How to Calculate Gravitational Force

Calculates the attraction between two bodies as F = G × mass₁ × mass₂ ÷ distance², where G is the gravitational constant, 6.674×10⁻¹¹. Doubling the distance cuts the force to a quarter.

Every object attracts every other. That attraction is gravitation.

F = G\dfrac{m_1 m_2}{r^2}

$G$ is the gravitational constant, $6.674 \times 10^{-11}$ , $m_1$ and $m_2$ are the two masses and $r$ is the distance between their centres. The force falls with the square of the distance, so twice as far apart means a quarter of the force.

Example

Take the Earth, of mass $5.972 \times 10^{24}$ kg and radius 6371 km, and a person of 70 kg. The force works out at $F = 687$ N, and the acceleration of the person is $687 \div 70 = 9.82$ m/s². That is nothing other than $g$ . Weight is simply gravitation between you and the Earth.

Why it is not felt between everyday objects

$G$ is extraordinarily small. Two people of 60 kg standing a metre apart attract each other with about $2.4 \times 10^{-7}$ N, far too little to notice. Only when one of the masses is planet-sized does the force become something you can feel.

Notes

The distance is measured between centres, not between surfaces. For an object on the ground, use the radius of the Earth.

The force on each body is the same size. The Earth pulls on you exactly as hard as you pull on it; only the accelerations differ, because the masses do.