The value of the acceleration due to gravity is at a height

The value of the acceleration due to gravity is $g_{1}$ at a height $h=\frac{R}{2}$ ( $R=$ radius of the earth) from the surface of the earth. It is again equal to $g_{1}$ at a depth $d$ below the suface of the earth. The ratio $\left ( \frac{d}{R} \right )$ equals :
Option: 1 $\frac{4}{9}$
Option: 2 $\frac{5}{9}$
Option: 3 $\frac{1}{3}$
Option: 4 $\frac{7}{9}$

Answers (1)

D Deependra Verma

$g_{1}=\frac{G M}{\left(R+\frac{R}{2}\right)^{2}} \ldots (1)$

$\begin{aligned} &g_{2}=\frac{G M(R-d)}{R^{3}} \ldots (2) \\ &g_{1}=g_{2}\\ &\frac{\mathrm{GM}}{\left(\frac{3 \mathrm{R}}{2}\right)^{2}}=\frac{\operatorname{GM}(\mathrm{R}-\mathrm{d})}{\mathrm{R}^{3}}\\ &\Rightarrow \frac{4}{9}=\frac{(\mathrm{R}-\mathrm{d})}{\mathrm{R}}\\ &4 R=9 R-9 d\\ &5 \mathrm{R}=9 \mathrm{~d} \Rightarrow \frac{\mathrm{d}}{\mathrm{R}}=\frac{5}{9} \end{aligned}$

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The value of the acceleration due to gravity is at a height ( radius of the earth) from the surface of the earth. It is again equal to at a depth below the suface of the earth. The ratio equals : Option: 1 Option: 2 Option: 3 Option: 4

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