If M is mass of a planet and R is its radius then in order to become black hole [ c is speed of light ]

Options

A. `sqrt((GM)/(R)) le c `
B. `sqrt((GM)/(2R) ) ge c`
C. `sqrt((2GM)/(R))ge c`
D. `sqrt((2GM)/R) le c `

Answer

To determine the condition under which a planet becomes a black hole, we need to analyze the escape speed from the planet's surface. The escape speed (v) is given by the formula:

\[ v = \sqrt{\frac{2GM}{R}} \]

where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the planet,
- \( R \) is the radius of the planet.

For a planet to become a black hole, the escape speed must be greater than or equal to the speed of light \( c \). Therefore, we set up the inequality:

\[ \sqrt{\frac{2GM}{R}} \geq c \]

Now, squaring both sides of the inequality to eliminate the square root gives us:

\[ \frac{2GM}{R} \geq c^2 \]

Next, we can rearrange this inequality to express it in terms of the mass and radius of the planet:

\[ 2GM \geq c^2 R \]

Finally, we can express the condition for a planet to become a black hole as:

\[ M \geq \frac{c^2 R}{2G} \]

This equation shows that for a planet with mass \( M \) and radius \( R \), if the mass is greater than or equal to \( \frac{c^2 R}{2G} \), then the planet will become a black hole.