JEE Main · 2024 · 29 Jan 2024 · Shift 1 · Gravitation

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

(A) $5 R - R$
(B) $\frac{3 R - R}{2}$
(C) $\frac{R}{2}$
(D) $\frac{5 R - R}{2}$ ✓ Correct

Answer: (D)

Solution

Gravity at height

h

above the surface:

g_{p} = \frac{g R ^{2}}{( R + h ) ^{2}}

.

Gravity at depth

h

below the surface:

g_{q} = g (1 - \frac{h}{R})

.

Setting

g_{p} = g_{q}

and writing

x = h / R

\frac{1}{( 1 + x ) ^{2}} = 1 - x ⟹ 1 = (1 - x) (1 + x)^{2} = 1 + x - x^{2} - x^{3}

x^{3} + x^{2} - x = 0 ⟹ x (x^{2} + x - 1) = 0

The non-trivial root is

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

, so

h = R \cdot \frac{5 - 1}{2} = \frac{5 R - R}{2}

Key confirmed: D.

JEE Main · 2024 · 29 Jan 2024 · Shift 1 · Single correct option

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