EBK PHYSICS FOR SCIENTISTS AND ENGINEER
9th Edition
ISBN: 8220100654428
Author: Jewett
Publisher: Cengage Learning US
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Textbook Question
Chapter 13, Problem 13.62AP
(a) Show that the rate of change of the free-fall acceleration with vertical position near the Earth’s surface is
This rate of change with position is called a gradient.
(b) Assuming h is small in comparison to the radius of the Earth, show that the difference in free-fall acceleration between two points separated by vertical distance h is
(c) Evaluate this difference for h = 6.00 m, a typical height for a two-story building.
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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 13 Solutions
EBK PHYSICS FOR SCIENTISTS AND ENGINEER
Ch. 13 - A planet has two moons of equal mass. Moon 1 is in...Ch. 13 - Superman stands on top of a very tall mountain and...Ch. 13 - An asteroid is in a highly eccentric elliptical...Ch. 13 - Prob. 13.4QQCh. 13 - A system consists of five particles. How many...Ch. 13 - Rank the following quantities of energy from...Ch. 13 - Prob. 13.3OQCh. 13 - Suppose the gravitational acceleration at the...Ch. 13 - Imagine that nitrogen and other atmospheric gases...Ch. 13 - An object of mass m is located on the surface of a...
Ch. 13 - Prob. 13.7OQCh. 13 - The vernal equinox and the autumnal equinox are...Ch. 13 - Rank the magnitudes of the following gravitational...Ch. 13 - The gravitational force exerted on an astronaut on...Ch. 13 - Prob. 13.11OQCh. 13 - Each Voyager spacecraft was accelerated toward...Ch. 13 - In his 1798 experiment, Cavendish was said to have...Ch. 13 - Prob. 13.3CQCh. 13 - Prob. 13.4CQCh. 13 - Prob. 13.5CQCh. 13 - Prob. 13.6CQCh. 13 - Prob. 13.7CQCh. 13 - Prob. 13.8CQCh. 13 - A satellite in low-Earth orbit is not truly...Ch. 13 - In introductory physics laboratories, a typical...Ch. 13 - Determine the order of magnitude of the...Ch. 13 - A 200-kg object and a 500-kg object are separated...Ch. 13 - During a solar eclipse, the Moon, the Earth, and...Ch. 13 - Two ocean liners, each with a mass of 40 000...Ch. 13 - Three uniform spheres of masses m1 = 2.00 kg, m2 =...Ch. 13 - Two identical isolated particles, each of mass...Ch. 13 - Prob. 13.8PCh. 13 - Two objects attract each other with a...Ch. 13 - Review. A student proposes to study the...Ch. 13 - Prob. 13.11PCh. 13 - Prob. 13.12PCh. 13 - Review. Miranda, a satellite of Uranus, is shown...Ch. 13 - (a) Compute the vector gravitational field at a...Ch. 13 - Three objects of equal mass are located at three...Ch. 13 - A spacecraft in the shape of a long cylinder has a...Ch. 13 - An artificial satellite circles the Earth in a...Ch. 13 - Io, a satellite of Jupiter, has an orbital period...Ch. 13 - A minimum-energy transfer orbit to an outer planet...Ch. 13 - A particle of mass m moves along a straight line...Ch. 13 - Plasketts binary system consists of two starts...Ch. 13 - Two planets X and Y travel counterclockwise in...Ch. 13 - Comet Halley (Fig. P13.23) approaches the Sun to...Ch. 13 - Prob. 13.24PCh. 13 - Use Keplers third law to determine how many days...Ch. 13 - Neutron stars are extremely dense objects formed...Ch. 13 - A synchronous satellite, which always remains...Ch. 13 - (a) Given that the period of the Moons orbit about...Ch. 13 - Suppose the Suns gravity were switched off. The...Ch. 13 - A satellite in Earth orbit has a mass of 100 kg...Ch. 13 - How much work is done by the Moons gravitational...Ch. 13 - How much energy is required to move a 1 000-kg...Ch. 13 - Prob. 13.33PCh. 13 - An object is released from rest at an altitude h...Ch. 13 - A system consists of three particles, each of mass...Ch. 13 - Prob. 13.36PCh. 13 - A 500-kg satellite is in a circular orbit at an...Ch. 13 - Prob. 13.38PCh. 13 - Prob. 13.39PCh. 13 - Prob. 13.40PCh. 13 - Prob. 13.41PCh. 13 - Prob. 13.42PCh. 13 - Prob. 13.43PCh. 13 - Prob. 13.44PCh. 13 - Prob. 13.45PCh. 13 - Prob. 13.46PCh. 13 - Ganymede is the largest of Jupiters moons....Ch. 13 - Prob. 13.48PCh. 13 - At the Earths surface, a projectile is launched...Ch. 13 - Prob. 13.50APCh. 13 - Prob. 13.51APCh. 13 - Voyager 1 and Voyager 2 surveyed the surface of...Ch. 13 - A satellite is in a circular orbit around the...Ch. 13 - Why is the following situation impossible? A...Ch. 13 - Let gM represent the difference in the...Ch. 13 - Prob. 13.56APCh. 13 - Prob. 13.57APCh. 13 - Prob. 13.58APCh. 13 - Prob. 13.59APCh. 13 - Two spheres having masses M and 2M and radii R and...Ch. 13 - Two hypothetical planets of masses m1 and m2 and...Ch. 13 - (a) Show that the rate of change of the free-fall...Ch. 13 - A ring of matter is a familiar structure in...Ch. 13 - Prob. 13.64APCh. 13 - Review. As an astronaut, you observe a small...Ch. 13 - Prob. 13.66APCh. 13 - Studies of the relationship of the Sun to our...Ch. 13 - Review. Two identical hard spheres, each of mass m...Ch. 13 - Prob. 13.69APCh. 13 - Prob. 13.70APCh. 13 - Prob. 13.71APCh. 13 - Prob. 13.72APCh. 13 - Prob. 13.73APCh. 13 - Two stars of masses M and m, separated by a...Ch. 13 - Two identical particles, each of mass 1 000 kg,...Ch. 13 - Prob. 13.76APCh. 13 - As thermonuclear fusion proceeds in its core, the...Ch. 13 - The Solar and Heliospheric Observatory (SOHO)...Ch. 13 - The oldest artificial satellite still in orbit is...Ch. 13 - Prob. 13.80CP
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