Question

109. Escape velocity and black holes The work required to launch an object from the surface of Earth to outer space is given by W = ∫ from R to ∞ of F(x) dx, where R = 6370 km is the approximate radius of Earth, F(x) = (GMm)/x² is the gravitational force between Earth and the object, G is the gravitational constant, M is the mass of Earth, m is the mass of the object, and GM = 4 × 10¹⁴ m³/s².
c. The French scientist Laplace anticipated the existence of black holes in the 18th century with the following argument: If a body has an escape velocity that equals or exceeds the speed of light, c = 300,000 km/s, then light cannot escape the body and it cannot be seen. Show that such a body has a radius R ≤ 2GM/c². For Earth to be a black hole, what would its radius need to be?

Accepted Answer

Start with the expression for the work required to escape Earth's gravity, which is the integral of the gravitational force from the radius R to infinity:

$$ W = \int_{R}^{\infty} F(x) \, dx = \int_{R}^{\infty} \frac{GMm}{x^2} \, dx $$ Evaluate the integral to find the work done (or energy required) to escape Earth's gravitational field:

$$ W = GMm \int_{R}^{\infty} x^{-2} \, dx = GMm \left[-\frac{1}{x}\right]_{R}^{\infty} = \frac{GMm}{R} $$ Recall that the escape velocity $$v_e is $$the velocity needed so that the kinetic energy equals the work done against gravity:

$$ \frac{1}{2} m v_e^2 = \frac{GMm}{R} $$ Solve for the escape velocity $$v_e$$:

$$ v_e = \sqrt{\frac{2GM}{R}}  To $$find the radius $$R$$ for which the escape velocity equals the speed of light $$c$$, set $$v_e = c$$ and solve for $$R$$:

$$ c = \sqrt{\frac{2GM}{R}} \implies R = \frac{2GM}{c^2} $$

This shows that if a body's radius is less than or equal to $$\frac{2GM}{c^2$}$$, its escape velocity is at least $$c$$, making it a black hole.

For Earth, substitute the given values of $$GM$$ and $$c to $$find the radius it would need to become a black hole.

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Key Concepts

Gravitational Force and Work Done by a Variable Force

Escape Velocity

Schwarzschild Radius and Black Holes