Question

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 $= 5.98 \times 10^{24} \, kg$) have to be compressed to be a black hole?

• A. $10^{-9} \, m$
• B. $10^{-6} \, m$
• C. $10^{-2} \, m$
• D. $100\, m$

Answer 1

The Correct Option is C

 To determine the radius to which Earth must be compressed to become a black hole, we need to calculate its Schwarzschild radius. The Schwarzschild radius (\(R_s\)) is given by the formula:

\(R_s = \frac{2GM}{c^2}\)

where:

• \(G\) is the gravitational constant, approximately \(6.674 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2}\).
• \(M\) is the mass of the Earth, \(5.98 \times 10^{24} \, \text{kg}\).
• \(c\) is the speed of light in a vacuum, \(3 \times 10^8 \, \text{m/s}\).

Substitute these values into the formula:

\(R_s = \frac{2 \times 6.674 \times 10^{-11} \times 5.98 \times 10^{24}}{(3 \times 10^8)^2}\)

Simplifying the equation step-by-step:

1. Calculate \(2 \times 6.674 \times 10^{-11} \times 5.98 \times 10^{24}\):
2. Calculate \((3 \times 10^8)^2\):
3. Divide the results:

Therefore, the Schwarzschild radius is approximately \(8.87 \times 10^{-3} \, \text{m}\), which rounded is approximately \(10^{-2} \, \text{m}\). Thus, Earth would need to be compressed to a radius of about \(10^{-2} \, \text{m}\) to form a black hole.

Conclusion: The correct answer is \(10^{-2} \, m\).