Question

If $M$ is mass of a planet and $R$ is its radius, then for a planet in order to become black hole $(c$ is speed of light$)$

• A. $\sqrt{\frac{GM}{R}}\le c$
  
• B. $\sqrt{\frac{GM}{R}}\ge c$
  
• C. $\sqrt{\frac{2GM}{R}}\ge c$
  
• D. $\sqrt{\frac{2GM}{R}}\le c$
  

Answer 1

The correct option is C $\sqrt{\frac{2GM}{R}}\ge c$


A planet can become a black hole if its mass and radius are such that it has an immense force of gravity on its surface. The force of attraction has to be a so large that even light cannot escape from its surface.

Given, speed of light $=c$.

We know that, escape velocity on the surface of a planet having mass $M$ and radius $R$ is given by,$$v_e=\sqrt{\frac{2GM}{R}}$$ If $v_e\ge c$ .i.e,

$\sqrt{\frac{2GM}{R}}\ge c$

$\Rightarrow$ Even light can't escape from the surface of such planet making it appear black. The radius of such a planet (technically called as event horizon) is known as schwarzschild radius.

Any object that has radius less than or equal to schwarzschild radius is termed as a black hole.

Hnece, option (c) is the correct answer.