Gravity of Earth - Estimating g From The Law of Universal Gravitation

Estimating g From The Law of Universal Gravitation

From the law of universal gravitation, the force on a body acted upon by Earth's gravity is given by

where r is the distance between the centre of the Earth and the body (see below), and here we take m₁ to be the mass of the Earth and m₂ to be the mass of the body.

Additionally, Newton's second law, F = ma, where m is mass and a is acceleration, here tells us that

Comparing the two formulas it is seen that:

So, to find the acceleration due to gravity at sea level, substitute the values of the gravitational constant, G, the Earth's mass (in kilograms), m₁, and the Earth's radius (in metres), r, to obtain the value of g:

Note that this formula only works because of the mathematical fact that the gravity of a uniform spherical body, as measured on or above its surface, is the same as if all its mass were concentrated at a point at its centre. This is what allows us to use the Earth's radius for r.

The value obtained agrees approximately with the measured value of g. The difference may be attributed to several factors, mentioned above under "Variations":

• The Earth is not homogeneous
• The Earth is not a perfect sphere, and an average value must be used for its radius
• This calculated value of g only includes true gravity. It does not include the reduction of constraint force that we perceive as a reduction of gravity due to the rotation of Earth, and some of gravity being "used up" in providing the centripetal acceleration

There are significant uncertainties in the values of r and m₁ as used in this calculation, and the value of G is also rather difficult to measure precisely.

If G, g and r are known then a reverse calculation will give an estimate of the mass of the Earth. This method was used by Henry Cavendish.