Physics for Scientists and Engineers, Technology Update (No access codes included)
9th Edition
ISBN: 9781305116399
Author: Raymond A. Serway, John W. Jewett
Publisher: Cengage Learning
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Chapter 1, Problem 1.23P
A section of land has an area of 1 square mile and contains 640 acres. Determine the number of square meters in 1 acre.
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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 1 Solutions
Physics for Scientists and Engineers, Technology Update (No access codes included)
Ch. 1 - In a machine shop, two cams are produced, one of...Ch. 1 - True or False: Dimensional analysis can give you...Ch. 1 - The distance between two cities is 100 mi. What is...Ch. 1 - One student uses a meterstick to measure the...Ch. 1 - A house is advertised as having 1 420 square feet...Ch. 1 - Answer each question yes or no. Must two...Ch. 1 - The price of gasoline at a particular station is...Ch. 1 - Rank the following five quantities in order from...Ch. 1 - Prob. 1.6OQCh. 1 - Prob. 1.7OQ
Ch. 1 - Prob. 1.8OQCh. 1 - Newtons second law of motion ((Chapter 5) says...Ch. 1 - A calculator displays a result as 1.365 218 0 X...Ch. 1 - Suppose the three fundamental standards of the...Ch. 1 - Why is the metric system of units considered...Ch. 1 - What natural phenomena could serve as alternative...Ch. 1 - Express the following quantities using the...Ch. 1 - Prob. 1.1PCh. 1 - The standard kilogram (Fig. 1.1a) is a...Ch. 1 - Prob. 1.3PCh. 1 - Prob. 1.4PCh. 1 - Prob. 1.5PCh. 1 - Prob. 1.6PCh. 1 - A crystalline solid consists of atoms stacked up...Ch. 1 - The mass of a copper atom is 1 06 10-25 kg, and...Ch. 1 - Prob. 1.9PCh. 1 - Figure P1.10 shows a frustum of a cone. Match each...Ch. 1 - Kinetic energy K (Chapter 7) has dimensions kg ...Ch. 1 - Prob. 1.12PCh. 1 - The position of a particle moving under uniform...Ch. 1 - (a) Assume the equation x = At3 + Bt describes the...Ch. 1 - A solid piece of lead has a mass of 23.94 g and a...Ch. 1 - An ore loader moves 1 200 tom/h from a mine to the...Ch. 1 - A rectangular building lot has a width of 75.0 ft...Ch. 1 - Suppose your hair grows at the rate 1/32 in. per...Ch. 1 - Why is the following situation impossible? A...Ch. 1 - A pyramid has a bright of 481 ft, and its base...Ch. 1 - The pyramid described in Problem 20 contains...Ch. 1 - Assume it takes 7.00 min to fill a 30.0-gal...Ch. 1 - A section of land has an area of 1 square mile and...Ch. 1 - A house is 50.0 ft long and 26 ft wide and has...Ch. 1 - Prob. 1.25PCh. 1 - Let AI represent the density of aluminum and Fe...Ch. 1 - One gallon of paint (volume = 3.78 103 m3) covers...Ch. 1 - Prob. 1.28PCh. 1 - Prob. 1.29PCh. 1 - Prob. 1.30PCh. 1 - Find the order of magnitude of the number of...Ch. 1 - (a) Compute the order of magnitude of the mass of...Ch. 1 - To an order of magnitude, how many piano tuners...Ch. 1 - Prob. 1.34PCh. 1 - A rectangular plate has a length of (21.310.2) cm...Ch. 1 - How many significant figures are in the following...Ch. 1 - The tropical year, the time interval from one...Ch. 1 - Carry out the arithmetic operations (a) the sum of...Ch. 1 - Review. In a community college parking lot, the...Ch. 1 - Review. While you arc on a trip to Europe, you...Ch. 1 - Review. A child is surprised that because of sales...Ch. 1 - Prob. 1.42PCh. 1 - Review. The ratio of the number of sparrows...Ch. 1 - Prob. 1.44PCh. 1 - Prob. 1.45PCh. 1 - Review. Prove that one solution of the equation...Ch. 1 - Prob. 1.47PCh. 1 - Review. A highway curve forms a section of a...Ch. 1 - Prob. 1.49PCh. 1 - Prob. 1.50PCh. 1 - Review. A student is supplied with a stack of copy...Ch. 1 - The radius of a uniform solid sphere is measured...Ch. 1 - A sidewalk is to be constructed around a swimming...Ch. 1 - Collectible coins are sometimes plated with gold...Ch. 1 - Prob. 1.55APCh. 1 - (a) What is the order of magnitude of the number...Ch. 1 - The diameter of our disk-shaped galaxy, the Milky...Ch. 1 - Why is the following situation impossible? In an...Ch. 1 - A high fountain of water is located at the center...Ch. 1 - A water fountain is at the center of a circular...Ch. 1 - The data in the following table represent...Ch. 1 - The distance from the Sun to the nearest star is...Ch. 1 - Assume there are 100 million passenger cars in the...Ch. 1 - A spherical shell has an outside radius of 2.60 cm...Ch. 1 - Bacteria and other prokaryotes are found deep...Ch. 1 - Air is blown into a spherical balloon so that,...Ch. 1 - A rod extending between x = 0 and x = 14.0 cm has...Ch. 1 - In physics, it is important to use mathematical...Ch. 1 - The consumption of natural gas by a company...Ch. 1 - A woman wishing to know the height of a mountain...Ch. 1 - Prob. 1.71APCh. 1 - Prob. 1.72APCh. 1 - You stand in a flat meadow and observe two cows...
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