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Ch 15: Oscillations
Chapter 15, Problem 13

Nothing can escape the event horizon of a black hole, not even light. You can think of the event horizon as being the distance from a black hole at which the escape speed is the speed of light, 3.00 ✕ 10⁸ m/s, making all escape impossible. What is the radius of the event horizon for a black hole with a mass 5.0 times the mass of the sun? This distance is called the Schwarzschild radius.

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Identify the mass of the black hole in terms of the mass of the sun. Given that the mass of the black hole is 5.0 times the mass of the sun, denote the mass of the sun as M_sun and the mass of the black hole as M = 5.0 * M_sun.
Recall the formula for the Schwarzschild radius, R_s, which describes the radius of the event horizon of a black hole. The formula is R_s = \frac{2GM}{c^2}, where G is the gravitational constant (6.67430 x 10^-11 m^3 kg^-1 s^-2), M is the mass of the black hole, and c is the speed of light in vacuum (3.00 x 10^8 m/s).
Substitute the mass of the black hole (M = 5.0 * M_sun) into the Schwarzschild radius formula. Replace M_sun with the actual mass of the sun, which is approximately 1.989 x 10^30 kg.
Perform the calculation step by step. First, calculate the product 5.0 * M_sun to find the mass of the black hole. Then, substitute this value into the Schwarzschild radius formula along with the constants G and c.
Simplify the expression to find the Schwarzschild radius, R_s, in meters. This will give you the radius of the event horizon for the black hole.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Event Horizon

The event horizon is the boundary surrounding a black hole beyond which no information or matter can escape. It represents the point at which the escape velocity equals the speed of light, making it impossible for anything, including light, to break free from the gravitational pull of the black hole.
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Schwarzschild Radius

The Schwarzschild radius is the radius of the event horizon for a non-rotating black hole. It is directly proportional to the mass of the black hole and can be calculated using the formula r_s = 2GM/c², where G is the gravitational constant, M is the mass of the black hole, and c is the speed of light.
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Escape Velocity

Escape velocity is the minimum speed needed for an object to break free from the gravitational attraction of a celestial body without any additional propulsion. For a black hole, this speed reaches the speed of light at the event horizon, illustrating the extreme gravitational forces at play.
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