FUND.OF PHYSICS (LL)-PRINT COMP.VOL.2
12th Edition
ISBN: 9781119801269
Author: Halliday
Publisher: WILEY
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Textbook Question
Chapter 10, Problem 2P
What is the angular speed of (a) the second hand, (b) the minute hand, and (c) the hour hand of a smoothly running analog watch? Answer in radians per second.
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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
FUND.OF PHYSICS (LL)-PRINT COMP.VOL.2
Ch. 10 - Prob. 1QCh. 10 - A force is applied to the rim of a disk that can...Ch. 10 - A good baseball pitcher can throw a baseball...Ch. 10 - What is the angular speed of a the second hand, b...Ch. 10 - When a slice of buttered toast is accidentally...Ch. 10 - The angular position of a point on a rotating...Ch. 10 - ILW A diver makes 2.5 revolutions on the way from...Ch. 10 - The angular position of a point on the rim of a...Ch. 10 - The angular acceleration of a wheel is = 6.0t4 ...Ch. 10 - A drum rotates around its central axis at an...
Ch. 10 - Starting from rest, a disk rotates about its...Ch. 10 - A disk, initially rotating at 120 rad/s, is slowed...Ch. 10 - The angular speed of an automobile engine is...Ch. 10 - ILW A flywheel turns through 40 rev as it slows...Ch. 10 - GO A disk rotates about its central axis starling...Ch. 10 - SSM Starting from rest, a wheel has constant =...Ch. 10 - A merry-go-round rotates from rest with an angular...Ch. 10 - At t = 0, a flywheel has an angular velocity of...Ch. 10 - A pulsar is a rapidly rotating neutron star that...Ch. 10 - What are the magnitudes of a the angular velocity,...Ch. 10 - An object rotates about a fixed axis, and the...Ch. 10 - Between 1911 and 1990, the top of the leaning bell...Ch. 10 - An astronaut is tested in a centrifuge with radius...Ch. 10 - SSM WWW A flywheel with a diameter of 1.20 m is...Ch. 10 - A vinyl record is played by rotating the record so...Ch. 10 - SSM a What is the angular speed about the polar...Ch. 10 - The flywheel of a steam engine runs with a...Ch. 10 - A seed is on a turntable rotating at 3313 rev/min,...Ch. 10 - Figure 10-32 shows an early method of measuring...Ch. 10 - A gyroscope flywheel of radius 2.83 cm is...Ch. 10 - GO A disk, with a radius of 0.25 m. is to be...Ch. 10 - A car starts from rest and moves around a circular...Ch. 10 - SSM Calculate the rotational inertia of a wheel...Ch. 10 - SSM Two uniform solid cylinders, each rotating...Ch. 10 - SSM Calculate the rotational inertia of a meter...Ch. 10 - Trucks can be run on energy stored in a rotating...Ch. 10 - Four identical particles of mass 0.50 kg each are...Ch. 10 - SSM A small ball of mass 0.75 kg is attached to...Ch. 10 - The length of a bicycle pedal arm is 0.152 m, and...Ch. 10 - SSM ILW During the launch from a board, a divers...Ch. 10 - If a 32.0 N m torque on a wheel causes angular...Ch. 10 - GO A pulley, with a rotational inertia of 1.0 103...Ch. 10 - Prob. 58PCh. 10 - An automobile crankshaft transfers energy from the...Ch. 10 - A thin rod of length 0.75 m and mass 0.42 kg is...Ch. 10 - A 32.0 kg wheel, essentially a thin hoop with...Ch. 10 - SSM ILW A meter stick is held vertically with one...Ch. 10 - A uniform cylinder of radius 10 cm and mass 20 kg...Ch. 10 - GO A tall, cylindrical chimney fall;; over when...Ch. 10 - Prob. 68PCh. 10 - A wheel, starling from rest, rotates with a...Ch. 10 - Prob. 72PCh. 10 - A uniform helicopter rotor blade is 7.80 m long,...Ch. 10 - Prob. 75PCh. 10 - Starting from rest at t = 0, a wheel undergoes a...Ch. 10 - SSM A record turntable rotating at 3313 rev/min...Ch. 10 - Prob. 79PCh. 10 - A disk rotates al constant angular acceleration,...Ch. 10 - Prob. 82PCh. 10 - Prob. 84PCh. 10 - Prob. 85PCh. 10 - Prob. 86PCh. 10 - Prob. 87PCh. 10 - Prob. 88PCh. 10 - Prob. 89PCh. 10 - Prob. 90PCh. 10 - Prob. 91PCh. 10 - Prob. 92PCh. 10 - Prob. 93PCh. 10 - Prob. 94PCh. 10 - Prob. 95PCh. 10 - Prob. 96PCh. 10 - Prob. 97P
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