Universal Gravitation

The Universal Law and Reciprocal Forces

For two point masses m₁ and m₂ separated by a center-to-center distance r, the magnitude of the mutual attractive gravitational force F is defined by Newton's equation:

F = G · (m₁ · m₂) / r²

According to Newton's third law of motion, the force exerted by m₁ on m₂ is equal in magnitude and opposite in direction to the force exerted by m₂ on m₁ (F₁₂ = −F₂₁). The gravitational potential energy stored in the two-body configuration is defined as U = −G · (m₁ · m₂) / r, taking zero potential energy at infinite separation.

Worked example: 100 kg masses at 10 meters and 20 meters

Consider two masses m₁ = 100 kg and m₂ = 100 kg separated by distance r = 10 m with gravitational scale G = 1 N·m²/kg².

1. Calculate the product of the masses: m₁ · m₂ = 100 × 100 = 10,000 kg².
2. Square the separation distance: r² = 10² = 100 m².
3. Evaluate the gravitational force: F = 1 × 10,000 / 100 = 100.0 N. Both masses experience this exact 100.0 N pull towards each other.
4. Evaluate the gravitational potential energy: U = −1 × 10,000 / 10 = −1,000.0 J.
5. Now double the distance to r = 20 m. The new distance squared is r² = 20² = 400 m².
6. Evaluate the new force: F = 1 × 10,000 / 400 = 25.0 N. Increasing the distance by a factor of 2 reduces the force by a factor of 2² = 4, verifying the inverse-square law.

As distance quadruples from 10 m to 40 m, force drops by a factor of 16 (from 100.0 N to 6.25 N), illustrating that gravitational pull diminishes rapidly over large astronomical scales.

Interactive experiment: verify inverse-square scaling

Open the Universal Gravitation module. The module provides sliders for Mass 1 (m1), Mass 2 (m2), Distance (r), and Gravitational Constant (G Scale), alongside toggles for force vectors and gravitational field lines.

1. Set Mass 1 (m1) to 100 kg, Mass 2 (m2) to 100 kg, Distance (r) to 10 m, and Gravitational Constant (G Scale) to 1. Check the readout card to confirm Force (F) reads exactly 100.00 N and both vector arrows F₁₂ and F₂₁ point towards the opposite mass with equal magnitude.
2. Drag Distance (r) from 10 m to 20 m. Confirm that the force readout decreases from 100.00 N to 25.00 N, demonstrating the factor of 4 reduction predicted by the inverse-square law.
3. At Distance (r) = 20 m, increase Mass 1 (m1) from 100 kg to 200 kg. Observe that Force (F) doubles from 25.00 N to 50.00 N, confirming linear scaling with mass.
4. Adjust Distance (r) to the boundary minimum of 2 m while keeping m₁ = 100 kg, m₂ = 100 kg, and G Scale = 1. Observe that Force (F) reaches 2500.00 N without graphical clipping or numerical overflow.

The on-screen vector arrows resize dynamically to indicate force magnitude while clamping their display length so that arrows never overlap when the masses are positioned close together.

Assumptions and physical limitations

Newton's universal gravitation law treats bodies as ideal point masses or spherically symmetric mass distributions where the shell theorem applies (the gravitational attraction outside a spherically symmetric body is identical to that of a point mass at its center). For irregular or non-spherical bodies, higher-order multipole gravitational moments must be integrated.

Furthermore, Newton's law is a classical non-relativistic approximation that assumes instantaneous action-at-a-distance. In Einstein's General Theory of Relativity, gravitational interaction propagates at the speed of light c, and gravitation is modeled as the curvature of four-dimensional spacetime caused by mass-energy density. Relativistic phenomena such as gravitational time dilation, gravitational wave radiation, and the perihelion precession of Mercury fall outside this classical two-body model.

Primary sources and educational references

OpenStax University Physics Volume 1, Newton's Law of Universal Gravitation covers the formal derivation of the inverse-square law, spherical mass distributions, and gravitational potential energy.

Cavendish, H. (1798). Experiments to Determine the Density of the Earth. Philosophical Transactions of the Royal Society of London details the first laboratory measurement of the gravitational attraction between known masses using a torsion balance.