Question

If 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 it. When both d and h are much smaller than the radius of earth, then

• A. d = h
• B. d = h²
• C. d = h/2
• D. d=2h

Answer 1

The correct option is D

d=2h

Step1: Given

Height and depth at which acceleration due to gravity is equal.

Step2: Variation of $g$ with height

$g_h=g\left(1-\frac{2h}{R}\right)[where{g}_h=gravityatheighth,h=height,R=radius]$

Step3: Variation of $g$ with depth

$g_d=g\left(1-\frac{d}{R}\right)[where{g}_d=gravityatdepthd,d=depth,R=radius]$

Step4: Solving the above two equations

$$\begin{gathered} g_h=g\left(1-\frac{2h}{R}\right) \\ {g}_h=g-\frac{2hg}{R} \\ g-g_h=\frac{2hg}{R}........equation\left(i\right) \\ {g}_d=g\left(1-\frac{d}{R}\right) \\ {g}_d=g-\frac{gd}{R} \\ g-g_d=\frac{gd}{R}........equation\left(ii\right) \end{gathered}$$

We'll equate equations (i) and (ii) as it is asked at what both will be equal.

$$\begin{gathered} g-g_h=g-g_d \\ \frac{2hg}{R}=\frac{gd}{R} \\ 2h=d \end{gathered}$$

Hence, option D is the correct answer.