The value of acceleration due to gravity at a height of \(10 \mathrm{~km}\) from the surface of earth is \(x\). At what depth inside the earth is the value of the acceleration due to gravity has the same value \(x\) ?

The value of acceleration due to gravity at height \((h=10 \mathrm{~km})\) is \(x\).
\(\text { i.e., } g_{h}=x \)
\( \Rightarrow \frac{g}{\left(1+\frac{h}{R_{e}}\right)^{2}}=x \)
\( \Rightarrow g\left(1-\frac{2 h}{R_{e}}\right)=x\quad ...(i)\)
The value of acceleration due to gravity at depth \(d\) below the surface of earth,
\({r} g_{d}=x\)
\(g\left(1-\frac{d}{R_{e}}\right)=x\quad ...(ii)\)
From Eqs. (i) and (ii), we have
\(\Rightarrow g\left(1-\frac{2 h}{R_{e}}\right)=g\left(1-\frac{d}{R_{e}}\right)\)
\(\Rightarrow 1-\frac{2 h}{R_{e}}=1-\frac{d}{R_{e}} \Rightarrow \frac{2 h}{R_{e}}=\frac{d}{R_{e}}\)
\(\Rightarrow d=2 h=2 \times 10=20 \mathrm{~km}\)