Question

In the reported figure of earth, the value of acceleration due to gravity is same at point $A$ and $C$ but it is smaller than that of its value at point $B$ (surface of the earth). The value of $OA : AB $ will be $x: y$. The value of $x$ is $\ldots \ldots \ldots$

Answer 1

$g_{ A }=\frac{ GM ( r )}{ R ^{3}}$
$g _{ C }=\frac{ GM }{\left( R +\frac{ R }{2}\right)^{2}}$
$g_A = g_C$
$\frac{ r }{ R ^{3}}=\frac{1}{\frac{9}{4} R ^{2}} \Rightarrow r =\frac{4 R }{9}$
so $OA =\frac{4 R }{9} ; AB = R - r =\frac{5 R }{9}$
$OA : AB =4: 5$