Question

The value of 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 $g_1$ and a depth $d$ below the surface. The ratio $\left(\frac{d}{R}\right)$ equals

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

Answer 1

The correct option is B $\frac{5}{9}$

According to question, $g_h=g_d=g_1$

$g_h=\frac{GM}{{(R+\frac{R}{2})}^2}$ and
$g_d=\frac{GM(R-d)}{R^3}$

$\frac{GM}{{\left(\frac{3R}{2}\right)}^2}=\frac{GM(R-d)}{R^3}\Rightarrow \frac{4}{9}=\frac{(R-d)}{R}$

$\Rightarrow 4R=9R-9d\Rightarrow 5R=9d$

$\therefore \frac{d}{R}=\frac{5}{9}$