If the acceleration due to gravity experienced by a point mass at a height h above the surface of earth is same as that of the acceleration due to gravity at a depth \alpha h(h\lt \lt R_e) from the earth surface. The value of \alpha will be \underline\hspace2cm . (use R_e=6400 ~km )

Gravitation PYQ — JEE Main 2022 · 29 Jul · Shift 1 · Q68 with Solution

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Correct answer: 2

For a point at height $h$ above the Earth's surface,

g_h = g\left(\frac{R_e}{R_e+h}\right)^2

Since $h \ll R_e$ , use binomial approximation:

g_h = g\left(1+\frac{h}{R_e}\right)^{-2} \approx g\left(1-\frac{2h}{R_e}\right)

At a depth $d$ from the surface,

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

Here, depth is given as

d = \alpha h

So,

g_d = g\left(1-\frac{\alpha h}{R_e}\right)

The acceleration due to gravity at height $h$ is equal to that at depth $\alpha h$ :

g_h = g_d

Substitute the approximations:

g\left(1-\frac{2h}{R_e}\right) = g\left(1-\frac{\alpha h}{R_e}\right)

Cancel $g$ :

1-\frac{2h}{R_e} = 1-\frac{\alpha h}{R_e}

\frac{2h}{R_e} = \frac{\alpha h}{R_e}

Thus,

\alpha = 2

\boxed{2}

Stored correct answer = $2$

Our derived answer also = $2$ , so they agree.