Question

Derive the expression for the acceleration due to gravity at a depth d under the earth of radius r and mass m

Answer 1

Let us consider, mass of the earth be M and its radius= R

We know, density= mass/volume

So mass M= ρ×V

                 M  =ρ× (4/3)π³----- (1)

               

Acceleration due to gravity, g= GM/R²

From (1), g= {G×(4/3)πR³ρ}/R²= (4/3)GπRρ----- (2)

If a body is kept under the earth surface at depth (say) d, The acceleration due to gravity on that body will be

g'= GM'/(R-d)²

where M' is the mass of the inner solid sphere of mass at that depth

So, g'= {G/(R-d)²}×M'= {G/(R-d)²}× (4/3)π(R-d)³ρ

  g'= (4/3)Gπ(R-d)ρ----- (3)

Now, (3)÷(2)=> g'/g= {(4/3)πG(R-d)ρ}/ {(4/3)GRρ}

                              = (R-d)/R = 1- (d/R)

 g'= g{1-(d/R)}