What mass would have to be compressed to a radius of $1.0 \times 10 ^ { - 15 } \mathrm {~m}$ (the order of magnitude of the radius of an atomic nucleus) in order for it to become a black hole? ( $G = 6.67 \times 10 ^ { - 11 } \mathrm {~N}$.
$\mathrm { m } ^ { 2 } / \mathrm { kg } ^ { 2 } , c = 3.0 \times 10 ^ { 8 } \mathrm {~m} / \mathrm { s }$ )
A) $6.7 \times 10 ^ { 11 } \mathrm {~kg}$
B) $2.2 \times 10 ^ { 3 } \mathrm {~kg}$
C) $1.1 \times 10 ^ { - 6 } \mathrm {~kg}$
D) $8.3 \times 10 ^ { 15 } \mathrm {~kg}$
E) $4.1 \times 10 ^ { - 8 } \mathrm {~kg}$

Question

Quizplus · Accepted Answer

The Schwarzschild radius is given by $r_s = \frac{2GM}{c^2}$, where $G$ is the gravitational constant, $M$ is the mass, and $c$ is the speed of light. Plugging in the given values, we get $r_s = \frac{2(6.67 \times 10^{-11} \mathrm{~N~m^2/kg^2})(M)}{(3.0 \times 10^8 \mathrm{~m/s})^2}$. Solving for $M$, we get $M = \frac{r_s c^2}{2G} = \frac{(1.0 \times 10^{-15} \mathrm{~m})(3.0 \times 10^8 \mathrm{~m/s})^2}{2(6.67 \times 10^{-11} \mathrm{~N~m^2/kg^2})} = 6.7 \times 10^{11} \mathrm{~kg}$.

What Mass Would Have to Be Compressed to a Radius 1.0×10−15 m1.0 \times 10 ^ { - 15 } \mathrm {~m}1.0×10−15 m

What Mass Would Have to Be Compressed to a Radius $1.0 \times 10 ^ { - 15 } \mathrm {~m}$