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 11, Problem 11.2QQ
Recall the skater described at the beginning of this section. Let her mass be m. (i) What would be her
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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 11 Solutions
Physics for Scientists and Engineers, Technology Update (No access codes included)
Ch. 11 - Which of the following statements about the...Ch. 11 - Recall the skater described at the beginning of...Ch. 11 - A solid sphere and a hollow sphere have the same...Ch. 11 - A competitive diver leaves the diving board and...Ch. 11 - An ice skater starts a spin with her arms...Ch. 11 - A pet mouse sleeps near the eastern edge of a...Ch. 11 - Let us name three perpendicular directions as...Ch. 11 - Let the four compass directions north, east,...Ch. 11 - Answer yes or no 10 (he following questions, (a)...Ch. 11 - Prob. 11.6OQ
Ch. 11 - Two ponies of equal mass are initially at...Ch. 11 - Consider an isolated system moving through empty...Ch. 11 - Stars originate as large bodies of slowly rotating...Ch. 11 - A scientist arriving at a hotel asks a bellhop to...Ch. 11 - Prob. 11.3CQCh. 11 - Two children are playing with a roll of paper...Ch. 11 - Both torque and work are products of force and...Ch. 11 - In some motorcycle races, the riders drive over...Ch. 11 - If the torque acting on a particle about an axis...Ch. 11 - A ball is thrown in such a way that it does not...Ch. 11 - If global warming continues over the next one...Ch. 11 - A cat usually lands on its feet regardless of the...Ch. 11 - Prob. 11.11CQCh. 11 - Prob. 11.1PCh. 11 - The displacement vectors 42.0 cm at 15.0 and 23.0...Ch. 11 - Prob. 11.3PCh. 11 - Use the definition of the vector product and the...Ch. 11 - Calculate the net torque (magnitude and direction)...Ch. 11 - Prob. 11.6PCh. 11 - If AB=AB, what is the angle between A and B?Ch. 11 - A particle is located at a point described by the...Ch. 11 - Two forces F1 and F2 act along the two sides of an...Ch. 11 - Prob. 11.10PCh. 11 - A light, rigid rod of length l = 1.00 m joins two...Ch. 11 - A 1.50-kg particle moves in the xy plane with a...Ch. 11 - A particle of mass m moves in the xy plane with a...Ch. 11 - Heading straight toward the summit of Pikes Peak,...Ch. 11 - Review. A projectile of mass m is launched with an...Ch. 11 - Review. A conical pendulum consists of a bob of...Ch. 11 - A particle of mass m moves in a circle of radius R...Ch. 11 - A counterweight of mass m = 4.00 kg is attached to...Ch. 11 - The position vector of a particle of mass 2.00 kg...Ch. 11 - A 5.00-kg particle starts from the origin at time...Ch. 11 - A ball having mass m is fastened at the end of a...Ch. 11 - A uniform solid sphere of radius r = 0.500 m and...Ch. 11 - Big Ben (Fig. P10.27, page 281), the Parliament...Ch. 11 - Show that the kinetic energy of an object rotating...Ch. 11 - A uniform solid disk of mass m = 3.00 kg and...Ch. 11 - Model the Earth as a uniform sphere. (a) Calculate...Ch. 11 - Prob. 11.27PCh. 11 - The distance between the centers of the wheels of...Ch. 11 - A space station is coast me ted in the shape of a...Ch. 11 - A disk with moment of inertia I1 rotates about a...Ch. 11 - A playground merry-go-round of radius R = 2.00 m...Ch. 11 - Prob. 11.32PCh. 11 - A 60.0-kg woman stands at the western rim of a...Ch. 11 - A student sits on a freely rotating stool holding...Ch. 11 - A uniform cylindrical turntable of radius 1.90 m...Ch. 11 - Prob. 11.36PCh. 11 - A wooden block of mass M resting on a...Ch. 11 - Review. A thin, uniform, rectangular signboard...Ch. 11 - A wad of sticky clay with mass m and velocity vi...Ch. 11 - Why is the following situation impossible? A space...Ch. 11 - A 0.005 00-kg bullet traveling horizontally with...Ch. 11 - Section 11.5 The Motion of Gyroscopes and Tops A...Ch. 11 - The angular momentum vector of a precessing...Ch. 11 - A light rope passes over a light, frictionless...Ch. 11 - Prob. 11.45APCh. 11 - Review. Two boys are sliding toward each other on...Ch. 11 - We have all complained that there arent enough...Ch. 11 - Prob. 11.48APCh. 11 - A rigid, massless rod has three particles with...Ch. 11 - Prob. 11.50APCh. 11 - A projectile of mass m moves to the right with a...Ch. 11 - Two children are playing on stools at a restaurant...Ch. 11 - Prob. 11.53APCh. 11 - Prob. 11.54APCh. 11 - Two astronauts (Fig. P11.39), each having a mass...Ch. 11 - Two astronauts (Fig. P11.39), each having a mass...Ch. 11 - Native people throughout North and South America...Ch. 11 - A uniform rod of mass 300 g and length 50.0 cm...Ch. 11 - Global warming is a cause for concern because even...Ch. 11 - The puck in Figure P11.46 has a mass of 0.120 kg....Ch. 11 - A uniform solid disk of radius R is set into...Ch. 11 - In Example 11.8, we investigated an elastic...Ch. 11 - Prob. 11.63CPCh. 11 - A solid cube of wood of side 2a and mass M is...
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