Question

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