Physics for Scientists and Engineers: Foundations and Connections
1st Edition
ISBN: 9781133939146
Author: Katz, Debora M.
Publisher: Cengage Learning
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Chapter 15, Problem 4PQ
Why is the Earth’s atmosphere denser near sea level than it is at a high altitude? Be sure to explain why the atmosphere’s density is not uniform and why the air isn’t all in contact with the Earth’s surface.
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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
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(a) Show explicitly that F is a tensor, that is, show that it transforms appropriately under a
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(b)…
Chapter 15 Solutions
Physics for Scientists and Engineers: Foundations and Connections
Ch. 15.1 - Imagine an airplane flying at constant velocity....Ch. 15.3 - Prob. 15.2CECh. 15.3 - Prob. 15.3CECh. 15.3 - Prob. 15.4CECh. 15.4 - Prob. 15.5CECh. 15.4 - Prob. 15.6CECh. 15 - Prob. 1PQCh. 15 - Prob. 2PQCh. 15 - Dry air is primarily composed of nitrogen. In a...Ch. 15 - Why is the Earths atmosphere denser near sea level...
Ch. 15 - Crater Lake in Oregon is the deepest lake in the...Ch. 15 - Prob. 6PQCh. 15 - Prob. 7PQCh. 15 - One study found that the dives of emperor penguins...Ch. 15 - Prob. 9PQCh. 15 - Prob. 10PQCh. 15 - Suppose you are at the top of Mount Everest and...Ch. 15 - Prob. 12PQCh. 15 - Prob. 13PQCh. 15 - Prob. 14PQCh. 15 - A 20.0-kg child sits on a four-legged stool. The...Ch. 15 - Prob. 16PQCh. 15 - The dolphin tank at an amusement park is...Ch. 15 - Prob. 18PQCh. 15 - A block of an unknown material floats in water...Ch. 15 - Prob. 20PQCh. 15 - Prob. 21PQCh. 15 - A spherical submersible 2.00 m in radius, armed...Ch. 15 - What fraction of an iceberg floating in the ocean...Ch. 15 - Prob. 24PQCh. 15 - A hollow copper (Cu = 8.92 103 kg/m3) spherical...Ch. 15 - Prob. 26PQCh. 15 - You have probably noticed that carrying a person...Ch. 15 - A straw is in a glass of juice. Peter puts his...Ch. 15 - Prob. 29PQCh. 15 - Prob. 30PQCh. 15 - Prob. 31PQCh. 15 - Prob. 32PQCh. 15 - A rectangular block of Styrofoam 25.0 cm in...Ch. 15 - Prob. 34PQCh. 15 - Prob. 35PQCh. 15 - A manometer is shown in Figure P15.36. Rank the...Ch. 15 - The gauge pressure measured on a cars tire is 35...Ch. 15 - Prob. 38PQCh. 15 - Prob. 39PQCh. 15 - To allow a car to slow down or stop, hydraulic...Ch. 15 - Prob. 41PQCh. 15 - Prob. 42PQCh. 15 - Prob. 43PQCh. 15 - Water enters a smooth, horizontal tube with a...Ch. 15 - Prob. 45PQCh. 15 - Prob. 46PQCh. 15 - Prob. 47PQCh. 15 - A fluid flows through a horizontal pipe that...Ch. 15 - Water is flowing through a pipe that has a...Ch. 15 - Prob. 50PQCh. 15 - Prob. 51PQCh. 15 - Figure P15.52 shows a Venturi meter, which may be...Ch. 15 - At a fraternity party, drinking straws have been...Ch. 15 - Liquid toxic waste with a density of 1752 kg/m3 is...Ch. 15 - Water is flowing in the pipe shown in Figure...Ch. 15 - Prob. 56PQCh. 15 - Water flows through a pipe that gradually descends...Ch. 15 - Air flows horizontally with a speed of 108 km/h...Ch. 15 - Prob. 59PQCh. 15 - Prob. 60PQCh. 15 - Prob. 61PQCh. 15 - Prob. 62PQCh. 15 - Prob. 63PQCh. 15 - Prob. 64PQCh. 15 - Prob. 65PQCh. 15 - Prob. 66PQCh. 15 - Prob. 67PQCh. 15 - Prob. 68PQCh. 15 - Prob. 69PQCh. 15 - Prob. 70PQCh. 15 - The density of air in the Earths atmosphere...Ch. 15 - A manometer containing water with one end...Ch. 15 - Prob. 73PQCh. 15 - Prob. 74PQCh. 15 - Prob. 75PQCh. 15 - Prob. 76PQCh. 15 - Prob. 77PQCh. 15 - Case Study Shannon uses the example of a helium...Ch. 15 - Prob. 79PQCh. 15 - Prob. 80PQCh. 15 - A uniform wooden board of length L and mass M is...
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