Calculate the gravitational pull between two masses with Newton’s law, plus the gravitational acceleration, circular orbital speed and escape speed from a planet.
kg
Earth’s mass is 5.972e24 kg
kg
m
Earth’s radius is 6.371e6 m
Enter the values to see the result here.
Formula
Newton’s law of universal gravitation, with G=6.6743×10−11Nm2kg−2:
F=r2Gm1m2g=r2Gm1vorb=rGm1vesc=r2Gm1
Worked example
Find the gravitational acceleration at Earth’s surface.
m1=5.972×1024 kg and r=6.371×106 m.
g=6.6743×10−11×5.972×1024/(6.371×106)2=9.82m/s2.
A 70 kg person is pulled with a force of 70×9.82=687 N, which is their weight. The Moon, at 384 400 km, pulls Earth with about 1.98×1020 N.
Common mistakes
Using the height above the surface instead of the distance from the centre. Add the planet’s radius.
Entering the Earth’s mass in tonnes or grams. It must be in kilograms.
Forgetting the exponent: 5.972 and 5.972e24 differ by 24 orders of magnitude.
Treating the result as a weight in kilograms. It is a force in newtons.
Frequently asked questions
Why is the distance measured between centres?
Newton showed that a spherical body attracts as if all its mass were at its centre. So for a person standing on Earth, the distance is Earth's radius, not the person's height above the ground.
Why do you feel weightless in orbit?
Because the orbiting body is in free fall around Earth. Gravity is still about 90 percent as strong at the height of the space station; it is the constant falling, not the absence of gravity, that causes the sensation.
What is escape speed?
The minimum speed for an object to leave a body's gravity without further propulsion, ignoring air resistance. It is the square root of 2 times the circular orbital speed at the same distance: about 11.2 km/s from Earth's surface.