Variation of g with Depth

Effect of Depth on g:

Consider the earth to be a sphere of mass M, radius R and centre O. The acceleration due to gravity at any point A on the surface of the earth will be

g =

Assuming the earth to be a homogeneous sphere of average density , then its total mass will be

M = Volume × density

=

g =

g =

Effect of depth on g

Let be the acceleration due to gravity at a point B at depth d below the surface of the earth. A body at B is situated at the surface of inner solid sphere and lies inside the spherical shell of thickness d. The gravitational force of attraction on a body inside a spherical shell is always zero. Therefore, a body at B experiences gravitational force due to inner shaded sphere of radius (R − d) and mass M', where

=

=

=

=

=

=

=

=

Clearly, the acceleration due to gravity decreases with the increase in depth d. That is why the acceleration due to gravity is less in mines than that on earth's surface.

Relation between Height h and Depth d for the Same Change in g:

Acceleration due to gravity at a height h above the earth's surface,

=

Acceleration due to gravity at a depth d below the earth's surface,

=

For the same change in g, we have

=

∴ = or or

Hence the acceleration due to gravity at a height h above the earth's surface will be same as that at depth d = , below the earth's surface. But this fact holds only when .