FUNDAMENTALS OF PHYSICS V.1
11th Edition
ISBN: 9781119573913
Author: Halliday
Publisher: WILEY
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Chapter 8, Problem 86P
GO In Fig. 8-67, a small block is sent through point A with a speed of 7.0 m/s. Its path is without friction until it reaches the section of length L = 12 m, where the coefficient of kinetic friction is 0.70. The indicated heights are h1 = 6.0 m and h2 = 2.0 m. What are the speeds of the block at (a) point B and (b) point C? (c) Does the block reach point D? If so, what is its speed there; if not, how far through the section of friction does it travel?
Figure 8-67 Problem 86.
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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
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[5 marks]
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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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