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Chapter 18: Problem 8

As the mass of a black hole increases, its Schwarzschild radius a. increases as the square of the mass. b. increases proportionately. c. stays the same. d. decreases proportionately. e. decreases as the square of the mass.

Short Answer

Expert verified
b. increases proportionately.

Step by step solution

01

Understanding Schwarzschild Radius

The Schwarzschild radius (Rs) is defined as Rs = 2GM/c^2, where G is the gravitational constant, M is the mass of the black hole, and c is the speed of light.
02

Analyzing the Formula

From the formula Rs = 2GM/c^2, it can be seen that Rs is directly proportional to M (mass). This means that if the mass of the black hole increases, the Schwarzschild radius also increases.
03

Finding the Proportionality

Since Rs is directly proportional to M, the relationship between the Schwarzschild radius and the mass of the black hole is linear.
04

Selecting the Correct Answer

Among the options given, the one that correctly describes this relationship is: b. increases proportionately.

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Key Concepts

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

Black Hole
A black hole is a region in space where the gravitational pull is so strong that not even light can escape from it. This intense gravitational field is caused by a very dense mass concentrated in a tiny space.
Black holes can form when massive stars collapse at the end of their life cycle. They can vary in size from small 'stellar' black holes, which are formed by collapsing stars, to 'supermassive' black holes, found at the centers of galaxies.

Some important characteristics of black holes include:
  • Event Horizon: The boundary beyond which nothing can escape.
  • Singularity: The point at the center with infinite density.
  • Accretion Disk: The disk of matter that gets pulled into the black hole.
Understanding these concepts is fundamental to studying how black holes interact with their surroundings and affect space-time.
Gravitational Constant
The gravitational constant, denoted as G, is a key quantity in Newton's law of universal gravitation. It describes the strength of the gravitational force between two masses. The value of G is approximately \(6.67430 \times 10^{-11} \ \text{m}^3 \ \text{kg}^{-1} \ \text{s}^{-2}\).
In the formula for the Schwarzschild radius \(R_s = \frac{2GM}{c^2}\), G helps determine how gravitational forces scale with mass and distance.

This constant is crucial because:
  • It applies universally to any pair of masses.
  • It helps calculate gravitational effects in both small-scale (earthbound) and large-scale (astronomical) scenarios.
  • It is essential for understanding the general theory of relativity and black hole physics.
By understanding G, you gain insight into how gravity influences objects across the universe.
Mass of Black Hole
The mass of a black hole plays a critical role in its properties, especially the Schwarzschild radius.
The Schwarzschild radius is given by the formula \(R_s = \frac{2GM}{c^2}\), where M is the mass of the black hole.
From this equation, it's clear that as the mass (M) of the black hole increases, the Schwarzschild radius (Rs) also increases.

Here are some key points about the mass of black holes:
  • The larger the mass of the black hole, the larger its Schwarzschild radius.
  • This radius is directly proportional to the mass, meaning if the mass doubles, the radius also doubles.
  • Supermassive black holes have enormous masses, up to billions of times the mass of our sun, leading to very large Schwarzschild radii.
Understanding the relationship between mass and the Schwarzschild radius helps in grasping how the size and gravitational pull of black holes operate.

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Most popular questions from this chapter

The Moon has a mass equal to \(3.7 \times 10^{-8} M_{\text {sun. }}\) Suppose the Moon suddenly collapsed into a black hole. a. What would be the Schwarzschild radius of the black-hole Moon? b. What effect would this collapse have on tides raised by the Moon on Earth? Explain. c. Do you think this event would generate gravitational waves? Explain.

Suppose astronomers discover a \(3-M_{\text {sun }}\) black hole located a few light-years from Earth. Should they be concerned that its tremendous gravitational pull will lead to Earth's untimely demise?

Relative motion between two objects is apparent a. even at everyday speeds, such as \(10 \mathrm{km} / \mathrm{h}\). b. only at very large speeds, such as \(0.8 \mathrm{c}\) c. only near very large masses. d. only when both objects are in the same reference frame.

A car approaches you at \(50 \mathrm{km} / \mathrm{h}\). A fly inside the car is flying toward the back of the car at \(7 \mathrm{km} / \mathrm{h}\). From your point of view by the side of the road, the fly is moving at _______ \(\mathrm{km} / \mathrm{h}.\) a. 7 b. 28.5 c. 43 d. 57

A car approaches you at \(50 \mathrm{km} / \mathrm{h}\). The driver turns on the headlights. From your point of view, the light from the headlights is moving at a. \(c+50 \mathrm{km} / \mathrm{h}\) b. \(c-50 \mathrm{km} / \mathrm{h}\) c. \((c+50 \mathrm{km} / \mathrm{h}) / 2\) d. \(c\)
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