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Question
Derive an expression for acceleration due to gravity at depth ‘d’ below the earth’s surface.
Solution
Let M be mass of the earth, R be the radius of the earth
gd be gravitational acceleration at depth 'd ' from the earth surface
g be gravitational acceleration on the earth surfaces.
ρ be the density of the earth.
‘P’ be the point inside the earth at depth 'd ' from earth surfaces.
∴ CS-CP=d, ∴ CP=R-d .............(1) (since CS=R)
`g=(GM)/R^2`,
`therefore g=(G4/3piR^3rho)/R^2`
`therefore g=(4GpiRrho)/3` ....(2)
gd = acceleration due to gravity at depth 'd '
`g_d=(Gxx"Mass of the sphere with radius CP")/(CP^2)`
`thereforeg_d=(G4/3piCP^3rho)/(CP^2)`
`thereforeg_d=(4GpiCPrho)/3` .... (3)
Dividing eq. (3) by eq. (2)
`g_d/g=(CP)/R=(R-d)/R`
`therefore g_d=g(1-d/R)`
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