depends on mass and distance but could not tell us how strong it actually is. G is what anchors the equation to physical reality. G has a very precisely measured value: approximately 6.674 × 10⁻¹¹ N m² kg⁻². The tiny size of this number is deeply meaningful — it tells us that gravity is by far the weakest of the four fundamental forces. Two 1 kg masses placed 1 metre apart attract each other with a force of only 6.674 × 10⁻¹¹ newtons, which is far too small to feel. Yet because gravity has infinite range and always attracts (never repels), it dominates on astronomical scales where masses are enormous, shaping galaxies and governing planetary orbits. G is a universal constant, meaning it has the same value everywhere in the universe and at all times — from the surface of the Earth to the edge of the observable cosmos. This universality was a revolutionary idea: the same law, with the same constant, governs the fall of an apple and the orbit of the Moon. Measuring G precisely remains one of the most difficult experiments in physics, and our knowledge of it is less precise than almost any other fundamental constant.

Key takeaways

G as a proportionality constant
Deriving the units of G
Why G is so small
Measuring G: the Cavendish experiment
Physical significance: G and the scale of the universe

G as a proportionality constant

Newton's law states that the gravitational force between two masses is proportional to the product of the masses and inversely proportional to the square of the distance between them: F ∝ m₁m₂/r². To turn a proportionality into an equation, we need a constant of proportionality. G is exactly that constant: F = G·m₁m₂/r². Think of G as a conversion factor, similar to how speed converts time into distance. It converts the combination (kg × kg / m²) into newtons of force.

Deriving the units of G

We can find the units of G by rearranging the equation: G = F·r²/(m₁m₂). Substituting SI units: G = N·m²/kg² = (kg·m·s⁻²)·m²/kg² = m³·kg⁻¹·s⁻². Both expressions — N m² kg⁻² and m³ kg⁻¹ s⁻² — are correct and used in different contexts. The unit N m² kg⁻² is more intuitive when working with forces, while m³ kg⁻¹ s⁻² appears naturally in orbital mechanics. Knowing these units allows you to check dimensional consistency in any gravitation problem.

Why G is so small

The value 6.674 × 10⁻¹¹ N m² kg⁻² reflects the genuine weakness of gravity at the human scale. Compare it with the electrostatic constant k ≈ 8.99 × 10⁹ N m² C⁻²: the electromagnetic force between typical charged particles is roughly 10³⁶ times stronger than the gravitational force between the same particles. Gravity only 'wins' on planetary and stellar scales because planets contain roughly 10²⁵ kg of matter, all of which contributes positively to gravitational attraction, whereas electric charges tend to cancel out in bulk matter.

Measuring G: the Cavendish experiment

G cannot be derived from other constants — it must be measured experimentally. Henry Cavendish achieved this in 1798 using a torsion balance: two small lead spheres were suspended on a rod by a thin wire. Two large lead spheres were placed nearby, and the gravitational attraction twisted the wire by a tiny, measurable angle. By knowing the masses, distances, and the wire's restoring force, Cavendish calculated G. His result was remarkably accurate — within about 1% of the modern value. Despite 225 years of improvements, G remains the least precisely known fundamental constant, currently known to about 22 parts per million (compared to parts per billion for most others).

Physical significance: G and the scale of the universe

G determines the minimum mass a celestial body needs to pull itself into a sphere (a concept called hydrostatic equilibrium), the radius of black holes (via the Schwarzschild radius r_s = 2GM/c²), and the rate at which the universe expands and structure forms. It also links to the concept of gravitational potential energy: U = −Gm₁m₂/r. If G were even slightly larger, stars would burn through their fuel far faster and life as we know it would be impossible — a fact that features prominently in discussions of fine-tuning in cosmology.

The Key Equation

F = \frac{G m_1 m_2}{r^2}

F — Gravitational force (Newton (N)): The attractive force acting on each mass directed towards the other.
G — Gravitational constant (N m² kg⁻²): Universal constant equal to 6.674 × 10⁻¹¹ N m² kg⁻²; sets the strength of gravity.
m_1, m_2 — Masses (Kilogram (kg)): The two masses attracting each other gravitationally.
r — Separation distance (Metre (m)): The distance between the centres of mass of the two objects.

Worked Example

Problem. Calculate the gravitational force between the Earth (mass = 5.97 × 10²⁴ kg) and the Moon (mass = 7.35 × 10²² kg), given that the average distance between their centres is 3.84 × 10⁸ m. Use G = 6.674 × 10⁻¹¹ N m² kg⁻².

F \approx 1.98 \times 10^{20} \text{ N}

Answer. The gravitational force between the Earth and the Moon is approximately 1.98 × 10²⁰ N — roughly 200 quadrillion newtons, which is what keeps the Moon in its orbit.

Figure

Diagram

Origins · 1 minHenry Cavendish and how this was discovered

Henry Cavendish

1731–1810 · British

Cavendish was the first person to measure the gravitational constant G experimentally. In 1798, using a brilliantly designed torsion balance apparatus, he measured the tiny gravitational attraction between lead spheres and determined the value of G — effectively 'weighing the Earth' by calculating its mean density. His result has stood the test of time and remained close to the modern accepted value for over two centuries. Cavendish was extraordinarily reclusive — he communicated with his female servants only by written note and reportedly spoke to almost nobody. Yet he was one of the most gifted experimental scientists of his age, also discovering hydrogen and anticipating Ohm's Law decades before Ohm.

Universal Law of Gravitation — G appears directly in Newton's Universal Law of Gravitation — understanding the law is essential context for understanding the role and significance of G.
Acceleration Due to Gravity (g) — The surface gravitational acceleration g is derived from G, the Earth's mass, and its radius: g = GM/R². Connecting G to g bridges the universal constant to everyday experience.
Gravitational Potential Energy — The formula for gravitational potential energy U = −Gm₁m₂/r also contains G, and understanding G helps explain why potential energy is negative and approaches zero at infinite separation.
Orbital Motion and Kepler's Laws — G determines orbital speeds and periods. Kepler's third law in its Newtonian form T² = (4π²/GM)r³ explicitly contains G, making it central to all orbital calculations.