Why value of g is zero at the center of earth ?

 Why value of acceleration due to gravity (g) is zero at the center of the earth ?

that is, 

Fig. 1 Earth show with its mass and radius for calculation of its gravity at its center.

The acceleration due to gravity of earth (g) acting on a body is defined as the gravitational force between the earth and the body per unit mass of that body.

Consider a body of mass m with radius r and the earth has mass M and radius R. 

Suppose d be the distance between the center of earth and the center of body.. Then gravitational force between these two objects is, 

 F = GMm / d²

if the body is at the surface of the earth, then d = R+r.

Since R>>r, then d ≅ R.

so , F = GMm / R². 

Now, by definition, 

 g = F/m

So,  g = GM/R². ..................................(1)

This relation shows that g depends on M and R, because G is a fixed quantity.

At the center of Earth, R= 0 , then from (1), g =  ∞, which is not acceptable.

So, to show g = 0 at R = 0, we need to consider the density (ρ) of earth and replace M by density and volume terms. It is because mass varies  with size of object or volume of the object.

For earth,

Volume = (4/3)πR³

^{Mass = M= density x volume =  (4/3)πρR3}

^{Using value of M in (1),} 

^{g = [Gρπ(4/3R3)]/R2}

So, g = (4/3)GπρR. ............................ (2)

From (2), we see that g is directly proportional to R.

So, At the center of earth, R=0, which gives from (2), g =0.

This shows that the acceleration due gravity acting on the body of mass becomes zero if the body is, somehow, taken to the point of center of the earth.

This completes the proof. 

Video on this Link 

may help you to understand this easily.

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