The pull between two masses, from Newton's law of gravitation.
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The Coddy physics course teaches the method: you solve each problem in steps on an interactive board, and it tells you which step went wrong.
Every mass pulls on every other mass. Newton's law of universal gravitation gives the size of that pull: F = Gm₁m₂/r², where m₁ and m₂ are the two masses in kilograms, r the distance between their centres in metres, and G the gravitational constant, 6.6743 × 10⁻¹¹ N·m²/kg². The force is in newtons and acts on both masses equally, each towards the other.
G is tiny, so the pull between everyday objects is too small to feel: two 70 kg people 1 m apart attract each other with about 3.3 × 10⁻⁷ N. It takes a planet's mass to make a force you notice. The distance is squared, which makes this an inverse square law: double the distance and the force drops to a quarter, triple it and it drops to a ninth.
The calculator solves the law for whichever of F, m₁, m₂ and r you leave blank. Masses can be given in kilograms, tonnes, Earth masses (M⊕ = 5.9722 × 10²⁴ kg) or solar masses (M☉ = 1.98847 × 10³⁰ kg), and distances in metres, kilometres or astronomical units (1 AU = 1.495978707 × 10¹¹ m). Each is converted to SI first, as its own step, so you can type 1 Earth mass and 6371 km instead of the powers of ten.
One Earth mass is 5.9722 × 10²⁴ kg, one solar mass 1.98847 × 10³⁰ kg, one tonne 1000 kg. In the calculator you can choose these units instead.
Kilometres multiply by 1000; 1 AU is 1.495978707 × 10¹¹ m. For an object on a planet's surface, use the planet's radius; above it, add the height to the radius.
F = G · m₁ · m₂ / r². Multiply G by both masses, then divide by r². For the Earth and a 70 kg person: 6.6743 × 10⁻¹¹ · 5.9722 × 10²⁴ · 70 / (6.371 × 10⁶)² = 687 N.
Very large or very small forces come out in powers of ten. The pull between the Sun and the Earth is about 3.54 × 10²² N.
Values from F = Gm₁m₂/r² with G = 6.6743 × 10⁻¹¹ N·m²/kg², rounded. Each distance is centre to centre; the person is 70 kg.
| Pair | Masses | Distance r | Force |
|---|---|---|---|
| Two 1 kg masses | 1 kg and 1 kg | 1 m | 6.67 × 10⁻¹¹ N |
| Two people | 70 kg and 70 kg | 1 m | 3.27 × 10⁻⁷ N |
| Person and the Moon overhead | 70 kg and 7.342 × 10²² kg | 378,029 km | 0.0024 N |
| Person and the Sun | 70 kg and 1 M☉ | 1 AU | 0.415 N |
| Person and the Earth, on the ground | 70 kg and 1 M⊕ | 6,371 km | 687 N |
| Person and the Earth, at the height of the ISS | 70 kg and 1 M⊕ | 6,771 km | 609 N |
| Earth and the Moon | 1 M⊕ and 7.342 × 10²² kg | 384,400 km | 1.98 × 10²⁰ N |
| Sun and the Earth | 1 M☉ and 1 M⊕ | 1 AU | 3.54 × 10²² N |
m₁ = 1 M⊕, m₂ = 70 kg, r = 6371 kmConvert first: 1 M⊕ = 5.9722 × 10²⁴ kg and 6371 km = 6.371 × 10⁶ m. F = 6.6743 × 10⁻¹¹ · 5.9722 × 10²⁴ · 70 / (6.371 × 10⁶)² = 687 N. That is the person's weight: 70 · 9.81 = 686.7 N, the same within rounding.
m₁ = 1 M⊕, m₂ = 7.342 × 10²² kg, r = 384,400 kmWith r = 3.844 × 10⁸ m, F = 6.6743 × 10⁻¹¹ · 5.9722 × 10²⁴ · 7.342 × 10²² / (3.844 × 10⁸)² = 1.98 × 10²⁰ N. This pull keeps the Moon in its orbit and the Moon's pull on the Earth, the same size, raises the tides.
m₁ = 1 M☉, m₂ = 1 M⊕, r = 1 AU1 M☉ = 1.98847 × 10³⁰ kg and 1 AU = 1.495978707 × 10¹¹ m. F = 6.6743 × 10⁻¹¹ · 1.98847 × 10³⁰ · 5.9722 × 10²⁴ / (1.496 × 10¹¹)² = 3.54 × 10²² N.
m₁ = 1 M⊕, m₂ = 70 kg, F = 171.86 NLeave r blank. r = √(Gm₁m₂/F) = √(6.6743 × 10⁻¹¹ · 5.9722 × 10²⁴ · 70 / 171.86) = 1.274 × 10⁷ m, about 12,742 km: two Earth radii. A quarter of the 687 N pull at the surface, at twice the distance from the centre.
m₁ = 70 kg, m₂ = 70 kg, r = 1 mF = 6.6743 × 10⁻¹¹ · 70 · 70 / 1² = 3.27 × 10⁻⁷ N, the weight of a speck of dust of about 0.03 mg. Gravity between ordinary objects is real but far too weak to notice.
F = Gm₁m₂/r², Newton's law of universal gravitation. m₁ and m₂ are the masses in kilograms, r the distance between their centres in metres, and G = 6.6743 × 10⁻¹¹ N·m²/kg². The force comes out in newtons.
G is the constant that sets the strength of gravity everywhere: 6.6743 × 10⁻¹¹ N·m²/kg², the CODATA value the calculator uses. It is the force in newtons between two 1 kg masses 1 m apart.
Rearrange the formula: r = √(Gm₁m₂/F). Multiply G by both masses, divide by the force, then take the square root. The answer is the distance between the centres.
Rearrange for the unknown mass: m₁ = Fr² / (Gm₂). Multiply the force by the distance squared, then divide by G times the other mass. Leave that mass's box blank and the calculator does it.
A round planet or star pulls as if all its mass sat at its centre. So for an object on or above the Earth, r is the Earth's radius, 6371 km, plus the height above the ground.
The force falls with the square of the distance, because r is squared in the formula. Double the distance and the force is a quarter; triple it and the force is a ninth.
Yes, your weight is the Earth's gravitational force on you. Near the surface it is simpler to use W = mg, which gives the same answer: 70 kg times 9.81 m/s² is 687 N, as is G times the Earth's mass times 70 kg over the radius squared.
Because G is so small. Two 70 kg people 1 m apart pull on each other with about 3.3 × 10⁻⁷ N. A mass as large as a planet is needed to produce a force big enough to feel.
