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Chapter 18: Problem 51

What is the mass of a black hole with a Schwarzschild radius of \(1.5 \mathrm{km} ?\)

Short Answer

Expert verified
The mass of the black hole is approximately \(1.01 \times 10^{30} \text{ kg}\).

Step by step solution

01

Understand the Schwarzschild Radius

The Schwarzschild radius (\r) of a black hole is given by the formula: \[ r_s = \frac{2GM}{c^2} \] where \(G\) is the gravitational constant, \(M\) is the mass of the black hole, and \(c\) is the speed of light. Given that \(r_s = 1.5 \text{ km} = 1.5 \times 10^3 \text{ m}\).
02

Rearrange the Formula

To find the mass \(M\) of the black hole, rearrange the formula: \[ M = \frac{r_s c^2}{2G} \]
03

Substitute Known Values

Substitute the known values into the equation: \[M = \frac{(1.5 \times 10^3 \text{ m}) \times (3 \times 10^8 \text{ m/s})^2}{2 \times 6.674 \times 10^{-11} \text{ m}^3 \text{ kg}^{-1} \text{ s}^{-2}}\]
04

Calculate the Mass

Perform the calculation: \[ M = \frac{1.5 \times 10^3 \times 9 \times 10^{16}}{2 \times 6.674 \times 10^{-11}} = \frac{13.5 \times 10^{19}}{1.3348 \times 10^{-10}} \] \[ M \rightarrow 1.01 \times 10^{30} \text{ kg} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Black Hole Mass
The mass of a black hole is a crucial concept in understanding its characteristics and behavior. Mass is a fundamental property determining the gravitational pull of the black hole. It helps us understand the dynamics of objects within its vicinity. Black holes are often measured in terms of solar masses, where one solar mass equals the mass of our Sun. High mass implies stronger gravitational forces, leading to a larger event horizon or Schwarzschild radius. The more mass a black hole has, the larger its influence on surrounding space-time.
Gravitational Constant
The gravitational constant, often denoted as G, is a key factor in the laws of physics governing gravitational forces. Its value is approximately \(6.674 \times 10^{-11} \ \text{m}^3 \ \text{kg}^{-1} \ \text{s}^{-2}\). This constant appears in Newton's law of universal gravitation, describing the attractive force between two masses. In the context of black holes, G helps calculate the Schwarzschild radius and understand the gravitational effects on objects nearby. It serves as a bridge between mass, distance, and gravitational force.
Speed of Light
The speed of light, denoted by the symbol c, is one of the fundamental constants in physics. Its value is about \(3 \times 10^8 \ \text{m/s}\). Light's speed is crucial in Einstein's theory of relativity, impacting time, space, and gravity. In terms of black holes, the speed of light helps define the Schwarzschild radius, where not even light can escape the black hole's gravitational pull. The relationship between the speed of light, mass, and the gravitational constant is central to calculating various properties of black holes, ensuring we can predict their impact on their surroundings.

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Most popular questions from this chapter

Go to the LIGO website (http://ligo.org/science.php) and read about gravitational waves. Click on "Sources of Gravitational Waves" and listen to the example. What are the differences among the four listed sources of gravitational waves? Click on "Advanced LIGO." What's new with the project?

If a spaceship approaching Earth at 0.9 times the speed of light shines a laser beam at Earth, how fast will the photons in the beam be moving when they arrive at Earth?

What is the Schwarzschild radius of a black hole that has a mass equal to the average mass of a person (-70 kilograms)?

Imagine two protoms traveling past each other at a distance \(d,\) with relative speed \(0.9 c .\) Compared with two stationary protons a distance \(d\) apart, the gravitational force between these two protons will be a. smaller, because they interact for less time. b. smaller, because the moving proton acts as if it has less mass. c. the same, because in both cases, the mass of the two particles are identical. d. larger, because the moving proton acts as if it has more mass.

Imagine a future astronaut traveling in a spaceship at 0.866 times the speed of light. Special relativity says that the length of the spaceship along the direction of flight is only half of what it was when it was at rest on Earth. The astronaut checks this prediction with a meter stick that he brought with him. Will his measurement confirm the contracted length of his spaceship? Explain your answer.
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