FUNDAMENTALS OF PHYSICS V.1
11th Edition
ISBN: 9781119573913
Author: Halliday
Publisher: WILEY
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Textbook Question
Chapter 8, Problem 6Q
In Fig. 8-23a, you pull upward on a rope that is attached to a cylinder on a vertical rod. Because the cylinder fits tightly on the rod, the cylinder slides along the rod with considerable friction. Your force does work W = +100 J on the cylinder–rod–Earth system (Fig. 8-23b). An “energy statement” for the system is shown in Fig. 8-23c: the kinetic energy K increases by 50 J, and the gravitational potential energy Ug increases by 20 J. The only other change in energy within the system is for the thermal energy Eth. What is the change ∆Eth?
Figure 8-23 Question 6.
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Question B3
Consider the following FLRW spacetime:
t2
ds² = -dt² +
(dx²
+ dy²+ dz²),
t2
where t is a constant.
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[2 marks]
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[3 marks]
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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)
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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
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Question A2
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Suppose that A, is a covector field, and consider the object
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Chapter 8 Solutions
FUNDAMENTALS OF PHYSICS V.1
Ch. 8 - In Fig. 8-18, a horizontally moving block can take...Ch. 8 - Figure 8-19 gives the potential energy function of...Ch. 8 - Figure 8-20 shows one direct path and four...Ch. 8 - In Fig. 8-21, a small, initially stationary block...Ch. 8 - In Fig. 8-22, a block slides from A to C along a...Ch. 8 - In Fig. 8-23a, you pull upward on a rope that is...Ch. 8 - The arrangement shown in Fig. 8-24 is similar to...Ch. 8 - In Fig. 8-25, a block slides along a track that...Ch. 8 - Figure 8-26 shows three situations involving a...Ch. 8 - Figure 8-27 shows three plums that are launched...
Ch. 8 - When a particle moves from f to i and from j to i...Ch. 8 - SSM What is the spring constant of a spring that...Ch. 8 - In Fig. 8-29, a single frictionless roller-coaster...Ch. 8 - You drop a 2.00 kg book to a friend who stands on...Ch. 8 - Figure 8-31 shows a ball with mass m = 0.341 kg...Ch. 8 - SSM In Fig. 8-32, a 2.00 g ice flake is released...Ch. 8 - In Fig. 8-33, a small block of mass m = 0.032 kg...Ch. 8 - Figure 8-34 shows a thin rod, of length L = 2.00 m...Ch. 8 - A 1.50 kg snowball is fired from a cliff 12.5 m...Ch. 8 - GO In Problem 2, what is the speed of the car at a...Ch. 8 - a In Problem 3, what is the speed of the book when...Ch. 8 - SSM WWW a In Problem 5, what is the speed of the...Ch. 8 - a In Problem 8, using energy techniques rather...Ch. 8 - SSM A 5.0 g marble is fired vertically upward...Ch. 8 - a In Problem 4, what initial speed must be given...Ch. 8 - SSM In Fig. 8-35, a runaway truck with failed...Ch. 8 - A 700 g block is released from rest at height h0...Ch. 8 - In Problem 6, what are the magnitudes of a the...Ch. 8 - a In Problem 7, what is the speed of the ball at...Ch. 8 - GO Figure 8-36 shows an 8.00 kg stone at rest on a...Ch. 8 - GO A pendulum consists