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Acceleration Due to Gravity Below and Above the Surface of Earth

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Acceleration due to gravity below the surface of earth

  • To calculate acceleration due to gravity below the surface of the earth (between the surface and centre of the earth).

Density of the earth is constant throughout. Therefore,

ρ = M/ (4/3π Re3)          ....(1)

where,

M= mass of the earth

Volume of sphere= `4/3π "R"_e^2`

Re = radius of the earth.

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

Therefore density can be written as

 `rho = M_s(4/3π "R"_s^3)`      ....(2)

Comparing equation (1) and (2)

`"M"_e/"M"_s="R"_e^3/"R"_s^3      "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="R"_e^3/("R"_e-"d")^3`

`"M"_s = "M"_e(("R"_e-"d")^3)/"R"_e^3`        ...(3)

To 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.

`"F" ="GmM"_s/("R"_e-"d")^2`

`"g"="F"/"m"`  where g=acceleration due to gravity at point 'd' below the surface of the earth.

`"g"="GM"_s/("R"_e-"d")^2`

Putting the value of `"M"_s` from Eq.(3)

`="GM"_e("R"_e-"d")^3/("R"_e^3("R"_e-"d")^2)`

`="GM"_e(R_e-d)/R_e^3`

W.k.t.    `"g"="GM"_e/"R"_e^2`

`g(d)="GM"_e/R_e^3("R"_e-"d")`

`=g((R_e-d)/R_e)=g(1-d/R_e)`

Acceleration due to gravity above the surface of the earth:

  • To calculate the value of acceleration due to gravity of a point mass m at a height h above the surface of the earth.

  • Force of gravitation between the object and the earth will be 

`F(h)=("GM"_Em)/(R_E + h)^2`

The acceleration experienced by the point mass is `("F"("h"))/"m"≡"g"("h")`

`"g"(h)=("F"(h))/m="GM"_E/("R"_E + h)^2` ...(1)

This is cleary less than the value of g on the surface of earth: `"g"="GM"_E/"R"_E^2` for h<<RE, we can expand the RHS of Eq.(1)

`g(h)="GM"_E/(R_E^2 (1+h/R_E)^2)`

`=g(1+h/"R"_E)^-2`

For `"h"/"R"_E`<<1. Using binomial expression,

`"g"(h)cong"g"(1-(2h)/R_E)`

Conclusion: The value of acceleration due to gravity varies on the surface, above surface and below the surface of the earth.

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