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Step-by-Step Solution
Step 1: Recall the Schwarzschild Radius Formula
A black hole is characterized by its Schwarzschild radius, which is the radius within which the mass of the object must be compressed so that not even light can escape. The formula for the Schwarzschild radius is:
$R = \frac{2GM}{c^2}$,
where:
$G$ is the universal gravitational constant, $6.67 \times 10^{-11}\,\mathrm{N\,m^2\,kg^{-2}}$.
$M$ is the mass of the object (in this case, Earth).
$c$ is the speed of light ($3 \times 10^8\,\mathrm{m/s}$).
Step 2: Identify the Given Data
For Earth,
Mass, $M = 5.98 \times 10^{24}\,\mathrm{kg}$.
Speed of light, $c = 3 \times 10^8\,\mathrm{m/s}$.
Gravitational constant, $G = 6.67 \times 10^{-11}\,\mathrm{N\,m^2\,kg^{-2}}$.
Step 3: Substitute Values into the Formula
Substitute $M$, $G$, and $c$ into $R = \frac{2GM}{c^2}$:
$R = \frac{2 \times 6.67 \times 10^{-11} \times 5.98 \times 10^{24}}{(3 \times 10^8)^2}\,\mathrm{m}.$
Step 4: Compute the Radius
Perform the calculation step by step:
$R = \frac{2 \times 6.67 \times 10^{-11} \times 5.98 \times 10^{24}}{(3 \times 10^8)^2}.$
Simplifying, we obtain:
$R \approx 10^{-2}\,\mathrm{m}.$
final Answer
The Earth would have to be compressed to a radius of approximately $10^{-2}$ m to become a black hole.
