Physics
Universal gravitation calculator
Newton's law of gravitation, gravity at a planet's surface, circular orbits, escape velocity, Kepler's third law and potential energy. Pick the relation, leave blank the field the exercise asks for, and the formula rearranges itself.
One law, from the apple to the orbits
The law of universal gravitation says that any two masses attract with a force proportional to the product of the masses and inversely proportional to the square of the distance between their centres, F = G·m₁·m₂/r². The constant G is tiny, 6.67 × 10⁻¹¹ N·m²/kg², which is why you cannot feel the pull between two people: it takes the mass of a planet to turn it into weight.
Almost everything else follows from this law. At a planet's surface the force on a mass m is m·g, so g = G·M/R², and the 9.81 m/s² in the problems is just that sum for the Earth. In a circular orbit gravity supplies the centripetal force: setting G·M·m/r² equal to m·v²/r gives v = √(G·M/r), with no satellite mass in it. The period 2πr/v, squared, is Kepler's third law: T² = 4π²·a³/(G·M), which Kepler found in Tycho Brahe's data and Newton explained.
Escape velocity comes from energy instead. A body launched from the surface escapes if its kinetic energy can pay for all of the potential energy −G·M·m/R, that is if ½·m·v² ≥ G·M·m/R: hence v = √(2·G·M/R), √2 times the speed of an orbit skimming the surface. For the Earth that is 11.2 km/s, against 7.9 for the lowest possible orbit.
Common mistakes
- Using the altitude instead of the orbit radius: the distance in the formula is from the planet's centre, so for the space station it is 6371 + 420 km, not 420.
- Forgetting the square on the distance: doubling the distance divides the force by four, not by two.
- Mixing kilometres and metres with G: the constant is in SI units, so distances must be in metres when working by hand. This page converts on its own.
- Thinking there is no gravity in orbit: at the space station's altitude it is still about 90 % of what it is on the ground. Astronauts float because they fall together with the station.
Frequently asked questions
Why are astronauts in orbit weightless?
Gravity is very much there: at the space station's altitude it is about 8.6 m/s². But the station and everything in it fall towards the Earth together, missing it all the time thanks to their sideways speed. Falling with the same acceleration, nothing presses on the floor: that is the absence of apparent weight, not of gravity.
Does escape velocity depend on the rocket's mass?
No. Both the kinetic and the potential energy are proportional to the mass of what leaves, so it cancels. A stone and a spacecraft need the same speed, 11.2 km/s from the Earth's surface. Mass matters for how much energy is needed in total, not for the speed.
How high is a geostationary satellite?
It has to go round once per sidereal day, 23 h 56 min. Kepler's third law with the Earth's mass gives a radius of about 42,164 km from the centre, which is about 35,786 km of altitude. Set a period of 0.99727 days in the Kepler relation to see it.
Why is gravitational potential energy negative?
Because zero is set at infinity, where two bodies no longer attract. As they approach, gravity does work and the potential energy drops below zero. The minus sign says the pair is bound: separating it takes energy. The m·g·h of mechanics problems is the approximation of this near the surface.
What is G and how was it measured?
6.6743 × 10⁻¹¹ N·m²/kg². Henry Cavendish first measured it in 1798 with a torsion balance, watching lead spheres attract each other. It is the least precisely known fundamental constant, precisely because gravity between laboratory masses is so weak.
How this calculation works
Law of gravitation: F = G·m₁·m₂/r², with G = 6.6743 × 10⁻¹¹ N·m²/kg². Surface gravity: g = G·M/R². Circular orbit: with gravity as the centripetal force, v = √(G·M/r), period 2πr/v and gravity at that altitude G·M/r². Escape velocity: from energy, ½·v² = G·M/R, so v = √(2·G·M/R), which is √2 times the grazing orbital speed. Kepler's third law: T² = 4π²·a³/(G·M), with mean speed 2πa/T; solving for a takes a cube root. Potential energy: U = −G·m₁·m₂/r, zero at infinity; the binding energy is −U. The sums are done in SI units: distances in km are multiplied by 1000, periods in days by 86,400.
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