Question

At what depth below the surface of the earth acceleration due to gravity \[\prime g\prime \] will be half of its value at \[1600{\text{ }}km\] above the surface of the earth?
A. $1.6 \times {10^6}m$
B. $2.4 \times {10^6}m$
C. $3.2 \times {10^6}m$
D. $4.8 \times {10^6}m$

Answer 1

Hint:The acceleration due to gravity is less for an object positioned at a height h than for one placed on the surface. The value of acceleration due to gravity \[\left( g \right)\] decreases as depth increases. At the poles, the value of \[g\] is higher, while at the equator, it is lower.

Complete step by step answer:
Let us consider that the value of \[g\] outside the earth is ${g^1}$ . Therefore, variation of \[g\] outside the earth is given by;
${g^1} = g\left( {1 - \dfrac{{2h}}{R}} \right)$
Here, \[g = \] Acceleration due to gravity $\left( {10\,m{s^{ - 1}}} \right)$, $h = $ Height above the earth surface $\left( {1600km} \right)$ and $R = $Radius of the earth $\left( {6400} \right)$.
Now, putting all the given values in the equation we will find the acceleration due to gravity outside the earth.
$\Rightarrow {g^1} = 10\left( {1 - \dfrac{{2 \times 1600}}{{6400}}} \right) \\
\Rightarrow {g^1} = 5m{s^{ - 2}} \\ $
Now, let us consider the variation of acceleration due to gravity \[g\] inside the earth be ${g^{11}}$. Again, putting the same formula we will find the variation inside the earth.
${g^{11}} = g\left( {1 - \dfrac{{2h}}{R}} \right)$
Now, it is said to us that, acceleration due to gravity inside the earth will be half of acceleration outside the earth.
${g^{11}} = \dfrac{{{g^1}}}{2} \\
\Rightarrow {g^{11}} = \dfrac{5}{2} = 2.5\,m{s^{ - 2}} \\ $
Now, putting all the value in the equation we will find the acceleration due to gravity inside the earth.
$2.5m{s^{ - 2}} = 10\left( {1 - \dfrac{h}{R}} \right)$
After evaluating the value we will get it as
$\dfrac{1}{4} = 1 - \dfrac{h}{{6400}}$
Now, from here we will find the $'h'$
$h = \dfrac{3}{4} \times 6400\,km \\
\therefore h = 4.8 \times {10^6}m \\ $
Hence, the correct option is D.

Note:One thing to keep in mind about gravity acceleration at different heights and depths from the earth's surface is that the value of gravity acceleration at a small height from the earth's surface drops faster than the value of gravity acceleration at a depth below the earth's surface.