Last updated on April 16th, 2021 at 09:44 am
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.
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]
