Gravitational Force Calculator

How the Gravitational Force Calculator Works

This tool computes the gravitational attraction between any two masses using Newton’s law of universal gravitation. Enter the two masses in kilograms and the distance between their centres in metres. Because gravitational forces can be astronomically large or vanishingly small, the result is shown in scientific notation in newtons (N), with conversions to pound-force (lbf) and the resulting acceleration on the second mass.

The Formula

Newton’s law states:

F = G × m₁ × m₂ ÷ r²

where:

• F is the gravitational force (N)
• G is the universal gravitational constant, 6.674 × 10⁻¹¹ N·m²/kg²
• m₁ and m₂ are the two masses (kg)
• r is the distance between the centres of the two masses (m)

The force is an inverse-square relationship: doubling the distance quarters the force, while tripling it cuts the force to one-ninth.

Worked Example

Find the force between Earth (m₁ = 5.972 × 10²⁴ kg) and a 70 kg person standing on its surface, at a distance of Earth’s radius (r = 6.371 × 10⁶ m):

• Numerator: G · m₁ · m₂ = 6.674 × 10⁻¹¹ × 5.972 × 10²⁴ × 70 ≈ 2.790 × 10¹⁶
• Denominator: r² = (6.371 × 10⁶)² ≈ 4.059 × 10¹³
• F = 2.790 × 10¹⁶ ÷ 4.059 × 10¹³ ≈ 687 N

Dividing by the person’s mass gives 687 ÷ 70 ≈ 9.8 m/s², exactly the acceleration due to gravity at Earth’s surface — confirming that weight is just gravitational force.

How Distance Affects the Force

Because of the inverse-square term, distance has a powerful effect. The table shows the Earth-person force from the example above at multiples of Earth’s radius:

Distance (× Earth radius)	r (m)	Force (N)
1× (surface)	6.371 × 10⁶	≈ 687
2×	1.274 × 10⁷	≈ 172
3×	1.911 × 10⁷	≈ 76
10×	6.371 × 10⁷	≈ 6.9

Why Gravitational Force Matters

Newton’s law explains an enormous range of phenomena:

• Orbits — it keeps moons around planets and planets around stars, balancing the inward gravitational pull against orbital motion.
• Weight — your weight on any planet is the gravitational force between you and that body.
• Tides — differences in the Moon’s gravitational pull across Earth raise the oceans.
• Spacecraft — mission planners use this formula to calculate trajectories and gravitational assists.

Because the constant G is so small, gravity is the weakest of the fundamental forces and only becomes noticeable when one of the masses is planet-sized.