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)

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

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