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Heat capacities are normally positive, but there is an

An Introduction to Thermal Physics | 1st Edition | ISBN: 9780201380279 | Authors: Daniel V. Schroeder ISBN: 9780201380279 40

Solution for problem 55P Chapter 1

An Introduction to Thermal Physics | 1st Edition

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An Introduction to Thermal Physics | 1st Edition | ISBN: 9780201380279 | Authors: Daniel V. Schroeder

An Introduction to Thermal Physics | 1st Edition

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Problem 55P

Heat capacities are normally positive, but there is an important class of exceptions: systems of particles held together by gravity, such as stars and star clusters.

(a) Consider a, system of just two particles, with identical masses, orbiting in circles about their center of mass. Show that the gravitational potential energy of this system is −2 times the total kinetic energy.

(b) The conclusion of part (a) turns out to be true, at least on average, for any system of particles held together by mutual gravitational attraction:

Here each Ū refers to the total energy (of that type) for the entire system, averaged over some sufficiently long time period. This result is known as the virial theorem. (For a proof, see Carroll and Ostlie (1996), Section 2.4.) Suppose, then, that you add some energy to such a system and then wait for the system to equilibrate. Does the average total kinetic energy increase or decrease? Explain.

(c) A star can be modeled as a gas of particles that interact with each other only gravitationally. According to the equipartition theorem, the average kinetic energy of the particles in such a star should be where T is the average temperature. Express the total energy of a star in terms of its average temperature, and calculate the heat capacity. Note the sign.

(d) Use dimensional analysis to argue that a star of mass M and radius R should have a total potential energy of −GM2/R, times some constant of order 1.

(e) Estimate the average temperature of the sun, whose mass is 2 × 1030 kg and whose radius is 7 × 108 m. Assume, for simplicity, that the sun is made entirely of protons and electrons.

Step-by-Step Solution:

Part (a)

Step 1:

        Consider a, system of just two particles, with identical masses, orbiting in circles about their center of mass. The mass of one particle be M. The distance between the center of mass of the system and the particles is . and the distance between the two masses is r. To show that the potential energy of the system is -2 times as that of total the kinetic energy of the system.

        The gravitational potential energy of one particle comparing with other

        UP = ----(1)

Step 2 of 3

Chapter 1, Problem 55P is Solved
Step 3 of 3

Textbook: An Introduction to Thermal Physics
Edition: 1
Author: Daniel V. Schroeder
ISBN: 9780201380279

This full solution covers the following key subjects: Energy, average, system, particles, star. This expansive textbook survival guide covers 10 chapters, and 454 solutions. The answer to “Heat capacities are normally positive, but there is an important class of exceptions: systems of particles held together by gravity, such as stars and star clusters.(a) Consider a, system of just two particles, with identical masses, orbiting in circles about their center of mass. Show that the gravitational potential energy of this system is ?2 times the total kinetic energy.________________(b) The conclusion of part (a) turns out to be true, at least on average, for any system of particles held together by mutual gravitational attraction: Here each ? refers to the total energy (of that type) for the entire system, averaged over some sufficiently long time period. This result is known as the virial theorem. (For a proof, see Carroll and Ostlie (1996), Section 2.4.) Suppose, then, that you add some energy to such a system and then wait for the system to equilibrate. Does the average total kinetic energy increase or decrease? Explain.________________(c) A star can be modeled as a gas of particles that interact with each other only gravitationally. According to the equipartition theorem, the average kinetic energy of the particles in such a star should be where T is the average temperature. Express the total energy of a star in terms of its average temperature, and calculate the heat capacity. Note the sign.________________(d) Use dimensional analysis to argue that a star of mass M and radius R should have a total potential energy of ?GM2/R, times some constant of order 1.________________(e) Estimate the average temperature of the sun, whose mass is 2 × 1030 kg and whose radius is 7 × 108 m. Assume, for simplicity, that the sun is made entirely of protons and electrons.” is broken down into a number of easy to follow steps, and 279 words. Since the solution to 55P from 1 chapter was answered, more than 347 students have viewed the full step-by-step answer. The full step-by-step solution to problem: 55P from chapter: 1 was answered by , our top Physics solution expert on 07/05/17, 04:29AM. An Introduction to Thermal Physics was written by and is associated to the ISBN: 9780201380279. This textbook survival guide was created for the textbook: An Introduction to Thermal Physics , edition: 1.

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Heat capacities are normally positive, but there is an