Newton's Gravitational Force Calculator

Isaac Newton published the law of universal gravitation in his Principia (1687). It says every mass in the universe attracts every other mass, and the force depends only on the two masses and the distance between their centers.

The formula:

F = G × (m₁ × m₂) / r²

Where:

• G = 6.674 × 10⁻¹¹ N·m²/kg² (the gravitational constant)
• m₁, m₂ = the two masses in kilograms
• r = distance between their centers in meters
• F = attractive force in newtons

The constant G is one of the most fundamental and most poorly measured numbers in physics. Henry Cavendish measured it in 1798 with a torsion balance to 1% accuracy. We have done better in the 200+ years since, but only modestly. Current best value is uncertain in the 5th significant figure.

Why gravity is weak: Two 1,000 kg satellites floating 10 m apart attract each other with a force of:

F = 6.674 × 10⁻¹¹ × (1000 × 1000) / 10² = 6.674 × 10⁻⁷ N

That is about the weight of a grain of sand. Gravity is 10³⁶ times weaker than the electromagnetic force between two electrons. The reason it dominates at large scales is that gravity is always attractive (charge can cancel out, mass cannot) and the universe is enormous.

The inverse-square law: Double the distance, force drops by 4×. Triple it, drops by 9×. Halve the distance, force grows by 4×. This sharp falloff is why the Moon has a strong tidal pull on Earth’s oceans (close-ish) while distant stars exert no measurable effect.

Worked example, Earth and Moon: Mass of Earth = 5.972 × 10²⁴ kg Mass of Moon = 7.342 × 10²² kg Average distance = 3.844 × 10⁸ m

F = 6.674 × 10⁻¹¹ × (5.972 × 10²⁴ × 7.342 × 10²²) / (3.844 × 10⁸)² F = 6.674 × 10⁻¹¹ × 4.385 × 10⁴⁷ / 1.478 × 10¹⁷ F ≈ 1.98 × 10²⁰ N

That is the total gravitational pull holding the Moon in its orbit. Spread across a 7 × 10²² kg Moon, it produces a centripetal acceleration of about 2.7 × 10⁻³ m/s², exactly what is needed for the Moon’s 27.3-day orbital period.

The fudge that everyone uses: F = mg near Earth’s surface: If you set m₁ = Earth’s mass and r = Earth’s radius (6.371 × 10⁶ m), the gravitational force on any mass m₂ near the surface comes out to:

F = G·M_Earth / R_Earth² × m₂ = 9.81 × m₂

The factor in front, 9.81 m/s², is what we call g. Every “F = mg” problem in introductory physics is Newton’s gravitational law in this special case. The reason g is treated as a constant is that for any object near Earth’s surface, the distance r barely changes — even at the top of Mount Everest, r is only 0.14% larger than at sea level, so g drops by only 0.3%.

Newton’s third law and falling apples: By Newton’s third law, when Earth pulls down on a falling apple with force F, the apple pulls up on Earth with exactly the same force F. Earth does not appear to move because its acceleration is F/M_Earth, which is utterly negligible compared to the apple’s F/m_apple. But the symmetry is real: every interaction involves two equal-and-opposite forces.

Reference masses and distances:

Where Newton breaks down: This formula is an approximation. For very strong gravitational fields (near black holes, neutron stars) or for orbits where small precession effects matter (Mercury), Einstein’s general relativity is required. General relativity reduces to Newton’s law in the weak-field limit, and the corrections are typically tiny except in extreme conditions.