Q121 / 180 · KCET 2020 · Physics
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 $x$ ?
- A$5 \mathrm{~km}$
- Correct answerB$20 \mathrm{~km}$
- C$10 \mathrm{~km}$
- D$15 \mathrm{~km}$
Solution
The value of acceleration due to gravity at height $(h=10 \mathrm{~km})$ is $x$.
$\begin{array}{ll}
\text { i.e., } & g_{h}=x \\
\Rightarrow \quad & \frac{g}{\left(1+\frac{h}{R_{e}}\right)^{2}}=x \\
\Rightarrow \quad & g\left(1-\frac{2 h}{R_{e}}\right)=x ...(i)
\end{array}$
The value of acceleration due to gravity at depth $d$ below the surface of earth,
$\begin{array}{r}
g_{d}=x \\
g\left(1-\frac{d}{R_{e}}\right)=x ...(ii)
\end{array}$
From Eqs. (i) and (ii), we have
$\Rightarrow \quad g\left(1-\frac{2 h}{R_{e}}\right)=g\left(1-\frac{d}{R_{e}}\right)$
$\Rightarrow \quad 1-\frac{2 h}{R_{e}}=1-\frac{d}{R_{e}} \Rightarrow \frac{2 h}{R_{e}}=\frac{d}{R_{e}}$
$\Rightarrow \quad d=2 h=2 \times 10=20 \mathrm{~km}$
Hence, the correct answer is option: 'B'.