Variation of g with Height

Every object in this universe attracts other objects with an invisible force of attraction called the force of gravitation. It is the weakest force in nature. The value of the gravitational force is directly proportional to the product of the two masses and is inversely proportional to the square of the distance between them.

History 

The theory of gravity, which started in an apple orchard, has now expanded to explain the existence of huge and mysterious black holes. The journey began in the 17th century with Galileo, who, after numerous trials and errors, came up with the first-ever quantitative calculation of the value of acceleration due to gravity. 

The equation stated that when an object is in freefall, then the distance travelled by the falling object freely without any obstruction is given by

S=ut+1/2at²   — (equation 1)

Where S = distance covered by the object in time ‘t’

u = initial velocity at the time of release

t = time taken

a = acceleration due to gravity (equals to g in case of free fall)

For freefall, we consider

U = 0. U is the initial velocity at the time of release or velocity after attaining the maximum height of a throw, which is always zero.

Therefore from equation 1, we get:

H = ½ gt²

This implies that height increases as the factor of time squared multiplied with the acceleration due to gravity.

Due to the lack of the right technology at the time, Galileo could not produce a general theory. Newton entered this debate a century later and revolutionised the world with a new gravitational force concept.

Formula to measure how g changes with height

How does the value of acceleration due to gravity varies with height?

Proof: The acceleration due to gravity is given by,

g = F/m

Where F is the force exerted by the Earth on the object of mass, m. This force depends on several factors, and a change in height is one of those factors.

Let’s assume an object is placed at a height, h, above the surface of the Earth. The forces of gravity on it because of the Earth is given by,

F = GMm / (R+h)²

where M is the mass of the Earth and R is its radius.

Thus, g = F/m = GM/(R+h)²

From this equation, we can conclude that the value of acceleration due to gravity decreases as we go up.

We can write, g = GM/(R+h)² = GM/R²(1+h/R)² = g’/(1+h/R)²

Where g’ = GM/R² is the value on the surface of the Earth.

If h << R, then through the application of Binomial expansion

g = g’/(1+ h/R)² 

≈g'( 1- 2h/R)

Conclusion 

The value of acceleration due to gravity is maximum on the surface of the Earth and further decreases as we move upward and tends to zero when h= R/2. The value of acceleration due to gravity is maximum at the poles and minimum at the equator. The value of acceleration due to gravity in various units is: 

SI : 9.8 m/ s²

CGS : 980 cm/s²

FPS : 32.2 feet/s²