of a 2.0 kg stone swinging...Ch. 8 - Figure 8-34 shows a pendulum of length L = 1.25 m....Ch. 8 - A 60 kg skier starts from rest at height H = 20 m...Ch. 8 - ILW The string in Fig. 8-38 is L = 120 cm long,...Ch. 8 - A block of mass m = 2.0 kg is dropped from height...Ch. 8 - At t = 0 a 1.0 kg ball is thrown from a tall tower...Ch. 8 - A conservative force F=(6.0x12)i N, where x is in...Ch. 8 - Tarzan, who weighs 688 N, swings from a cliff at...Ch. 8 - Figure 8-41a applies to the spring in a cork gun...Ch. 8 - SSM WWW In Fig. 8-42, a block of mass m = 12 kg is...Ch. 8 - GO A 2.0 kg breadbox on a frictionless incline of...Ch. 8 - ILW A block with mass m = 2.00 kg is placed...Ch. 8 - In Fig. 8-45, a chain is held on a frictionless...Ch. 8 - GO In Fig. 8-46, a spring with k = 170 N/m is at...Ch. 8 - GO A boy is initially seated on the top of a...Ch. 8 - GO In Fig. 8-42, a block of mass m = 3.20 kg...Ch. 8 - GO Two children are playing a game in which they...Ch. 8 - A uniform cord of length 25 cm and mass 15 g is...Ch. 8 - Figure 8-49 shows a plot of potential energy U...Ch. 8 - GO Figure 8-50 shows a plot of potential energy U...Ch. 8 - The potential energy of a diatomic molecule a...Ch. 8 - A single conservative force Fx acts on a 1.0 kg...Ch. 8 - A worker pushed a 27 kg block 9.2 m along a level...Ch. 8 - A collie drags its bed box across a floor by...Ch. 8 - A horizontal force of magnitude 35.0 N pushes a...Ch. 8 - SSM A rope is used to pull a 3.57 kg block at...Ch. 8 - An outfielder throws a baseball with an initial...Ch. 8 - A 75 g Frisbee is thrown from a point 1.1 m above...Ch. 8 - In Fig. 8-51, a block slides down an incline. As...Ch. 8 - SSM ILW A 25 kg bear slides, from rest, 12 m down...Ch. 8 - A 60 kg skier leaves the end of a ski-jump ramp...Ch. 8 - During a rockslide, a 520 kg rock slides from rest...Ch. 8 - A large fake cookie sliding on a horizontal...Ch. 8 - GO In Fig. 8-52, a 3.5 kg block is accelerated...Ch. 8 - A child whose weight is 267 N slides down a 6.1 m...Ch. 8 - ILW In Fig. 8-53, a block of mass m = 2.5 kg...Ch. 8 - You push a 2.0 kg block against a horizontal...Ch. 8 - GO In Fig. 8-54, a block slides along a track from...Ch. 8 - A cookie jar is moving up a 40 incline. At a point...Ch. 8 - A stone with a weight of 5.29 N is launched...Ch. 8 - Prob. 60PCh. 8 - When a click beetle is upside down on its back, it...Ch. 8 - GO In Fig. 8-55, a block slides along a path that...Ch. 8 - The cable of the 1800 kg elevator cab in Fig. 8-56...Ch. 8 - GO In Fig. 8-57, a block is released from rest at...Ch. 8 - GO A particle can slide along a track with...Ch. 8 - A 3.2 kg sloth hangs 3.0 m above the ground. a...Ch. 8 - SSM A spring k = 200 N/m is fixed at the top of a...Ch. 8 - From the edge of a cliff, a 0.55 kg projectile is...Ch. 8 - SSM In Fig. 8-60, the pulley has negligible mass,...Ch. 8 - GO In Fig. 8-38, the string is L = 120 cm long,...Ch. 8 - SSM In Fig. 8-51, a block is sent sliding down a...Ch. 8 - Two snowy peaks are at heights H = 850 m and h =...Ch. 8 - SSM The temperature of a plastic cube is monitored...Ch. 8 - A skier weighing 600 N goes over a frictionless...Ch. 8 - SSM To form a pendulum, a 0.092 kg ball is...Ch. 8 - Prob. 76PCh. 8 - Prob. 77PCh. 8 - At a certain factory, 300 kg crates are dropped...Ch. 8 - SSM A 1500 kg car begins sliding down a 