Introduction

We all have experienced that all objects tend to fall on the ground when thrown in the air. This phenomenon of any object getting attracted to the earth is known as gravity. The change in velocity while the object is getting attracted to the earth is known as acceleration due to gravity.

The value of acceleration due to gravity, g,  on earth surface is 9.8 m/s2. The gravity has magnitude and direction, and it is a vector quantity. This is always constant for all bodies.

Acceleration due to gravity 

The earth is a solid spherical object, which attracts other objects due to gravitational force. Due to the gravitational force of the earth, the motion of objects is influenced. The change in velocity with which the object is pulled towards the earth can be defined as the acceleration due to gravity. 

Gravity

Sir Issac Newton discovered gravity in 1687. When he was sitting under an apple tree and observed an apple falling from the tree.  This led him to question the force behind the fall, and finally, he gave Newton’s law of gravitation energy.

Gravity is the universal force of attraction in between all matter. It is the weakest attraction force and is not involved in defining the internal properties of matter. But, due to its universal long reach, it plays a crucial role in controlling the trajectories of the bodies in our solar system. 

Gravity creates the same effect on all objects. So, whether it is an iron nail or a feather, both fall at the same pace. The difference in speed is because of air resistance. If both the objects are dropped in a vacuum, both the objects will touch the ground at the same time.

Acceleration due to gravity formula

After understanding gravity, let us look at the formula to define acceleration due to gravity.

The force on the body, F = mg ——— (1)

As per the universal law of gravitation, F = GmM/(r+h)2

Considering the height to be negligible in front of the radius of the earth, the equation can be modified to

F = GmM/r2 ———- (2)

Now equating equations (1) and (2)

mg = GmM/r2

g = GM/r2

The  acceleration due to gravity formula, g = GM/r2

where F =force on the body

m =mass of the body

g =acceleration due to Gravity

G =Universal gravitational constant (6.67×10-11 Nm2/kg2)

m =mass of the particular object

M =mass of the Earth

r =radius of the Earth

h =height from which the body is attracted towards the earth

The acceleration due to the gravity formula provides certain inputs.

• The value of acceleration due to gravity remains the same for all objects irrespective of their mass.
• The value on earth is solely dependent on the mass of the earth, and not the mass of the object.

Acceleration due to gravity on the surface of our earth

As discussed above, regarding the earth made up of imaginary concentric spherical shells, let us understand the acceleration due to gravity on the surface of the earth.

Considering the earth as a uniform solid sphere, so density of the earth can be given as

ρ = M/V

Here M is the mass of earth and V is the volume of earth. Radius of earth is given by R.

ρ = M/[4/3 x πR3]

M = ρ x [4/3 x πR3] ———— (1)

We have known that, g = GM/r2 ———— (2)

Now substitute, the value of M deducted in (1) in the (2)

g =  {G x ρ x [4/3 x πR3]}/r2

g = 4/3 [π ρRG] 

The above equation is the acceleration due to the gravity formula on the surface of the earth.

The value of acceleration due to gravity is affected by certain factors. Let us look at these factors.

• Depth from the surface of the Earth.
• Altitude from the surface of the Earth.
• The Rotational motion of the Earth.
• The shape of the Earth.

Relationship of gravity with the height

The value of acceleration due to gravity at a given height, is mathematically described as 

gH = g(1-2h/R)

Let us look at the steps to deduct the equation mathematically.

F = GMm/(R+h)2 ————- (1)

mg = GMm/(R+h)2 

gH = GM/[R2(1+ h/R)2] ——————- (2)

g = GM/R2 ——————- (3)

Divide (3) by (2), we get

gH = g(1+h/R)2 ———— (4)

So, the value of acceleration due to gravity decreases with the increase in height, and the value becomes zero at a certain point from the surface of the earth. So, considering it for (4), we get

gH = g(1-2h/R)

Relationship of gravity with the depth

The acceleration due to gravity on the surface of the earth is mathematically defined as 

g = 4/3 [π ρRG]

From a particular depth d from the earth’s surface,  the  acceleration due to gravity formula is given as 

gd = g (R-d)/R

Relationship of gravity due to shape of the earth

The acceleration due to gravity is always inversely proportional to the radius of the earth. The gravity differs as per the shape of the earth and can be defined from the below-mentioned formula.

gp/ge = R2e/R2p

gp  =acceleration due to gravity at the poles

ge =acceleration due to gravity at the equator

Re =Radius of the Earth near the equator

Rp =Radius of the Earth near the poles

So, according to the above formula, gravity is more at the equator and less near the poles.

Conclusion

Gravity plays a crucial role in our everyday lives. We discussed how the acceleration due to gravity differs because of the object placed at a certain height, how it changes as depth changes, and how the value of acceleration due to gravity changes in places near the poles and equator.