Draw graphs showing the variation of accleeration due to gravity with (a) height above the Earth's surface, (b) depth below the Earth's surface.

Explanation

Let M, R and `rho` be the mass, radius and density (assumed to be uniform) of the Earth. The gravitional acceleration at the Earth's surface is
`g = (GM)/(R^(2))" "…(1)`
where G is the universal gravitational constant.
Consider a body of mass at a depth d below the Earth's surface. Then, its distance from the Earth's centre is R-d. if `g_(d)` is the acceleration due to gravity at the depth d, the weight of the body there is `mg_(d)`.
It can be shown that the gravitational force on the body is only due to the inner sphere of radius R -d and mass M'.
`thereforemg_(d)=(GmM')/((R-d)^(2))`
`thereforeg_(d)=(GM)/((R-d)^(2))" "...(2)`
`rho = (M)/((4)/(3)piR^(3)) = (M')/((4)/(3)pi(R-d)^(3))`
`therefore M' = M((R-d)/(R))^(3)" "....(3)`
`therefore (g_(d))/(g) = (M')/(M)*(R^(2))/((R-d)^(2))`
`= (M)/(M)((R-d)/(R))^(3) *(R^(2))/((R-d)^(2))" "...`from Eq. (3)]
` = (R-d)/(R )`
`therefore g_(d) = (g(R-d))/(R) = g(1-(d)/(R))" "...(4)`
This is the required expression.