5.0...Ch. 8 - In Fig. 8-65, a 1400 kg block of granite is pulled...Ch. 8 - A particle can move along only an x axis, where...Ch. 8 - For the arrangement of forces in Problem 81, a...Ch. 8 - SSM A 15 kg block is accelerated at 2.0 m/s2 along...Ch. 8 - A certain spring is found not to conform to Hookes...Ch. 8 - SSM Each second, 1200 m3 of water passes over a...Ch. 8 - GO In Fig. 8-67, a small block is sent through...Ch. 8 - SSM A massless rigid rod of length L has a ball of...Ch. 8 - A 1.50 kg water balloon is shot straight up with...Ch. 8 - A 2.50 kg beverage can is thrown directly downward...Ch. 8 - A constant horizontal force moves a 50 kg trunk...Ch. 8 - GO Two blocks, of masses M = 2.0 kg and 2M, are...Ch. 8 - A volcanic ash flow is moving across horizontal...Ch. 8 - A playground slide is in the form of an arc of a...Ch. 8 - The luxury liner Queen Elizabeth 2 has a...Ch. 8 - A factory worker accidentally releases a 180 kg...Ch. 8 - If a 70 kg baseball player steals home by sliding...Ch. 8 - A 0.50 kg banana is thrown directly upward with an...Ch. 8 - A metal tool is sharpened by being held against...Ch. 8 - A swimmer moves through the water at an average...Ch. 8 - An automobile with passengers has weight 16 400 N...Ch. 8 - A 0.63 kg ball thrown directly upward with an...Ch. 8 - The summit of Mount Everest is 8850 m above sea...Ch. 8 - A sprinter who weighs 670 N runs the first 7.0 m...Ch. 8 - A 20 kg object is acted on by a conservative force...Ch. 8 - A machine pulls a 40 kg trunk 2.0 m up a 40 ramp...Ch. 8 - Prob. 106PCh. 8 - The only force acting on a particle is...Ch. 8 - In 1981, Daniel Goodwin climbed 443 m up the...Ch. 8 - A 60.0 kg circus performer slides 4.00 m down a...Ch. 8 - A 5.0 kg block is projected at 5.0 m/s up a plane...Ch. 8 - A 9.40 kg projectile is fired vertically upward....Ch. 8 - A 70.0 kg man jumping from a window lands in an...Ch. 8 - A 30 g bullet moving a horizontal velocity of 500...Ch. 8 - A 1500 kg car starts from rest on a horizontal...Ch. 8 - A 1.50 kg snowball is shot upward at an angle of...Ch. 8 - A 68 kg sky diver falls at a constant terminal...Ch. 8 - A 20 kg block on a horizontal surface is attached...Ch. 8 - Resistance to the motion of an automobile consists...Ch. 8 - SSM A 50 g ball is thrown from a window with an...Ch. 8 - A spring with a spring constant of 3200 N/m is...Ch. 8 - A locomotive with a power capability of 1.5 MW can...Ch. 8 - SSM A 0.42 kg shuffleboard disk is initially at...Ch. 8 - A river descends 15 m through rapids. The speed of...Ch. 8 - The magnitude of the gravitational force between a...Ch. 8 - Approximately 5.5 106 kg of water falls 50 m over...Ch. 8 - To make a pendulum, a 300 g ball is attached to...Ch. 8 - In a circus act, a 60 kg clown is shot from a...Ch. 8 - A 70 kg firefighter slides, from rest, 4.3 m down...Ch. 8 - The surface of the continental United States has...Ch. 8 - A spring with spring constant k = 200 N/m is...Ch. 8 - Fasten one end of a vertical spring to a ceiling,...Ch. 8 - The maximum force you can exert on an object with...Ch. 8 - Conservative force Fx acts on a particle that...Ch. 8 - Figure 8-73a shows a molecule consisting of two...Ch. 8 - Repeat Problem 83, but now with the block...Ch. 8 - A spring with spring constant k = 620 N/m is...
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