At what distance above and below the surface of the earth a body… — Solution | JEE Physics

MCQMediumJEE 2024Variation with Altitude & Depth

At what distance above and below the surface of the earth a body will have the same weight (take radius of earth as $R$ )?

A
$\sqrt{5}R - R$
B
$\frac{\sqrt{5}R - R}{2}$
C
$\frac{R}{2}$
D
$\frac{R}{2}(\sqrt{5} - 1)$

Answer

Correct answer:B

Step-by-step solution

Standard Method

Given: The radius of the earth is $R$ . We need the distance above and below the earth's surface where a body has the same weight.

Find: The value of the common distance $h$ .

For a point at height $h$ above the earth's surface, the gravitational acceleration is

g_{\text{above}} = \frac{gR^2}{(R+h)^2}

For a point at depth $h$ below the earth's surface, the gravitational acceleration is

g_{\text{below}} = g\left(1 - \frac{h}{R}\right)

Working from the extracted solution

Equating the weights at the two positions,

\frac{gR^2}{(R+h)^2} = g\left(1 - \frac{h}{R}\right)

Cancelling $g$ ,

\frac{R^2}{(R+h)^2} = 1 - \frac{h}{R}

Let

x = \frac{h}{R}

Then

\frac{1}{(1+x)^2} = 1 - x

From the extracted solution, solving gives

x = \frac{\sqrt{5}-1}{2}

Hence,

h = \frac{R}{2}(\sqrt{5}-1)

Therefore, the common distance is $\frac{R}{2}(\sqrt{5}-1)$ .

The solution explicitly states The Correct Option is B. The listed option value matching the derived result is written as the second option, $\frac{\sqrt{5}R - R}{2}$ , which is algebraically equal to $\frac{R}{2}(\sqrt{5}-1)$ . Therefore, the correct option is B.

Common mistakes

Using the same inverse-square formula for the point below the earth's surface is incorrect because inside a uniform earth, gravitational acceleration varies linearly with depth. Use $g\left(1-\frac{h}{R}\right)$ below the surface instead.
Treating the above-surface and below-surface distances as unrelated can complicate the setup. Here the question asks for the same distance above and below, so take the depth and height both as $h$ .
Missing the algebraic equivalence between $\frac{\sqrt{5}R-R}{2}$ and $\frac{R}{2}(\sqrt{5}-1)$ can lead to choosing the wrong option label. Factor out $R$ carefully before comparing options.

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Answer

Step-by-step solution

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