Question

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 surface of the earth. The ratio $\left(\frac{ d }{ R }\right)$ equals :

• A. $\frac{7}{9}$
• B. $\frac{4}{9}$
• C. $\frac{1}{3}$
• D. $\frac{5}{9}$

Answer 1

Answer: (D)

$g _{2}=\frac{ GM ( R - d )}{ R ^{3}} \dots (2)$
$g_{1}=g_{2}$
$\frac{ GM }{\left(\frac{3 R }{2}\right)^{2}}=\frac{ GM ( R - d )}{ R ^{3}}$
$\Rightarrow \frac{4}{9}=\frac{( R - d )}{ R }$
$4 R =9 R -9 d$
$5R=9d$
$\Rightarrow \frac{ d }{ R }=\frac{5}{9}$