# 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 / R**

^{2}and g = (4/3)*π*R ρ GSo 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/R^{2} ____________ (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) / ** R^{2}

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 /R^{2}______(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

earth’s surface using theanother expression of g on thedensity of theearth

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* R^{3} . ρ

(π or Pi = 22/7)**g = G.( (4/3) π R ^{3} ρ) / R^{2} **

*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)πR ρ G _____(4)

[2nd expression of g on earth’s surface]

Thus, you can derive the formulas of g.