Acceleration due to gravity on the surface of the earth

What is the g value of earth?

The g value of earth is the culmination of different types of forces on the earth, they include the gravitational force of attraction of the earth, Coriolis force, and centrifugal force due to rotation of the earth.

Now let M be the mass of earth, R be the radius of earth and m be the mass of an object on the surface of the earth

Now according to Newton’s law of universal gravitation, the gravitational force of attraction between the earth and the object will be given by F

⇒ F = (GMm)/R2

The acceleration experienced by the object due to gravitational attraction with earth will be given by g              

⇒  F= mg

On equating both equations we get g = GM/R2 

This value of acceleration due to gravity on the surface of the earth is not even throughout, it could vary depending on the altitude from the sea level or based on the latitude. So in order to find a more accurate g value, we have to take into account 2 other forces, the centrifugal force, and Coriolis force.

 Now let’s take into account these forces one by one.

Equation for coriolis force is given by

F = 2m*(Vr X ω )

Here m is the mass of earth ω is the angular velocity of the rotation of the planet and Vr is the velocity of a moving body on earth. 

On further vector analysis, we can see that the value of Coriolis force will be given by

F = 2mω2r

This force is directed in the direction which is perpendicular to both Vr and ω. It gives a free-falling body a horizontal component for downward acceleration.

 Another factor that affects the g value is the centrifugal force due to the rotation of the earth. The equation for centrifugal force is given by,

F = – m ωX(ωXR)

Here ω is the angular velocity due to earth’s rotation, m is the mass of earth, and R is the radius of earth in the radial direction. On simplifying the equation for the centrifugal force we get  

F = mω2R directed in the negative radial direction which is opposite to the direction of the force of gravitational attraction. As a result of this phenomenon, we get the effective gravity to be less than the value g that is purely due to gravitation. Thus the effective gravity is given by the equation

g’ = g – ω2Rcos2λ

Here λ is the angle of latitude. So from this equation, we can see that the g value will be maximum near the equator and minimum near the poles.

Note :

The g value depends heavily on the altitude of the surface. This is due to mass anomalies on the surface such as mountain ranges and water bodies.

Change in the g value above and below the surface.  

Inside: While we travel below the surface of the earth the effective amount of mass of the planets that attracts us will decrease as a function of the distance from the centre of the earth.

So let d be the depth below the surface, r be the distance from the centre, and let gin be the effective gravity inside the planet.

Therefore effective mass will  be given by the equation

m = ρ * (r/R)3

Now plugging this equation into Newton’s law of universal gravitation we get the following relation.  gin = g * (r/R)

Since r is at d depth from the surface of the earth we write this equation as,

gin = g * (R-d/R)

From this equation, we can see that the g value decreases with the increase in depth.

Outside : According to Newton’s law of universal gravitation we can say that the gravitational force of attraction experienced by a mass m at distance h from the surface of  the earth is given by F = G * (Mm/(h+R)2)       

Now the mass m will experience an acceleration gout , on plugging this in the above equation we  get,

mgout = G * (Mm/(h+R)2)  ( m on the either side of the relation cancels out ) 

Thus we get the equation, gout =  G * (M/(h+R)2)

From this equation, we can see that the g value decreases as we move away from the surface of the earth and it becomes almost zero as we reach an infinite distance from the surface of earth.

Conclusion :

We saw that the g value on the surface of earth depends on the mass and radius of the earth and the angular velocity of the rotation of the planet. The standard gravity of earth is at the sea level which is given by g = 9.806 m/s2. This value is maximum near the equator and minimum near the poles.

The gravity on the surface of earth is slightly uneven because of anomalies of mass on the surface such as a mountain.

The g value of earth is maximum on the surface of earth it decreases linearly below the surface as a function of depth and decreases inversely above the surface as function of the height from the surface.