Question

Discuss the variation of g with
(a) altitude (b) depth

Answer 1

( a ) The acceleration due to gravity on the earth is given by

$g=\frac{GM}{R^2}i.e.GM=R^{2}g$ . . . . . . .( 1 )

The acceleration due to gravity at height 'h' from the surface of the earth is given by

$g_h=\frac{GM}{(R+h{)}^2}i.e.GM=(R+h{)}^2{g}_h$ . . . . . . ( 2 )

From ( 1 ) and ( 2 ) we have,

$\frac{g_h}{g}=\frac{R^2}{(R+h{)}^2}$

$\frac{g_h}{g}=(1-\frac{2h}{R})$

( b ) The acceleration due to gravity on the surface of the earth is given by.

$g=\frac{GM}{R^2}$ . . . .( 1 )

let 'Q' be the density of the material of the earth.

Now, mass = volume $\times$ density

$M=\frac{4}{3}\pi R^3\times \rho$

Substituting in equation ( 1 ) we get

$g=\frac{G}{R^2}\times \frac{4}{3}\pi R^3\times \rho =\frac{4}{3}\pi GR\rho$

$g=\frac{4}{3}\pi GR\rho$ . . .. .. ( 2 )

$g_d=\frac{4}{3}\pi G(R-d)\rho$ . . . .. .( 3 )

Dividing equation( 3 ) by ( 2 ) we get

$\frac{g_d}{g}=\frac{R-d}{R}=(1-\frac{d}{R})$

$g_d=g(1-\frac{d}{R})$