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CGPET 2021 Solved Paper

Section: Physics

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Question : 3 of 150

Marks: +1, -0

A black hole is an object whose gravitational field is so strong that even light cannot escape from it. To what approximate radius would earth (mass

=5.98 \times 10^{24} \text{kg}

) have to be compressed to be a black hole?

$10^{-6} \text{m}$
$10^{-2} \text{m}$
$10^{2} \text{m}$
$10^{-9} \text{m}$

Solution:

Given, mass of earth,

m = 5.98 \times 10^{24} \text{kg}

speed of light,

c=3 \times 10^{8} \text{m/s}

Gravitational constant,

G = 6.67 \times 10^{-11} \text{N-m}^2/\text{kg}^2

For earth to be black hole, the escape velocity should be at least equal to speed of light.

∴ Escape velocity = speed of light

\sqrt{\frac{2Gm}{R}} = c

⇒

R = \frac{2Gm}{c^2}

= \frac{2 \times 6.67 \times 10^{-11} \times 5.9 \times 10^{24}}{(3 \times 10^{8})^{2}}

= \frac{2 \times 6.67 \times 598 \times 10^{-11} \times 10^{24}}{10^{4} \times 9 \times 10^{16}}

= \frac{1334 \times 598}{9} \times 10^{-31} \times 10^{24}

=10^{-2} \text{m}

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