Physics for Scientists and Engineers, Technology Update, Hybrid Edition (with Enhanced WebAssign Multi-Term LOE Printed Access Card for Physics)
9th Edition
ISBN: 9781305116429
Author: Raymond A. Serway, John W. Jewett
Publisher: Cengage Learning
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Chapter 10, Problem 10.10OQ
A toy airplane hangs from the ceiling at the bottom end of a string. Yon turn the airplane many times to wind up the string clockwise and release it. The airplane starts to spin counterclockwise, slowly at first and then faster and faster. Take counterclockwise as the positive sense and assume friction is negligible. When the string is entirely unwound, the airplane has its maximum rate of rotation, (i) At this moment, is its
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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
Physics for Scientists and Engineers, Technology Update, Hybrid Edition (with Enhanced WebAssign Multi-Term LOE Printed Access Card for Physics)
Ch. 10 - A rigid object rotates in a counterclockwise sense...Ch. 10 - Consider again the pairs of angular positions for...Ch. 10 - Ethan and Rebecca are riding on a merry-go-round....Ch. 10 - (i) If you are trying to loosen a stubborn screw...Ch. 10 - You turn off your electric drill and find that the...Ch. 10 - A section of hollow pipe and a solid cylinder have...Ch. 10 - A ball rolls without slipping down incline A,...Ch. 10 - A cyclist rides a bicycle with a wheel radius of...Ch. 10 - Consider an object on a rotating disk a distance r...Ch. 10 - A wheel is rotating about a fixed axis with...
Ch. 10 - A grindstone increases in angular speed from 4.00...Ch. 10 - Suppose a cars standard tires are replaced with...Ch. 10 - Figure OQ10.6 shows a system of four particles...Ch. 10 - As shown in Figure OQ10.7, a cord is wrapped onto...Ch. 10 - A constant net torque is exerted on an object....Ch. 10 - Prob. 10.9OQCh. 10 - A toy airplane hangs from the ceiling at the...Ch. 10 - A solid aluminum sphere of radius R has moment of...Ch. 10 - Is it possible to change the translational kinetic...Ch. 10 - Must an object be rotating to have a nonzero...Ch. 10 - Suppose just two external forces act on a...Ch. 10 - Explain how you might use the apparatus described...Ch. 10 - Example 10.6 Angular Acceleration of a Wheel A...Ch. 10 - Explain why changing the axis of rotation of an...Ch. 10 - Suppose you have two eggs, one hard-boiled and the...Ch. 10 - