Question

Explain why the value of g is zero at the centre of the earth.

Answer 1

The value of g changes as one travels deeper into the earth. It decreases as we move from the earth’s surface to its centre. We will assume the earth as a uniformly dense sphere. Consider a particle of mass m at point P at a distance (R − d) from the earth’s centre. the gravitational force on the particle due to the earth is

F = `(GmM')/((R - d)^2)`, where M' is the mass of the sphere of radius (R − d).

Density = `"Mass"/"Volume"`

∴ Mass = volume × density

∴ M' = `4/3 pi (R - d)^3 xx M/(4/3 pi R^3) = (M(R - d)^3)/R^3`

This is because the outer spherical shell is not effective. In this case, the acceleration due to gravity is 

g = `F/m = G/((R - d)^2)`,

`(M (R - d)^3)/R^3 = (GM (R - d))/R^3`

Where M is the mass of the earth. Thus, g decreases as d increases. It is less than that at the earth’s surface `((GM)/R^2)`. At the earth’s centre, d = R ∴ g = 0.