Acceleration due to Gravity: Formula, Unit and Values

Every charge produces its own gravitational field and other masses experience the force of gravitation in the vicinity of that body. This force is only attractive in nature.

Let us assume that a body is sitting on the surface of a planet. The planet has a large mass as compared to the small body on the surface. Thus, the force exerted by the entire planet on the body is very large as compared to that exerted by the body on the planet. This force is responsible for that body being stuck on the surface. If we divide the expression for the force exerted by the planet on the body by the mass of the body, we will get the expression for acceleration.

Derivation of formula – 

According to Newton’s law of gravitation, if two bodies have mass M and m and if they are separated by a distance x then they will experiences a force of attraction according to the following formula:

F=GMm/x2

Where,

G = Universal gravitational constant = 6.6710-11 Nm2/Kg2

The above expression gives the gravitational force of attraction between two masses.

In the case of a planet, when a body is sitting on the surface, it will experience an attractive force from the planet. If mass of body is m and mass of Earth is M and radius of Earth is R then the force on that body is,

F=GMm/R2 …………….(1)

By comparing this equation with Newton’s second law, 

F=ma ……(m=mass of body)……(2)

We will get the value of gravitational acceleration of a body on the surface as,

g=a=GM/R2…………….(3)

If we substitute the values for or planet (Earth) as Mass of Earth=5.972191024 Kg, Universal gravitational constant = G = 6.6710-11 Nm2/Kg2 and radius of Earth = R = 6378.1 Km, we will get the value of g as 9.8 m/s2

Some important points regarding ‘acceleration due to gravity’ –

1. ‘Acceleration due to gravity’ on any planet changes with the altitude and depth below the surface of the planet. Generally, for Earth, it decreases with altitude and depth below the surface
2. Formula for acceleration due to gravity on any planet – 

If M is the mass of planet and R is the radius of planet then the force gravitation experienced by the mass m is –

F=GMm/R2

Acceleration of that body is,

a=GM/R2………….(1)

Equation (1) gives the expression for ‘acceleration due to gravity’ on any planet

3. Constant of Gravitation is ‘universal constant’. It means that its value is fixed in any condition and in any part of the universe. That value is approximately 6.6710-11 Nm2/Kg2

Different units – 

1. SI Unit – SI unit for acceleration due to gravity is, metre/sec2
2. CGS Unit – CGS unit is cm/sec2
3. FPS Unit – feet/sec2

Different Values – 

• On the surface of planet Earth (equator) – 

Acceleration (a) changes with latitude as,

a’=a-R2cos2

At the equator, =0°. Hence, a’=a-R2=a-0.03386

• On the surface of planet Earth (poles) – 

 On the poles, =90° Hence, a’=a=9.83 m/sec2. The value of acceleration is 9.83 m/sec2

• At the centre of Earth – 

∴ a’=a[1-(d/R)]

This equation (9) describes how acceleration caused by gravity changes with depth (d) below the surface of the planet Earth. The equation (9) tells us that the value of acceleration due to gravity (a) decreases with depth inside the Earth’s surface.

 at d=R (at the centre of Earth),

 d/R=1 ∴ a’=0

Thus acceleration at centre of Earth is 0

Conclusion

In this article, we studied about the fundamental force named ‘Gravity’, how this force affects the bodies on our planet. Also, we calculated the acceleration of each body felt due to the force of gravitation. This acceleration is dependent only on the mass of the planet and not the body. The value of this acceleration is in general 9.8 m/sec2. But, this value changes with altitude, depth and latitude. Different units of acceleration caused by gravity are also studied.