COLLEGE PHYSICS
2nd Edition
ISBN: 9781711470832
Author: OpenStax
Publisher: XANEDU
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Chapter 21, Problem 22CQ
To determine
The loop rule to loop
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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 21 Solutions
COLLEGE PHYSICS
Ch. 21 - Prob. 1CQCh. 21 - Prob. 2CQCh. 21 - Prob. 3CQCh. 21 - Prob. 4CQCh. 21 - Prob. 5CQCh. 21 - Knowing that the severity of a shock depends on...Ch. 21 - Would your headlights dim when you start your...Ch. 21 - Some strings of holiday lights are wired in series...Ch. 21 - If two household lightbulbs rated 60 W and 100 W...Ch. 21 - Suppose you are doing a physics lab that asks you...
Ch. 21 - Before World War II, some radios got power through...Ch. 21 - Some light bulbs have three power settings (not...Ch. 21 - Is every emf a potential difference? Is every...Ch. 21 - Prob. 14CQCh. 21 - Given a battery, an assortment of resistors, and a...Ch. 21 - Two different 12-V automobile batteries on a store...Ch. 21 - What are the advantages and disadvantages of...Ch. 21 - Semitractor trucks use four large 12-V batteries....Ch. 21 - Prob. 19CQCh. 21 - Prob. 20CQCh. 21 - Prob. 21CQCh. 21 - Prob. 22CQCh. 21 - Prob. 23CQCh. 21 - Prob. 24CQCh. 21 - Suppose you are using a multimeter (one designed...Ch. 21 - Prob. 26CQCh. 21 - Prob. 27CQCh. 21 - Why can a null measurement be more accurate than...Ch. 21 - If a potentiometer is used to measure cell emfs on...Ch. 21 - Regarding the units involved in the relationship t...Ch. 21 - The RC time constant in heart defibrillation is...Ch. 21 - When making an ECG measurement, it is important to...Ch. 21 - Prob. 33CQCh. 21 - Prob. 34CQCh. 21 - Prob. 35CQCh. 21 - Prob. 36CQCh. 21 - A long, inexpensive extension cord is connected...Ch. 21 - Prob. 38CQCh. 21 - Prob. 39CQCh. 21 - (a) What is the resistance often 275-O resistors...Ch. 21 - (a) What is the resistance of a 1.00 102-O, a...Ch. 21 - What are the largest and smallest resistances you...Ch. 21 - An 1800-W toaster, a 1400-W electric frying pan,...Ch. 21 - Your car’s 30.0-W headlight and 2.40-kW starter...Ch. 21 - (a) Given a48.0-V battery and 24.0-O and 96.0-O...Ch. 21 - Referring to the example combining series and...Ch. 21 - Referring to Figure 21.6: (a) Calculate P3 and...Ch. 21 - Refer to Figure 21.7 and the discussion of lights...Ch. 21 - Prob. 10PECh. 21 - Show that if two resistors R1and R2are combined...Ch. 21 - Unreasonable Results Two resistors, one having a...Ch. 21 - Unreasonable Results Two resistors, one having a...Ch. 21 - Standard automobile batteries have six lead-acid...Ch. 21 - Car bon-zinc dry cells (sometimes referred to as...Ch. 21 - What is the output voltage of a 3.0000-V lithium...Ch. 21 - (a) What is the terminal voltage of a large 1.54-V...Ch. 21 - What is the internal resistance of an automobile...Ch. 21 - (a) Find the terminal voltage of a 12.0-V...Ch. 21 - A car battery with a 12-V emf and an internal...Ch. 21 - The hot resistance of a flashlight bulb is 2.30 ,...Ch. 21 - The label or a portable radio recommends the use...Ch. 21 - An automobile starter motor has an equivalent...Ch. 21 - A child’s electronic toy is supplied by three...Ch. 21 - (a) What is the internal resistance of a voltage...Ch. 21 - A person with body resistance between his hands of...Ch. 21 - Electric fish generate current with biological...Ch. 21 - Integrated Concepts A 12.0-V emf automobile...Ch. 21 - Unreasonable Results A 1.58-V alkaline cell with a...Ch. 21 - Unreasonable Results (a) What is the internal...Ch. 21 - Prob. 31PECh. 21 - Prob. 32PECh. 21 - Verify the second equation in Example 21.5 by...Ch. 21 - Verify the third equation in Example 21.5 by...Ch. 21 - Prob. 35PECh. 21 - Prob. 36PECh. 21 - Prob. 37PECh. 21 - Prob. 38PECh. 21 - Solve Example 21.5, but use loop abcdefgha instead...Ch. 21 - Prob. 40PECh. 21 - Prob. 41PECh. 21 - What is the sensitivity of the galvanometer (that...Ch. 21 - What is the sensitivity of the galvanometer (that...Ch. 21 - Find the resistance that must be placed in series...Ch. 21 - Find the resistance that must be placed in series...Ch. 21 - Find the resistance that must be placed in series...Ch. 21 - Find the resistance that must be placed in...Ch. 21 - Find the resistance that must be placed in series...Ch. 21 - Find the resistance that must be placed in...Ch. 21 - Prob. 50PECh. 21 - Suppose you measure the terminal voltage of a...Ch. 21 - A certain ammeter has a resistance of 5.00X10-5 ...Ch. 21 - A 1,00-?O voltmeter is placed in parallel with a...Ch. 21 - A 0.0200- ammeter is placed in series with a...Ch. 21 - Unreasonable Results Suppose you have a 40.0-...Ch. 21 - Unreasonable Results (a) What resistance would you...Ch. 21 - What is the emf x of a cell being measured in a...Ch. 21 - Calculate the emfx of a dry cell for which a...Ch. 21 - When an unknown resistance Rxis placed in a...Ch. 21 - To what value must you adjust R3to balance a...Ch. 21 - (a) What is the unknown emfx in a potentiometer...Ch. 21 - Suppose you want to measure resistances in the...Ch. 21 - The timing device in an automobile’s intermittent...Ch. 21 - A heart pacemaker fires 72 times a minute, each...Ch. 21 - The duration of a photographic flash is related to...Ch. 21 - A 2.00- and a 7.50-F capacitor can be connected in...Ch. 21 - After two time constants, what percentage of the...Ch. 21 - A 500- resistor, an uncharged 1.50-F capacitor and...Ch. 21 - A heart defibrillator being used on a patient has...Ch. 21 - An ECG monitor must have an RC time constant less...Ch. 21 - Prob. 71PECh. 21 - Using the exact exponential treatment, find how...Ch. 21 - Using the exact exponential treatment, find how...Ch. 21 - Integrated Concepts If you wish to take a picture...Ch. 21 - Integrated Concepts A flashing lamp in a Christmas...Ch. 21 - Integrated Concepts A 160F capacitor charged to...Ch. 21 - Unreasonable Results (a) Calculate the capacitance...Ch. 21 - Construct Your Own Problem Consider a camera's...Ch. 21 - Construe! Your Own Problem Consider a rechargeable...Ch. 21 - Prob. 1TPCh. 21 - Prob. 2TPCh. 21 - Prob. 3TPCh. 21 - Prob. 4TPCh. 21 - Prob. 5TPCh. 21 - Prob. 6TPCh. 21 - Prob. 7TPCh. 21 - Prob. 8TP
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