FUNDAMENTALS OF PHYSICS - EXTENDED
12th Edition
ISBN: 9781119773511
Author: Halliday
Publisher: WILEY
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Chapter 7, Problem 73P
To determine
To find: the average power of the motion in kilowatts and horsepower.
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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 7 Solutions
FUNDAMENTALS OF PHYSICS - EXTENDED
Ch. 7 - Prob. 1QCh. 7 - Figure 7-16a shows two horizontal forces that act...Ch. 7 - Is positive or negative work done by a constant...Ch. 7 - Spring A is stiffer than spring B kA kB. The...Ch. 7 - In three situations, a single force acts on a...Ch. 7 - Figure 7-23 shows three arrangements of a block...Ch. 7 - SSM A proton mass m = 1.67 1027 kg is being...Ch. 7 - If a Saturn V rocket with an Apollo spacecraft...Ch. 7 - On August 10, 1972, a large meteorite skipped...Ch. 7 - An explosion at ground level leaves a crater with...
Ch. 7 - A father racing his son has half the kinetic...Ch. 7 - A ice block floating in a river is pushed through...Ch. 7 - The only force acting on a 2.0 kg canister that is...Ch. 7 - A coin slides over a frictionless plane and across...Ch. 7 - A 12.0 N force with a fixed orientation does work...Ch. 7 - A luge and its rider, with a total mass of 85 kg,...Ch. 7 - SSM WWW A helicopter lifts a 72 kg astronaut 15 m...Ch. 7 - a In 1975 the roof of Montreals Velodrome, witha...Ch. 7 - 21 SSM A cord is used to vertically lower an...Ch. 7 - A cave rescue team lifts an injured spelunker...Ch. 7 - In Fig. 7-10, we must apply a force of magnitude...Ch. 7 - During spring semester at MIT, residents of the...Ch. 7 - In Fig. 7-10a, a block of mass m lies on a...Ch. 7 - SSM WWW The only force acting on a 2.0 kg body as...Ch. 7 - SSM WWW The force on a particle is directed along...Ch. 7 - A 1.5 kg block is initially at rest on a...Ch. 7 - GO A force F= cx3.00x2iacts on a particle as the...Ch. 7 - A can of sardines is made to move along an xaxis...Ch. 7 - A single force acts on a 3.0 kg particle-like...Ch. 7 - GO Figure 7-41 shows a cord attached to a cart...Ch. 7 - SSM A force of 5.0 N acts on a 15 kg body...Ch. 7 - A skier is pulled by a towrope up a frictionless...Ch. 7 - SSM ILW A 100 kg block is pulled at a constant...Ch. 7 - The loaded cab of an elevator has a mass of 3.0 ...Ch. 7 - A machine carries a 4.0 kg package from an initial...Ch. 7 - A 0.30 kg ladle sliding on a horizontal...Ch. 7 - Prob. 49PCh. 7 - a At a certain instant, a particle-like object is...Ch. 7 - A force F= 3.00 N i 7.00 N j 7.00 N k acts on...Ch. 7 - A funny car accelerates from rest through a...Ch. 7 - SSM A horse pulls a cart with a force of 40 lb at...Ch. 7 - An initially stationary 2.0 kg object accelerates...Ch. 7 - To pull a 50 kg crate across a horizontal...Ch. 7 - A frightened child is restrained by her mother as...Ch. 7 - How much work is done by a force F= 2x N i 3 N j,...Ch. 7 - 63 SSM To push a 25.0 kg crate up a frictionless...Ch. 7 - Boxes are transported from one location to another...Ch. 7 - If a car of mass 1200 kg is moving along a highway...Ch. 7 - An iceboat is at rest on a frictionless frozen...Ch. 7 - If a ski lift raises 100 passengers averaging 660...Ch. 7 - A force F= 4.0 N i cj acts on a particle as the...Ch. 7 - Prob. 71PCh. 7 - Prob. 72PCh. 7 - Prob. 73PCh. 7 - Prob. 74PCh. 7 - Prob. 75PCh. 7 - Prob. 76PCh. 7 - Prob. 77PCh. 7 - Prob. 78P
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