Physics for Scientists and Engineers
10th Edition
ISBN: 9781337553278
Author: Raymond A. Serway, John W. Jewett
Publisher: Cengage Learning
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Textbook Question
Chapter 14.4, Problem 14.4QQ
You are shipwrecked and floating in the middle of the ocean on a raft. Your cargo on the raft includes a treasure chest full of gold that you found before your ship sank, and the raft is just barely afloat. To keep you floating as high as possible in the water, should you (a) leave the treasure chest on top of the raft, (b) secure the treasure chest to the underside of the raft, or (c) hang the treasure chest in the water with a rope attached to the raft? (Assume throwing the treasure chest overboard is not an option you wish to consider.)
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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 14 Solutions
Physics for Scientists and Engineers
Ch. 14.1 - Suppose you are standing directly behind someone...Ch. 14.2 - The pressure at the bottom of a filled glass of...Ch. 14.3 - Several common barometers are built, with a...Ch. 14.4 - You are shipwrecked and floating in the middle of...Ch. 14.6 - You observe two helium balloons floating next to...Ch. 14 - A large man sits on a four-legged chair with his...Ch. 14 - Prob. 2PCh. 14 - Estimate the total mass of the Earths atmosphere....Ch. 14 - Prob. 4PCh. 14 - What must be the contact area between a suction...
Ch. 14 - Prob. 6PCh. 14 - Review. A solid sphere of brass (bulk modulus of...Ch. 14 - The human brain and spinal cord are immersed in...Ch. 14 - Blaise Pascal duplicated Torricellis barometer...Ch. 14 - A tank with a flat bottom of area A and vertical...Ch. 14 - Prob. 11PCh. 14 - A 10.0-kg block of metal measuring 12.0 cm by 10.0...Ch. 14 - A plastic sphere floats in water with 50.0% of its...Ch. 14 - The weight of a rectangular block of low-density...Ch. 14 - A wooden block of volume 5.24 104 m3 floats in...Ch. 14 - A hydrometer is an instrument used to determine...Ch. 14 - Refer to Problem 16 and Figure P14.16. A...Ch. 14 - On October 21, 2001, Ian Ashpole of the United...Ch. 14 - Prob. 19PCh. 14 - Water flowing through a garden hose of diameter...Ch. 14 - Water falls over a dam of height h with a mass...Ch. 14 - A legendary Dutch boy saved Holland by plugging a...Ch. 14 - Water is pumped up from the Colorado River to...Ch. 14 - In ideal flow, a liquid of density 850 kg/m3 moves...Ch. 14 - Review. Old Faithful Geyser in Yellowstone...Ch. 14 - You are working as an expert witness for the owner...Ch. 14 - A thin 1.50-mm coating of glycerin has been placed...Ch. 14 - A hypodermic needle is 3.00 cm in length and 0.300...Ch. 14 - What radius needle should be used to inject a...Ch. 14 - An airplane has a mass of 1.60 104 kg, and each...Ch. 14 - Prob. 31PCh. 14 - Decades ago, it was thought that huge herbivorous...Ch. 14 - Prob. 33APCh. 14 - The true weight of an object can be measured in a...Ch. 14 - Prob. 35APCh. 14 - Review. Assume a certain liquid, with density 1...Ch. 14 - Evangelista Torricelli was the first person to...Ch. 14 - A common parameter that can be used to predict...Ch. 14 - In 1983, the United States began coining the...Ch. 14 - Review. With reference to the dam studied in...Ch. 14 - The spirit-in-glass thermometer, invented in...Ch. 14 - A woman is draining her fish tank by siphoning the...Ch. 14 - Prob. 43APCh. 14 - Prob. 44APCh. 14 - Prob. 45APCh. 14 - Review. In a water pistol, a piston drives water...Ch. 14 - Prob. 47APCh. 14 - The hull of an experimental boat is to be lifted...Ch. 14 - Show that the variation of atmospheric pressure...Ch. 14 - Why is the following situation impossible? A barge...
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