Acceleration due to gravity below the surface of earth

The acceleration due to gravity at a point Q on the surface of the earth is different from the acceleration due to gravity at a point P inside the earth.

As said in the video, consider a point P below the surface of the earth as a distance d. Draw a circle around the point P so that we get the orange sphere as ahown in the diagram.

• To calculate acceleration due to gravity below the surface of the earth (between the surface and centre of the earth), the main fact to be used is that theDensity of the earth is constant throughout.
• As density is related to mass and volume, the next step is to find out the mass and volume of the blue sphere and the orange sphere; Volume of a sphere is 4/3 π r³ . Similarly, the blue circle is the earth as a whole having mass M_e the subscript e indicates earth; Similarly, M_s represents the mass of the orange sphere
• Therefore for earth,

                               ρ = M_e / (4/3π R_e³) ---------------------------------------equation(1)

where

M_e = mass of the earth

Volume of sphere= 4/3π R_e³

R_e = radius of the earth.

• As entire mass is concentrated at the centre of the earth.

Therefore density considering the orange sphere can be written as

ρ = M_s / (4/3π R_s³)   ---------------------------equation (2)

• Comparing equation (1) and (2)

M_e/M_s = R_e³_/R_s³ where R_s = (R_e-d) ³

d= distance of the body form the centre to the surface of the earth.

Therefore, M_e/M_s = Re³/(R_e-d)³

     M_s = M_e(R_e-d)³/R_e³ ------------------------------equation(3)

• Next we need to consider and calculate Gravitational force (F) between earth and point mass m at a depth d below the surface of the earth.

             Above figure shows the value of g at a depth d. In this case only the smaller sphere of radius (R_e–d) contributes to g(d)

F= G m M_s/(R_e-d)²

g(d) =F/m where g= acceleration due to gravity at point d below the surface of the earth.

g(d) = GM_s/(R_e-d)²  --------------------------------------equation (4)

• Putting the value of M_sfrom equation (3) on equation (4)

= GM_e (R_e-d)³/R_e³(R_e-d)²

=GM_e (R_e-d)/R_e³

• We know acceleration due to gravity on the surface of the earth, g=GM_e/R_e² ; Putting the same in equation (4) which is g(d) = GM_e/ R_e² (1-d/R_e)

Finally, we get

g (d) =  g(1-d/R_e)