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

Answer: (B)

Given : Height above the surface of earth
$h=1600\, km =1.6 \times 10^{6} m$
The acceleration due to gravity at a depth $d$, is given by,
$g'=g\left(1-\frac{d}{R}\right)$
Also, the value of $g$ at a height $h$ is
$g'' =g\left(1-\frac{2 h}{R}\right)$
$=g\left(1-\frac{2 \times 1.6 \times 10^{6}}{6.38 \times 10^{6}}\right)=0.5\, g$
Since, $g'$ must be equal to $\frac{g''}{2}$
Hence, $g\left(1-\frac{d}{R}\right) =0.5\, g$
$1-\frac{d}{R} =0.5$
or $d=0.5\, R =0.5 \times 6.38 \times 10^{6}$
$=3.19 \times 10^{6} m$