Question

At what depth below the surface of the earth, acceleration due to gravity $g$ will be half its value $1600\,km $ above the surface of the earth?

• A. $4.2 \times 10^{6} m$
• B. $3.19 \times 10^{6} m$
• C. $1.59 \times 10^{6} m$
• D. none of these

Answer 1

The Correct Option is A

Variation of g outside the earth $$ \begin{array}{l} g ^{1}= g \left(1-\frac{2 h }{ R }\right) \\ g ^{1}=10\left(1-\frac{2 \times 1600}{6400}\right) \\ g ^{1}=5 m / s ^{2} \end{array} $$ Variation of g inside the earth $$ g ^{11}= g \left(1-\frac{ h }{ R }\right) $$ Now, it is given that, $$ \begin{array}{l} g^{11}=\frac{g^{1}}{2} \\ 2.5 m / s ^{2}=10\left(1-\frac{ h }{6400}\right) \\ \frac{1}{4}=1-\frac{ h }{6400} \\ h =\frac{3}{4} \times 6400 Km \\ h =4.8 \times 10^{6} m \end{array} $$