# Derive the formula of Acceleration due to gravity on the earth’s surface | derive the formula for g

In this post, we will list down and derive the formula of Acceleration due to gravity on the earth’s surface. In other words, we will derive the formula or equation of g on the earth’s surface.

The 2 formulas we will derve for g (Acceleration due to gravity on the earth’s surface) are: g = GM / R2 and g = (4/3) π R ρ G

## Derive the formula of Acceleration due to gravity on the earth’s surface | derive the formula for g on the earth

In this section let’s find out the formula or equation of g on the earth’s surface.

Force of gravity (gravitational force value due to earth) acting on a body of mass m on the earth surface is expressed as:
F = GMm/R2 ____________ (1)

Here R is the radius of the earth (considering it a homogenous sphere)
and M here is the total mass of earth concentrated at its center. G is the gravitational constant.

Now, from Newton’s 2nd Law of Motion,
F = mg.___________________(2)

Just to recapitulate, as a body falls downwards it is continuously acted upon by a force of gravity. The body thus possesses an acceleration, called Acceleration Due to gravity(g).

From equation 1 and 2 we can write

mg = (GMm) / R2

Now, we get the equation or formula of g on earth’s surface as follows:

Acceleration due to gravity on the earth’s surface is represented as
g = GM / R2 ______(3)
[expression of g on earth’s surface]

Let’s find out another expression of g on the earth’s surface using the density of the earth

If the mean density of the earth is ρ then the mass of the earth is expressed as:
M = volume X density = (4/3) π R3 . ρ
(π or Pi = 22/7)
g = G.( (4/3) π R3 ρ) / R2

So we get the second expression or formula for g on earth’s surface:

Acceleration due to gravity of earth is represented as
g = (4/3) π R ρ G _____(4)
[2nd expression of g on earth’s surface]

Derive the formula of Acceleration due to gravity on the earth’s surface | derive the formula for g
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