Newtonian Gravity Basics: Inverse-Square Law & Formulas
Isaac Newton’s 1687 Philosophiae Naturalis Principia Mathematica unified the motion of falling apples on Earth with the orbits of the Moon and planets under a single mathematical law: the Universal Law of Gravitation.
The Universal Gravitation Formula
Gravitational Force:F = G · (m₁ · m₂) / r²
Where F is gravitational force in Newtons (N), G is the gravitational constant (6.6743 × 10⁻¹¹ N·m²/kg²), m₁, m₂ are masses in kg, and r is distance in meters.
1. The Inverse-Square Law
Gravitational force weakens rapidly with distance:
- Double the distance (2r) → Force drops to 1/4 (25%).
- Triple the distance (3r) → Force drops to 1/9 (11.1%).
- Ten times the distance (10r) → Force drops to 1/100 (1%).
| Distance Multiple | Relative Force | Orbital Context |
|---|---|---|
| 1 Earth Radius (6,371 km) | 100.0% (g = 9.81 m/s²) | Earth Sea Level |
| 1.06 Earth Radii (ISS Orbit, 400 km) | 88.8% (g = 8.71 m/s²) | Low Earth Orbit (Apparent weightlessness is due to free fall, not zero gravity) |
| 6.6 Earth Radii (35,786 km) | 2.3% (g = 0.22 m/s²) | Geostationary Orbit |
| 60.3 Earth Radii (384,400 km) | 0.027% (g = 0.0027 m/s²) | Lunar Orbit |
2. Deriving Surface Gravity (g)
By equating Newton’s Second Law (F = m · g) with Universal Gravitation (F = G · M · m / R²), the surface gravitational acceleration of any spherical planet is:
g = G · M / R²
For Earth: M = 5.972 × 10²⁴ kg, R = 6.371 × 10⁶ m → g ≈ 9.81 m/s².
Frequently Asked Questions
What is Newton’s Law of Universal Gravitation?
Formulated by Sir Isaac Newton in 1687, the law states that every point mass attracts every other point mass with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers: F = G · (m₁ · m₂) / r².
What is the value of the gravitational constant G?
The Newtonian gravitational constant G is approximately 6.67430 × 10⁻¹¹ N·m²/kg². It was first measured experimentally in the laboratory by Henry Cavendish in 1798 using a torsion balance.
Why does Newtonian gravity not allow antigravity?
In Newtonian gravity, mass (m) is always positive. Because G > 0, m₁ > 0, and m₂ > 0, the force F is strictly positive and attractive for all matter.