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

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

Answer 1

The correct option is C ${10}^{-2}\text{m}$

The escape speed on earth:
$V_e=\sqrt{\frac{2G{M}_e}{R_e}}$

In order for earth to behave as a black hole, $V_e\ge c$. Which can be achieved by decreasing the radius of the earth to $R^{\prime }$.

$\therefore c=\sqrt{\frac{2G{M}_e}{R^{\prime }}}$
$R^{\prime }=\frac{2G{M}_e}{c^2}$

$R^{\prime }=\frac{2\times 6.67\times {10}^{-11}\times 6\times {10}^{24}}{(3\times {10}^8{)}^2}$

$R^{\prime }=0.889\times {10}^{-2}m$
$R^{\prime }\approx {10}^{-2}m$