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The change in the value of g at a height h above the surface of the earth is the same as at a depth d below the surface of earth. When both d and h are much smaller than the radius of earth, then which of the following is correct ?

Option 1)
$d= 2h$

Option 2)
$d= h$

Option 3)
$d=h/2$

Option 4)
$d= 3h/2$

Answers (1)

best_answer

As we learnt in

At height,

$g_{h}=g\left ( 1-\frac{2h}{R} \right )where\; h< < R$

$or\; \; g-g_{h}=\frac{2hg}{R}\; \; or\; \; \Delta \; g_{h}=\frac{2hg}{R}..........(i)$

$At\; depth,\; g_{d}=g\left ( 1-\frac{d}{R} \right )where\; d< < R$

$or\; \; g-g_{d}=\frac{dg}{R}$

$or\; \; \Delta \; g_{d}=\frac{dg}{R}..........(ii)$

$From\; \; (i)\; and\; (ii),when\; \Delta g_{h}=\Delta g_{d}$

$\frac{2hg}{R}=\frac{dg}{R}\; or\; \; d=2h$

Value of g when h < < R -

Value of g decreases.

$g'=g\left [ 1-\frac{2h}{R} \right ]$

$R\rightarrow$ Radius of earth

$h\rightarrow$ height above surface of earth

- wherein

% decrease

$\frac{\Delta g}{g}\times 100 \; ^{0}\!\! /\! _{0}=\frac{2h}{R}\times 100 \; ^{0}\!\! /\! _{0}$

$\Delta g\rightarrow$ change in value of 'g'

$g_{h} = g |\frac{1-2h}{R}| -------- (i)$

$g_{d} = g|\frac{1-d}{R}| ------- (ii)$

$g_h=g_d$

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

$\Rightarrow d=2h$

$Ans (1)$

Option 1)

$d= 2h$

Correct

Option 2)

$d= h$

Incorrect

Option 3)

$d=h/2$

Incorrect

Option 4)

$d= 3h/2$

Incorrect

Posted by

prateek

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