JEE Main · 2023 · 31 Jan 2023 · Shift 1 · Gravitation

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At a certain depth "d" below surface of earth. value of acceleration due to gravity becomes four times that of its value at a height

3 R

above earth surface. Where R is Radius of earth (Take

R = 6400

km). The depth d is equal to

(A)5260 km
(B)640 km
(C)2560 km
(D)4800 km✓ Correct

Answer: (D)

Solution

At a depth

d

the acceleration due to gravity is

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

, while at a height

h = 3 R

it is

g_{h} = \frac{g}{( 1 + \frac{h}{R} ) ^{2}} = \frac{g}{( 1 + 3 ) ^{2}} = \frac{g}{16}

. The condition

g_{d} = 4 g_{h}

gives

g (1 - \frac{d}{R}) = \frac{4 g}{16} = \frac{g}{4}

, so

1 - \frac{d}{R} = \frac{1}{4}

and

\frac{d}{R} = \frac{3}{4}

. With

R = 6400

km the depth is

d = \frac{3}{4} \times 6400 = 4800

km.

JEE Main · 2023 · 31 Jan 2023 · Shift 1 · Single correct option

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