Newton's Law of Universal Gravitation
Reference for Newton gravitational force F = G*m1*m2/r^2.
Calculates attraction between any two masses and explains planetary orbits and tidal forces.
The Formula
F = G × (m₁ × m₂) / r²
Every object with mass attracts every other object with mass. The force is proportional to both masses and inversely proportional to the square of the distance between them.
Variables
| Symbol | Meaning |
|---|---|
| F | Gravitational force (Newtons) |
| G | Gravitational constant (6.674 × 10⁻¹¹ N⋅m²/kg²) |
| m₁, m₂ | Masses of the two objects (kg) |
| r | Distance between the centers of the two masses (meters) |
Example 1
Find the gravitational force between Earth and the Moon
m₁ = 5.972 × 10²⁴ kg (Earth), m₂ = 7.342 × 10²² kg (Moon)
r = 3.844 × 10⁸ m
F = 6.674 × 10⁻¹¹ × (5.972 × 10²⁴ × 7.342 × 10²²) / (3.844 × 10⁸)²
F ≈ 1.98 × 10²⁰ N
Example 2
Two 1,000 kg satellites are 10 m apart in space
F = 6.674 × 10⁻¹¹ × (1000 × 1000) / 10²
F = 6.674 × 10⁻¹¹ × 10⁶ / 100
F ≈ 6.674 × 10⁻⁷ N (extremely small — gravity is a weak force)
When to Use It
Use Newton's law of gravitation when:
- Calculating the force of gravity between celestial bodies
- Understanding why objects orbit planets and stars
- Comparing gravitational forces at different distances
- Determining the gravitational field strength on other planets
Key Notes
- Newton's law: F = Gm₁m₂ / r²: G = 6.674 × 10⁻¹¹ N·m²/kg². r is the distance between the centers of mass (not the surfaces). The force is always attractive — gravity never repels.
- Inverse-square law: Doubling the distance reduces gravitational force by a factor of 4. Tripling the distance reduces it by 9. At large distances, gravity drops off quickly but never reaches exactly zero.
- Near Earth's surface, simplifies to F = mg: Setting M₁ = M_Earth and r = R_Earth gives the surface acceleration g ≈ 9.81 m/s². The formula F = mg is just Newton's law in this special case.
- Symmetry of gravitational force: By Newton's third law, the force Earth exerts on a falling apple equals the force the apple exerts on Earth — but Earth's acceleration is imperceptibly small due to its enormous mass.
- Not valid in general relativity: Newton's gravitational law is an approximation. It breaks down in extreme conditions (near black holes or neutron stars), where Einstein's general relativity must be used.