Acceleration due to Gravity Formula
We know that all material objects have natural tendency to get attract towards the earth. That means earth exerts gravitational force. A force acting on body results in acceleration. And this is called acceleration due to gravity (‘g’). Acceleration is vector. The direction of gravitational force is towards the centre of earth and direction of ‘g’ is also directed towards centre of the earth i.e. vertically downwards.
Using Newton’s universal law of gravitation we can calculate the value of ‘g’.
Consider a object of mass m situated at a distance r from the centre of earth.
So force is,
F = GMm/r2 —— (1)
M = mass of earth
m = mass of object
and also
F = mg —– (2)
From equation (1) & (2)
GMm/r2 = mg —– (3)
From equation (3)
g = GM/r2 —— (4)
The object is at earth, r = R = radius of earth.
This value of g on the surface of earth is,
g = GM/R2 —— (5)
eqn (5) is the required equation for acceleration due to gravity ‘g’. The SI unit of ‘g’ is m/s2
Calculation of ‘g’:-
The mass and radius of earth are 6×1024 kg & 6.4×106m, respectively. Using equation (5)
g = 6.67×10-11×6×1024/(6.4×106)2
g = 9.8 m/s2 ——- (6)
This acceleration depends only mass M & radius R of earth and does not depends on mass of object. That means the acceleration due to gravity at a given point on earth is the same for all objects.
Numerical on Acceleration due to Gravity
A stone takes 7 sec to reach the ground from a height of 6m. What is the value of g on?
Given: Initial velocity, u = 0
S = 6m
t = 7 sec
from 2nd equation of motion, we have
S = ut + ½ gt2
6 = ½ g (7)2
g = 12/79 = 0.24 m/s2