Newton's Law of Universal Gravitation

Multiply the two masses in kilograms, divide by the square of their separation in metres and multiply by the gravitational constant G = 6.674×10⁻¹¹ N·m²/kg². The result is the attractive force in newtons. The distance r is always measured from centre of mass to centre of mass, not from surface to surface. Because r² sits in the denominator, the force falls with the square of the distance: doubling r reduces the force to one quarter. For the weight at Earth surface the formula with Earth mass and radius gives about 686 N for a 70 kg person.

For spherically symmetric bodies it can be shown mathematically that their entire mass acts as if concentrated at the centre. This result is the shell theorem and goes back to Newton. That is why for Earth you do not insert the distance to the surface but the Earth radius of about 6371 km plus the height above the ground. A common mistake is to use only the 400 km altitude of a satellite as r. The correct value is about 6771 km. The law holds in this form only outside the bodies; inside a sphere different relations apply.

First multiply both sides by r², then divide by the force and take the square root. This gives r = √(G·m₁·m₂/F_G). The distance is therefore the square root of the product of the gravitational constant and both masses divided by the force. Use only the magnitude of the force, since gravity is always attractive and has no sign reversal like electric charges. Consistently insert SI units, kilograms for the masses and newtons for the force, so that r comes out in metres. Finally check the order of magnitude: astronomical distances lie in the range of 10⁶ to 10¹¹ metres.

F_G = m·g is the approximation for a nearly uniform field right at Earth surface, where the local factor g ≈ 9.81 m/s² is almost constant. The general law of gravitation F = G·m₁·m₂/r² instead holds for any distance and describes how the force decreases with height. In fact g follows from the law of gravitation: g = G·M_Earth/r². At Earth surface both formulas give the same value. At large altitude, for example for satellites, or for other celestial bodies with different mass and radius, you must use the general law, because g deviates significantly there.

Because the distance appears as r² in the denominator, the force follows an inverse-square law. Doubling r puts (2r)² = 4r² in the denominator, so the force drops to one quarter. Tripling the distance leaves only one ninth. Conversely the force quadruples if you halve the distance. This strong distance dependence explains why the attraction of distant objects quickly becomes negligible, while it dominates near a surface. In problems a pure ratio calculation is often enough: you do not need G and the masses, you only square the factors of the distance change and take the reciprocal.