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 conditions under which a planet can become a black hole, we need to analyze the relationship between the escape velocity and the speed of light. Here’s a step-by-step solution:

### Step 1: Understand Escape Velocity
The escape velocity (\(V_E\)) is the minimum velocity an object must have to break free from the gravitational pull of a celestial body without any additional propulsion. The formula for escape velocity is given by:

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

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

### Step 2: Set Up the Condition for a Black Hole
For a planet to become a black hole, its escape velocity must be equal to or greater than the speed of light (\(c\)). Therefore, we set up the inequality:

\[ V_E \geq c \]

### Step 3: Substitute the Expression for Escape Velocity
Substituting the expression for escape velocity into the inequality, we have:

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

### Step 4: Square Both Sides
To eliminate the square root, we square both sides of the inequality:

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

### Step 5: Rearrange the Inequality
Now, we can rearrange the inequality to express the relationship between mass, radius, and the speed of light:

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

### Step 6: Final Formulation
This can be further simplified to find a condition for the mass and radius of the planet:

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

This equation indicates that for a planet of radius \(R\) to become a black hole, its mass \(M\) must be greater than or equal to \(\frac{c^2 R}{2G}\).

### Summary
In summary, the condition for a planet to become a black hole is given by:

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