At a height of 10 ,km above surface of earth, value of acceleration due to gravity is same as that o

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7.Gravitation

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At a height of $10 \,km$ above the surface of earth, the value of acceleration due to gravity is the same as that of a particular depth below the surface of earth. Assuming uniform mass density for the earth, the depth is ............. $km$

A

$1$

B

$5$

C

$10$

D

$20$

(KVPY-2020)

Solution

(d)

The value of acceleration due to gravity at a height $h$ is

$g_h=g\left(1-\frac{2 h}{R}\right)$

and at depth $d$ is

$g_d=g\left(1-\frac{d}{R}\right)$

It is given that, $g_h=g_d$

$\Rightarrow \quad g\left(1-\frac{2 h}{R}\right)=g\left(1-\frac{d}{R}\right)$

$\Rightarrow \quad \frac{2 h}{R}=\frac{d}{R}$

$\Rightarrow \quad d=2 h=2 \times 10 \quad(\because h=10 \,km )$

$=20 \,km$

Standard 11

Physics

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