Question

The acceleration due to gravity at a height 1 km above the earth is the same as at a depth d below the surface of earth. Then:
\[\begin{gathered}
  A.\;\;\;d{\text{ }} = {\text{ }}2{\text{ }}km \\
  B.\;\;\;\;d{\text{ }} = {\text{ }}\dfrac{1}{2}{\text{ }}km \\
  C.\;\;\;\;d{\text{ }} = {\text{ 1 }}km \\
  D.\;\;\;d{\text{ }} = {\text{ }}\dfrac{3}{2}{\text{ }}km \\
\end{gathered} \]

Answer 1

Hint: The acceleration due to gravity at a Height of H m is given by the formula:
\[{g_{height}} = g(1 - \dfrac{{2H}}{R})\] Equation 1
Similarly, the acceleration due to gravity at a Depth of D m is given by the formula:
\[{g_{depth}} = g(1 - \dfrac{D}{R})\] Equation 2
We simply have to equate the above two equations.


Complete step by step solution:
The acceleration due to gravity at a Height of H m is given by the formula:
\[{g_{height}} = g(1 - \dfrac{{2H}}{R})\]
Where,
g height is the acceleration due to gravity at a height of H m from Earth’s surface
g is the acceleration due to gravity at the surface of the Earth in ms-2
R is the radius of Earth in meters
Similarly, the acceleration due to gravity at a Depth of D m is given by the formula:
\[{g_{depth}} = g(1 - \dfrac{D}{R})\]
Where,
g depth is the acceleration due to gravity at a height of H m from Earth’s surface
Now, according to question, g height must be numerically equal to g depth. So, we will equate the Right-Hand Side (RHS) of equations 1 and 2 respectively.
We get,
\[ = > g(1 - \dfrac{{2H}}{R}) = g(1 - \dfrac{D}{R})\]
Here, we can observe in the above equation that Right – Hand Side and Left – Hand Side are similar.
Hence, we can conclude that,
$2H = D$
H is given as 1 km in the question
Inserting the value of H in above equation,
We get,
$D = 2km$

Hence Option (A) is correct.



Note: In such questions, care must be taken that we perform all calculations in SI units. Also, there is a high chance of committing silly mistakes in the calculations (since this question was calculation-intensive). Further, we need to know the value of the Radius of Earth since it was not provided in the question.