The value of acceleration due to gravity at the surface of the earth is `9.8 ms^(-2)` and the mean radius is about `6.4 xx 10^(6) m`. Assuming that we could get more soil some where, estimate how thick would an added uniform outer layer on the earth have to have the value of acceleration due to gravity `10 ms^(-2)` exactly ?

Explanation

Here, `g = 9.8 ms^(-2) , R = 6.4 xx 10^(6) m`,
`g' = 10 ms^(-2) , R' = ?`
Let `rho` be the density of earth's soil. Then
`g = (GM)/(R^(2)) = (G)/(R^(2)) xx (4)/(3) pi R^(3) rho = (4)/(3) pi GR rho`
`g' = (4)/(3) pi GR' rho`
`:. (g')/(g) = (R')/(R)` or `R' = (g')/(g) R = (10)/(9.8) xx (6.4 xx 10^(6))`
`= 6.53 xx 10^(6) m`
Thickness of the layer required on surface of earth
`= R' - R = 6.53 xx 10^(6) - 6.4 xx 10^(6)`
`= 0.13 xx 10^(6) m = 130 km`