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

Last updated on April 19th, 2023 at 04:07 pm

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
So let’s start with the step-by-step derivation process.

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.

derive the formula for g on the earth in terms of the mass of the earth

The force of gravity (gravitational force value due to earth) acting on a body of mass m on the earth’s 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 the earth concentrated at its center. G is the gravitational constant.

Now, from Newton’s 2^nd 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 equations 1 and 2, we can write

mg = (GMm) / R²

Now, we get the equation or formula of g on the 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 in terms of the mass of the earth]

derive the formula for g on the earth in terms of the density of the earth

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 the earth’s surface:

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

Thus, you can derive the formulas of g.