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 12CQ
Discuss three different common examples of natural processes that involve an increase in entropy. Be sure to account for all parts of each system under consideration.
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Question B3
Consider the following FLRW spacetime:
t2
ds² = -dt² +
(dx²
+ dy²+ dz²),
t2
where t is a constant.
a)
State whether this universe is spatially open, closed or flat.
[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
Fv=AAμ.
(a) Show explicitly that F is a tensor, that is, show that it transforms appropriately under a
coordinate transformation.
[5 marks]
(b)…
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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