Explain the variation of g with depth from the Earth's surface.

1.	Explain the variation of g with depth from the Earth's surface.
Answer» Solution :VARIATION of g with depth: Consider a particle of mass m which is in a deep mine on the Earth. (Example: coal mines - in NEYVELI). Assume the depth of the mine as d. To CALCULATE g' at a depth d, consider the follwoing points. The part of the Earth which is above the radius `(R_(E) - d)` do not contirbute to the ACCELERATION. The result is proved earlier and is given as `g^(') = (GM')/((R_(E) - d)^(2))` Here M' is the mass of the Earth of radius `(R_(E) - d)` Assuming the density of Earth `rho` to be constant, `rho = M/V` where M is the mass of the Earth and V its volume, Thys, `rho = (M')/(V') , (M')/(V') = M/V and M' = M/V V'` `M' = (M/4/3 pi R_(E)^(3)))(4/3 pi (R_(E) - d)^(3))` `M' = M/(R_E^3) (R_E - d)^(3)` `g' = G M/(R_E^3) (R_E - d)^3 cdot 1/((R_E - d)^(2))` `g' = GM(R_E(1-d/(R_E)))/(R_E^3) = GM ((1 - d/(R_E)))/(R_E^2)` THUS `g' = g(1 - d/ (R_E))` Here also g' < g. As depth increases, g' decreases.

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