Principles of Physics: A Calculus-Based Text
5th Edition
ISBN: 9781133104261
Author: Raymond A. Serway, John W. Jewett
Publisher: Cengage Learning
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Chapter 18, Problem 10OQ
Assume a sample of an ideal gas is at room temperature. What action will necessarily make the entropy of the sample increase? (a) Transfer energy into it by heat. (b) Transfer energy into it irreversibly by heat. (c) Do work on it. (d) Increase either its temperature or its volume, without letting the other variable decrease. (e) None of those choices is correct.
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Question B3
Consider the following FLRW spacetime:
t2
ds² = -dt² +
(dx²
+ dy²+ dz²),
t2
where t is a constant.
a)
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[2 marks]
b) Determine the Hubble factor H(t), and represent it in a (roughly drawn) plot as a function
of time t, starting at t = 0.
[3 marks]
c) Taking galaxy A to be located at (x, y, z) = (0,0,0), determine the proper distance to galaxy
B located at (x, y, z) = (L, 0, 0). Determine the recessional velocity of galaxy B with respect
to galaxy A.
d) The Friedmann equations are
2
k
8πG
а
4πG
+
a²
(p+3p).
3
a
3
[5 marks]
Use these equations to determine the energy density p(t) and the pressure p(t) for the
FLRW spacetime specified at the top of the page.
[5 marks]
e) Given the result of question B3.d, state whether the FLRW universe in question is (i)
radiation-dominated, (ii) matter-dominated, (iii) cosmological-constant-dominated, or (iv)
none of the previous. Justify your answer.
f)
[5 marks]
A conformally…
SECTION B
Answer ONLY TWO questions in Section B
[Expect to use one single-sided A4 page for each Section-B sub question.]
Question B1
Consider the line element
where w is a constant.
ds²=-dt²+e2wt dx²,
a) Determine the components of the metric and of the inverse metric.
[2 marks]
b) Determine the Christoffel symbols. [See the Appendix of this document.]
[10 marks]
c)
Write down the geodesic equations.
[5 marks]
d) Show that e2wt it is a constant of geodesic motion.
[4 marks]
e)
Solve the geodesic equations for null geodesics.
[4 marks]
Page 2
SECTION A
Answer ALL questions in Section A
[Expect to use one single-sided A4 page for each Section-A sub question.]
Question A1
SPA6308 (2024)
Consider Minkowski spacetime in Cartesian coordinates th
=
(t, x, y, z), such that
ds² = dt² + dx² + dy² + dz².
(a) Consider the vector with components V" = (1,-1,0,0). Determine V and V. V.
(b) Consider now the coordinate system x' (u, v, y, z) such that
u =t-x,
v=t+x.
[2 marks]
Write down the line element, the metric, the Christoffel symbols and the Riemann curvature
tensor in the new coordinates. [See the Appendix of this document.]
[5 marks]
(c) Determine V", that is, write the object in question A1.a in the coordinate system x'. Verify
explicitly that V. V is invariant under the coordinate transformation.
Question A2
[5 marks]
Suppose that A, is a covector field, and consider the object
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Chapter 18 Solutions
Principles of Physics: A Calculus-Based Text
Ch. 18.1 - The energy input to an engine is 3.00 times...Ch. 18.3 - Prob. 18.2QQCh. 18.4 - Prob. 18.3QQCh. 18.6 - (a) Suppose you select four cards at random from a...Ch. 18.7 - Which of the following is true for the entropy...Ch. 18.7 - An ideal gas is taken from an initial temperature...Ch. 18.8 - True or False: The entropy change in an adiabatic...Ch. 18 - Prob. 1OQCh. 18 - Prob. 2OQCh. 18 - A refrigerator has 18.0 kJ of work done on it...
Ch. 18 - Prob. 4OQCh. 18 - Consider cyclic processes completely characterized...Ch. 18 - Prob. 6OQCh. 18 - Prob. 7OQCh. 18 - Prob. 8OQCh. 18 - A sample of a monatomic ideal gas is contained in...Ch. 18 - Assume a sample of an ideal gas is at room...Ch. 18 - Prob. 11OQCh. 18 - Prob. 1CQCh. 18 - Prob. 2CQCh. 18 - Prob. 3CQCh. 18 - Prob. 4CQCh. 18 - Prob. 5CQCh. 18 - Prob. 6CQCh. 18 - Prob. 7CQCh. 18 - Prob. 8CQCh. 18 - Prob. 9CQCh. 18 - Prob. 10CQCh. 18 - Prob. 11CQCh. 18 - Discuss three different common examples of natural...Ch. 18 - The energy exhaust from a certain coal-fired...Ch. 18 - Prob. 1PCh. 18 - Prob. 2PCh. 18 - Prob. 3PCh. 18 - Prob. 4PCh. 18 - Prob. 5PCh. 18 - Prob. 6PCh. 18 - Prob. 7PCh. 18 - Prob. 8PCh. 18 - Prob. 9PCh. 18 - Prob. 10PCh. 18 - Prob. 11PCh. 18 - Prob. 12PCh. 18 - Prob. 13PCh. 18 - Prob. 14PCh. 18 - Argon enters a turbine at a rate of 80.0 kg/min, a...Ch. 18 - Prob. 16PCh. 18 - A refrigerator has a coefficient of performance...Ch. 18 - Prob. 18PCh. 18 - Prob. 19PCh. 18 - In 1993, the U.S. government instituted a...Ch. 18 - Prob. 21PCh. 18 - Prob. 22PCh. 18 - Prob. 23PCh. 18 - Prob. 24PCh. 18 - A heat pump used for heating shown in Figure...Ch. 18 - Prob. 26PCh. 18 - Prob. 27PCh. 18 - An ice tray contains 500 g of liquid water at 0C....Ch. 18 - Prob. 29PCh. 18 - Prob. 30PCh. 18 - Prob. 31PCh. 18 - (a) Prepare a table like Table 18.1 for the...Ch. 18 - Prob. 33PCh. 18 - Prob. 34PCh. 18 - Prob. 35PCh. 18 - Prob. 36PCh. 18 - Prob. 37PCh. 18 - Prob. 38PCh. 18 - Prob. 39PCh. 18 - Prob. 40PCh. 18 - Prob. 41PCh. 18 - Prob. 42PCh. 18 - (a) Find the kinetic energy of the moving air in a...Ch. 18 - Prob. 45PCh. 18 - Prob. 46PCh. 18 - Prob. 47PCh. 18 - An idealized diesel engine operates in a cycle...Ch. 18 - Prob. 49PCh. 18 - Prob. 50PCh. 18 - Prob. 51PCh. 18 - Prob. 52PCh. 18 - Prob. 53PCh. 18 - Prob. 54PCh. 18 - Prob. 55PCh. 18 - Prob. 56PCh. 18 - Prob. 57PCh. 18 - Prob. 58PCh. 18 - Prob. 59PCh. 18 - Prob. 60PCh. 18 - Prob. 61PCh. 18 - Prob. 62PCh. 18 - A 1.00-mol sample of an ideal monatomic gas is...Ch. 18 - Prob. 64PCh. 18 - Prob. 65P
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