The Actual Weight of the Earth

The question “How much does the Earth weigh?” may seem like a perfectly logical one to ask, but when the answer comes back as about 100g, then it is obvious that either the question or the answer is wrong.

The actual weight of the earth is based upon gravitational pull, and the earth as a body is impacted only to a tiny degree by the gravitational field of localised space. That is why in the strictest terms weight should be measured in Newtons. When people talk about how much they weigh though they are talking in reality about their mass, which is measured in kilograms or equivalent.

The simplest way of explaining this is to look at the example of apparent weightlessness on the moon. An astronaut will weigh one sixth of what they do on earth, not because they have lost five sixths of their mass, but because the gravitational pull is so much less on the moon, when compared to the earth. The astronaut’s mass has actually remained the same.

The question should therefore more likely be, “What is the mass of the Earth?” The answer is about 6 x 10^24kg, or six sequillion kilograms.  This figure is large enough to be meaningless to most people but it is fascinating to see the minds that have helped scientists to come up with this number. The calculation is based on the work of Eratosthenes, Galileo, Newton and Cavendish.

The formula for working out the mass of the earth starts out as –

F = GmM/r2 = ma

F is Gravitational force, G is the Gravitational Constant, m is the mass of a secondary object, M is the Mass of the earth, and r is the radius of the earth.

The formula is based on Newton’s Law of Gravity and also Newton’s Second Law of Motion

To work out M, the formula can be rearranged first to GM/ r2 = a, and then to M = ar2/G

Eratosthenes calculated the radius of the Earth in 230BC, calculating it to be 6.4 x 106m. Galileo calculated acceleration to be 9.8 m/sec2, whilst Henry Cavendish’s work determined the gravitational constant to be 6.67 x 10-11 m3/(kg sec2).

This therefore means that the Mass of the earth is –

M = 9.8 x (6.4 x 106)2/(6.67 x 10-11) = 6.0 x 1024 kg

Although weight and mass to most people mean the same thing, it is important to recognise that there is a huge difference between the two, and whilst the mass of the earth is extremely large, its weight is small.