Question

How much deep inside the earth (radius $R$ ) should a man go, so that his weight becomes one-fourth of that on the earth's surface?

• A. $\frac{R}{2}$
• B. $\frac{3 R}{4}$
• C. $\frac{R}{4}$
• D. $\frac{R}{3}$

Answer 1

Answer: (B)

Suppose depth inside the earth is $h$

As we know that, inside the earth
$g_{i} \propto\left(\frac{R-h}{R^{3}}\right)$ ...(i)
On the surface of the earth
$g_{s} \propto \frac{1}{R^{2}}$ ... (ii)
From Eqs. (i) and (ii), we get
$\frac{g_{i}}{g_{s}}=\frac{(R-h) R^{2}}{R^{3}}=\frac{R-h}{R}$
$\Rightarrow \frac{1}{4}=\frac{R-h}{R}$
$\Rightarrow R=4 R-4 h$
$\Rightarrow 3 R=4 h$
$\Rightarrow h=\frac{3}{4} R$