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 16, Problem 2OQ
A cylinder with a piston holds 0.50 m3 of oxygen at an absolute pressure of 4.0 atm. The piston is pulled outward, increasing the volume of the gas until the pressure drops to 1.0 atm. If the temperature stays constant, what new volume does the gas occupy? (a) 1.0 m3 (b) 1.5 m3 (c) 2.0 m3 (d) 0.12 m3 (e) 2.5 m3
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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
coordinate transformation.
[5 marks]
(b)…
Chapter 16 Solutions
Principles of Physics: A Calculus-Based Text
Ch. 16.2 - Consider the following pairs of materials. Which...Ch. 16.3 - Prob. 16.2QQCh. 16.4 - A common material for cushioning objects in...Ch. 16.4 - Prob. 16.4QQCh. 16.5 - Two containers hold an ideal gas at the same...Ch. 16.6 - Prob. 16.6QQCh. 16 - Prob. 1OQCh. 16 - A cylinder with a piston holds 0.50 m3 of oxygen...Ch. 16 - A hole is drilled in a metal plate. When the metal...Ch. 16 - When a certain gas under a pressure of 5.00 106...
Ch. 16 - Prob. 5OQCh. 16 - Prob. 6OQCh. 16 - What would happen if the glass of a thermometer...Ch. 16 - Prob. 8OQCh. 16 - A gas is at 200 K. If we wish to double the rms...Ch. 16 - Prob. 10OQCh. 16 - Prob. 11OQCh. 16 - A rubber balloon is filled with 1 L of air at 1...Ch. 16 - Prob. 13OQCh. 16 - An ideal gas is contained in a vessel at 300 K....Ch. 16 - Prob. 15OQCh. 16 - Prob. 16OQCh. 16 - Prob. 17OQCh. 16 - A sample of gas with a thermometer immersed in the...Ch. 16 - Prob. 19OQCh. 16 - Prob. 1CQCh. 16 - Prob. 2CQCh. 16 - Prob. 3CQCh. 16 - A piece of copper is dropped into a beaker of...Ch. 16 - Prob. 5CQCh. 16 - Prob. 6CQCh. 16 - Prob. 7CQCh. 16 - Prob. 8CQCh. 16 - Prob. 9CQCh. 16 - Prob. 10CQCh. 16 - Prob. 11CQCh. 16 - Prob. 12CQCh. 16 - Prob. 13CQCh. 16 - Prob. 1PCh. 16 - Convert the following to equivalent temperatures...Ch. 16 - Prob. 3PCh. 16 - Prob. 4PCh. 16 - Prob. 5PCh. 16 - Prob. 6PCh. 16 - Prob. 7PCh. 16 - Prob. 8PCh. 16 - Prob. 9PCh. 16 - A sample of a solid substance has a mass m and a...Ch. 16 - Each year thousands of children are badly burned...Ch. 16 - Prob. 12PCh. 16 - Prob. 13PCh. 16 - Prob. 14PCh. 16 - The active element of a certain laser is made of a...Ch. 16 - Prob. 16PCh. 16 - Prob. 17PCh. 16 - Prob. 18PCh. 16 - Prob. 19PCh. 16 - Prob. 20PCh. 16 - Prob. 21PCh. 16 - Prob. 22PCh. 16 - Prob. 23PCh. 16 - Prob. 24PCh. 16 - Prob. 25PCh. 16 - Prob. 26PCh. 16 - Prob. 27PCh. 16 - Prob. 28PCh. 16 - The mass of a hot-air balloon and its cargo (not...Ch. 16 - Prob. 30PCh. 16 - A popular brand of cola contains 6.50 g of carbon...Ch. 16 - Prob. 32PCh. 16 - At 25.0 m below the surface of the sea, where the...Ch. 16 - To measure how far below the ocean surface a bird...Ch. 16 - Prob. 35PCh. 16 - Prob. 36PCh. 16 - Prob. 37PCh. 16 - Prob. 38PCh. 16 - Prob. 39PCh. 16 - A cylinder contains a mixture of helium and argon...Ch. 16 - Prob. 41PCh. 16 - Prob. 42PCh. 16 - Prob. 43PCh. 16 - (a) How many atoms of helium gas fill a spherical...Ch. 16 - Fifteen identical particles have various speeds:...Ch. 16 - From the MaxwellBoltzmann speed distribution, show...Ch. 16 - Prob. 47PCh. 16 - Helium gas is in thermal equilibrium with liquid...Ch. 16 - Prob. 49PCh. 16 - Prob. 50PCh. 16 - Prob. 51PCh. 16 - Prob. 52PCh. 16 - A mercury thermometer is constructed as shown in...Ch. 16 - A liquid with a coefficient of volume expansion ...Ch. 16 - A clock with a brass pendulum has a period of...Ch. 16 - A vertical cylinder of cross-sectional area A is...Ch. 16 - Prob. 57PCh. 16 - Prob. 58PCh. 16 - Prob. 59PCh. 16 - The rectangular plate shown in Figure P16.60 has...Ch. 16 - In a chemical processing plant, a reaction chamber...Ch. 16 - Prob. 62PCh. 16 - Prob. 63PCh. 16 - Two concrete spans that form a bridge of length L...Ch. 16 - A 1.00-km steel railroad rail is fastened securely...Ch. 16 - Prob. 66PCh. 16 - Prob. 67PCh. 16 - Prob. 68PCh. 16 - Consider an object with any one of the shapes...Ch. 16 - Prob. 70PCh. 16 - Prob. 71PCh. 16 - Prob. 72PCh. 16 - Prob. 73PCh. 16 - A cylinder that has a 40.0-cm radius and is 50.0...Ch. 16 - Prob. 75P
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