Physics for Scientists and Engineers: Foundations and Connections
1st Edition
ISBN: 9781133939146
Author: Katz, Debora M.
Publisher: Cengage Learning
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Textbook Question
Chapter 10, Problem 15PQ
Center of Mass Revisited
N Find the center of mass of a system with three particles of masses 1.0 kg, 2.0 kg, and 3.0 kg kept at the vertices of an equilateral triangle of side 1.0 m (Fig. P10.15).
FIGURE P10.15
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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 10 Solutions
Physics for Scientists and Engineers: Foundations and Connections
Ch. 10.1 - What Do You Already Know About Rockets? Think...Ch. 10.3 - Prob. 10.2CECh. 10.3 - Prob. 10.3CECh. 10.3 - Prob. 10.4CECh. 10.5 - What is the purpose of the ropes attached to the...Ch. 10 - Prob. 1PQCh. 10 - Prob. 2PQCh. 10 - Prob. 3PQCh. 10 - A mother pushes her son in a stroller at a...Ch. 10 - Prob. 5PQ
Ch. 10 - Estimate the magnitude of the momentum of a car on...Ch. 10 - Prob. 7PQCh. 10 - Prob. 8PQCh. 10 - What is the magnitude of the Earths momentum...Ch. 10 - The velocity of a 10-kg object is given by...Ch. 10 - A particle has a momentum of magnitude 40.0 kg ...Ch. 10 - Prob. 12PQCh. 10 - Latoya, sitting on a sled, is being pushed by...Ch. 10 - A baseball is thrown vertically upward. The mass...Ch. 10 - Center of Mass Revisited N Find the center of mass...Ch. 10 - Prob. 16PQCh. 10 - Prob. 17PQCh. 10 - Two metersticks are connected at their ends as...Ch. 10 - A boy of mass 25.0 kg is sitting on one side of a...Ch. 10 - Prob. 20PQCh. 10 - Prob. 21PQCh. 10 - Prob. 22PQCh. 10 - Prob. 23PQCh. 10 - Prob. 24PQCh. 10 - Prob. 25PQCh. 10 - A person of mass m stands on a rope ladder that is...Ch. 10 - Prob. 27PQCh. 10 - Prob. 28PQCh. 10 - Two particles with masses 2.0 kg and 4.0 kg are...Ch. 10 - A billiard player sends the cue ball toward a...Ch. 10 - A crate of mass M is initially at rest on a...Ch. 10 - Prob. 32PQCh. 10 - Prob. 33PQCh. 10 - According to the National Academy of Sciences, the...Ch. 10 - Prob. 35PQCh. 10 - Prob. 36PQCh. 10 - Prob. 37PQCh. 10 - Usually, we do not walk or even stand on a...Ch. 10 - Prob. 39PQCh. 10 - There is a compressed spring between two...Ch. 10 - There is a compressed spring between two...Ch. 10 - A submarine with a mass of 6.26 106 kg contains a...Ch. 10 - A 44.0-kg child finds himself trapped on the...Ch. 10 - Problems 44 and 45 are paired. C A model rocket is...Ch. 10 - A model rocket is shot straight up and explodes at...Ch. 10 - An astronaut finds herself in a predicament in...Ch. 10 - Prob. 47PQCh. 10 - Prob. 48PQCh. 10 - Prob. 49PQCh. 10 - Prob. 50PQCh. 10 - The space shuttle uses its thrusters with an...Ch. 10 - Prob. 52PQCh. 10 - Prob. 53PQCh. 10 - Prob. 54PQCh. 10 - Prob. 55PQCh. 10 - The cryogenic main stage of a rocket has an...Ch. 10 - To lift off from the Moon, a 9.50 105 kg rocket...Ch. 10 - Prob. 58PQCh. 10 - Prob. 59PQCh. 10 - Prob. 60PQCh. 10 - Prob. 61PQCh. 10 - An astronaut out on a spacewalk to construct a new...Ch. 10 - Prob. 63PQCh. 10 - Prob. 64PQCh. 10 - A racquetball of mass m = 43.0 g, initially moving...Ch. 10 - Prob. 66PQCh. 10 - Prob. 67PQCh. 10 - Prob. 68PQCh. 10 - A comet is traveling through space with speed 3.33...Ch. 10 - A ballistic pendulum is used to measure the speed...Ch. 10 - Prob. 71PQCh. 10 - Prob. 72PQCh. 10 - Prob. 73PQCh. 10 - Figure P10.74 provides artists with human...Ch. 10 - Prob. 75PQCh. 10 - A single-stage rocket of mass 308 metric tons (not...Ch. 10 - Prob. 77PQCh. 10 - A light spring is attached to a block of mass 4m...Ch. 10 - Prob. 79PQCh. 10 - Prob. 80PQCh. 10 - A Show that the total momentum of a system of...Ch. 10 - Prob. 82PQCh. 10 - Prob. 83PQ
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