Gravitational Force Calculator (Newton’s Law)
Compute the attraction between two masses with F = G·m₁m₂ ÷ r².
G = 6.674×10⁻¹¹ N·m²/kg². Enter masses in kg and distance in metres; use e-notation for large values (e.g. 5.972e24).
Newton’s law of universal gravitation is F = G·m₁m₂ ÷ r², with G = 6.674×10⁻¹¹ N·m²/kg². For Earth (m₁ = 5.972×10²⁴ kg) and a 70 kg person at Earth’s radius (r = 6.371×10⁶ m), F ≈ 687 N — which is exactly that person’s weight.
Newton’s law of universal gravitation
Every pair of masses attracts along the line joining them with a force proportional to the product of the masses and inversely proportional to the square of the distance between their centres: F = G·m₁m₂ ÷ r². The force is mutual — Earth pulls you down with the same strength you pull Earth up. Because gravity always attracts, this is also what the F = ma calculator reports as your weight when you stand on the ground.
G = 6.674×10⁻¹¹ N·m²/kg², m₁ & m₂ in kg, r in metres, F in newtons (N)
Worked example
How hard does Earth pull on a 70 kg person standing at sea level? Use m₁ = 5.972×10²⁴ kg, m₂ = 70 kg, and r = 6.371×10⁶ m (Earth’s mean radius).
- 1 Write Newton’s law. F = G·m₁m₂ ÷ r², with G = 6.674×10⁻¹¹ N·m²/kg².
- 2 Substitute the values. F = 6.674×10⁻¹¹ × (5.972×10²⁴) × 70 ÷ (6.371×10⁶)².
- 3 Square the distance. r² = (6.371×10⁶)² ≈ 4.059×10¹³ m².
- 4 Compute the force. F ≈ 687 N — equal to the person’s weight, since this gravitational pull is what we call weight.
Key values and the inverse-square rule
G is fixed; the force scales with 1 ÷ r², so distance dominates.
| Quantity | Value / effect | Unit |
|---|---|---|
| Gravitational constant G | 6.674×10⁻¹¹ | N·m²/kg² |
| Earth’s mass | 5.972×10²⁴ | kg |
| Earth’s mean radius | 6.371×10⁶ | m |
| Double the distance (r → 2r) | Force × ¼ (quarters) | — |
| Triple the distance (r → 3r) | Force × 1⁄9 | — |
The inverse-square law and why gravity feels one-sided
Distance dominates. Because F depends on 1 ÷ r², doubling the separation cuts the force to a quarter and tripling it to a ninth. This is why gravity reaches across space yet fades quickly — orbits, tides, and the strength of g all trace back to that r² in the denominator.
Why everyday objects barely attract. G is minuscule (6.674×10⁻¹¹), so two 1 kg masses one metre apart attract with only about 6.7×10⁻¹¹ N — far too weak to feel. Gravity only becomes obvious when one of the masses is enormous, like a planet.
Link to g. Dividing the formula by the small mass gives the field strength g = G·M ÷ r². With Earth’s mass and radius this yields g ≈ 9.81 m/s², so weight W = m·g is just F = G·m₁m₂ ÷ r² rewritten for one big body.