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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Textbook Question
Chapter 13, Problem 17OQ
Suppose an observer and a source of sound are both at rest relative to the ground and a strong wind is blowing away from the source toward the observer. (i) What effect does the wind have on the observed frequency? (a) It causes an increase. (b) It causes a decrease. (c) It causes no change. (ii) What effect does the wind have on the observed wavelength? Choose from the same possibilities as in part (i). (iii) What effect does the wind have on the observed speed of the wave? Choose from the same possibilities as in part (i).
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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 13 Solutions
Principles of Physics: A Calculus-Based Text
Ch. 13.1 - (i) In a long line of people waiting to buy...Ch. 13.2 - Prob. 13.2QQCh. 13.2 - The amplitude of a wave is doubled, with no other...Ch. 13.3 - Suppose you create a pulse by moving the free end...Ch. 13.5 - Prob. 13.5QQCh. 13.7 - Consider detectors of water waves at three...Ch. 13.7 - Prob. 13.7QQCh. 13 - Prob. 1OQCh. 13 - Prob. 2OQCh. 13 - Rank the waves represented by the following...
Ch. 13 - Prob. 4OQCh. 13 - When all the strings on a guitar (Fig. OQ13.5) are...Ch. 13 - By what factor would you have to multiply the...Ch. 13 - A sound wave can be characterized as (a) a...Ch. 13 - Prob. 8OQCh. 13 - Prob. 9OQCh. 13 - A source vibrating at constant frequency generates...Ch. 13 - A source of sound vibrates with constant...Ch. 13 - Prob. 12OQCh. 13 - Prob. 13OQCh. 13 - Prob. 14OQCh. 13 - As you travel down the highway in your car, an...Ch. 13 - Prob. 16OQCh. 13 - Suppose an observer and a source of sound are both...Ch. 13 - Prob. 1CQCh. 13 - Prob. 2CQCh. 13 - Prob. 3CQCh. 13 - Prob. 4CQCh. 13 - When a pulse travels on a taut string, does it...Ch. 13 - Prob. 6CQCh. 13 - Prob. 7CQCh. 13 - Prob. 8CQCh. 13 - Prob. 9CQCh. 13 - Prob. 10CQCh. 13 - Prob. 11CQCh. 13 - How can an object move with respect to an observer...Ch. 13 - Prob. 13CQCh. 13 - Prob. 1PCh. 13 - Prob. 2PCh. 13 - Prob. 3PCh. 13 - Prob. 4PCh. 13 - The string shown in Figure P13.5 is driven at a...Ch. 13 - Prob. 6PCh. 13 - Prob. 7PCh. 13 - Prob. 8PCh. 13 - Prob. 9PCh. 13 - A transverse wave on a string is described by the...Ch. 13 - Prob. 11PCh. 13 - Prob. 12PCh. 13 - Prob. 13PCh. 13 - A transverse sinusoidal wave on a string has a...Ch. 13 - A steel wire of length 30.0 m and a copper wire of...Ch. 13 - Prob. 16PCh. 13 - Prob. 17PCh. 13 - Review. A light string with a mass per unit length...Ch. 13 - Prob. 19PCh. 13 - Prob. 20PCh. 13 - A series of pulses, each of amplitude 0.150 m, are...Ch. 13 - Prob. 22PCh. 13 - Prob. 23PCh. 13 - A taut rope has a mass of 0.180 kg and a length of...Ch. 13 - Prob. 25PCh. 13 - Prob. 26PCh. 13 - Prob. 27PCh. 13 - Prob. 28PCh. 13 - Prob. 29PCh. 13 - Prob. 30PCh. 13 - Write an expression that describes the pressure...Ch. 13 - Prob. 32PCh. 13 - Prob. 33PCh. 13 - Prob. 34PCh. 13 - Prob. 35PCh. 13 - Prob. 36PCh. 13 - A sound wave in air has a pressure amplitude equal...Ch. 13 - A rescue plane flies horizontally at a constant...Ch. 13 - A driver travels northbound on a highway at a...Ch. 13 - Prob. 40PCh. 13 - Prob. 41PCh. 13 - Prob. 42PCh. 13 - Prob. 43PCh. 13 - Prob. 44PCh. 13 - Review. A tuning fork vibrating at 512 Hz falls...Ch. 13 - Submarine A travels horizontally at 11.0 m/s...Ch. 13 - Prob. 47PCh. 13 - Prob. 48PCh. 13 - Prob. 49PCh. 13 - Review. A block of mass M, supported by a string,...Ch. 13 - Prob. 51PCh. 13 - Review. A block of mass M hangs from a rubber...Ch. 13 - Prob. 53PCh. 13 - The wave is a particular type of pulse that can...Ch. 13 - Prob. 55PCh. 13 - Prob. 56PCh. 13 - Prob. 57PCh. 13 - Prob. 58PCh. 13 - Prob. 59PCh. 13 - Prob. 60PCh. 13 - Prob. 61PCh. 13 - Prob. 62PCh. 13 - Prob. 63PCh. 13 - Prob. 64PCh. 13 - Prob. 65PCh. 13 - Prob. 66PCh. 13 - Prob. 67PCh. 13 - A sound wave moves down a cylinder as in Active...Ch. 13 - A string on a musical instrument is held under...Ch. 13 - A train whistle (f = 400 Hz) sounds higher or...Ch. 13 - The Doppler equation presented in the text is...
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