Inverse-Square Law

The gravitational force F between two objects depends on only three quantities: the mass of the first object m₁, the mass of the second object m₂, and the straight-line (centre-to-centre) distance r between them. All other properties — shape, colour, composition, charge — are irrelevant to the gravitational interaction.

Experiment and theory both show that if you double m₁ while keeping everything else constant, F doubles. Similarly, doubling m₂ doubles F. Mathematically, F ∝ m₁ m₂. This makes intuitive sense: a more massive object has more 'gravitational influence', and a more massive target is pulled more strongly.

The force weakens as the objects move apart, but not simply in proportion to distance — it weakens as the square of the distance. F ∝ 1/r². Geometrically, gravity radiates outward uniformly in all directions. The same total 'flux' of gravity passes through every concentric sphere around a source. Because the surface area of a sphere is 4πr², the intensity (force per unit area) must decrease as 1/r² to keep the total constant. This geometric argument is exact and does not depend on any special assumptions about gravity.

Combining the two proportionalities, F ∝ m₁ m₂ / r². To convert this into an equation we introduce the universal gravitational constant G, giving F = G m₁ m₂ / r². The constant G is the same everywhere in the universe. Henry Cavendish first measured G experimentally in 1798 using a sensitive torsion balance, obtaining a value remarkably close to the modern accepted value of G = 6.674 × 10⁻¹¹ N m² kg⁻².

Gravity is always attractive — m₁ pulls m₂ toward itself, and by Newton's Third Law, m₂ pulls m₁ back with an equal and opposite force. The two forces are equal in magnitude, opposite in direction, and act along the line joining the centres of the two masses. There is no 'gravitational repulsion' in Newtonian gravity.

Strictly, the formula F = G m₁ m₂ / r² applies to point masses (or objects whose sizes are negligible compared to the distance r). Newton proved — and this is non-trivial — that a uniform spherical shell of mass M attracts an external point mass exactly as if all of M were concentrated at the sphere's centre. This 'Shell Theorem' means we can treat Earth (radius 6.371 × 10⁶ m) as a point mass when calculating its pull on objects above its surface, using r = distance from Earth's centre.

• F — Gravitational Force (Newton (N)): The magnitude of the attractive force each mass exerts on the other. It is always positive (attractive); direction is along the line joining the two masses.
• G — Universal Gravitational Constant (N m² kg⁻²): A fundamental constant of nature, G = 6.674 × 10⁻¹¹ N m² kg⁻². Its extremely small value explains why gravity between ordinary objects is negligible; only planetary-scale masses produce noticeable forces.
• m_1 — Mass of first object (kilogram (kg)): The inertial and gravitational mass of the first object. Mass is a scalar quantity representing the amount of matter.
• m_2 — Mass of second object (kilogram (kg)): The inertial and gravitational mass of the second object.
• r — Separation distance (metre (m)): The straight-line distance between the centres of mass of the two objects. This is squared in the denominator, giving the law its inverse-square character.

Problem. The Earth has mass M_E = 5.972 × 10²⁴ kg and radius R_E = 6.371 × 10⁶ m. A student of mass 60 kg stands on Earth's surface. (a) Calculate the gravitational force the Earth exerts on the student. (b) If the student travels to a point 2R_E above Earth's surface (i.e., a distance of 3R_E from Earth's centre), by what factor does the gravitational force change, and what is the new force?

Answer. Surface force ≈ 589 N; at 3R_E from centre the force is 1/9 of this, approximately 65.4 N.

• Gravitational Constant (G) — G is the proportionality constant that converts the inverse-square proportionality into an equation. Understanding what G represents, its numerical value (6.674 × 10⁻¹¹ N m² kg⁻²), and how Cavendish measured it is the direct follow-up to learning the law itself.
• Gravitational Field Concept — The gravitational field g⃗ at a point in space is defined as the force per unit mass at that point, derived directly from the inverse-square law. This concept reframes gravitation in terms of a field rather than action-at-a-distance.
• Acceleration Due to Gravity (g) — The value g = 9.81 m s⁻² on Earth's surface is derived by applying the inverse-square law with Earth's mass and radius: g = GM_E / R_E². This concept shows the law's most important everyday consequence.
• Variation of g — Because the inverse-square law depends on distance from Earth's centre, g varies with altitude, depth, and latitude. This topic is a direct quantitative application of the inverse-square relationship.
• Free-Fall as 1D Motion — Free fall is motion under the gravitational force described by the inverse-square law. Near Earth's surface the force is approximately constant, giving uniform acceleration g, which connects kinematics to gravitation.
• Upward & Downward Motion Under Gravity — Projectile and vertical motion problems use the gravitational acceleration g that originates from the inverse-square force law. This is the kinematic application of the gravitational force near Earth's surface.
• Distinction Between Mass & Weight — Weight is defined as the gravitational force on a mass: W = mg = GMm/r². The inverse-square law shows why weight changes with location while mass remains constant, making this distinction physically meaningful.
• Weight on Earth, Moon & Space — Using the inverse-square law with different values of M and r for Earth, Moon, and other bodies directly explains why weights differ across celestial bodies, making this a concrete application of the law.
• Apparent Weight & Free-Fall — When an object is in free fall, the gravitational force still acts according to the inverse-square law, but there is no normal reaction force. Understanding apparent weightlessness requires a firm grasp of what the gravitational force actually is.
• Gravitational Potential Energy Near Earth — The near-Earth formula U = mgh is an approximation valid when r ≈ R_E. The exact form U = −GMm/r is obtained by integrating the inverse-square force, directly linking potential energy to the law of gravitation.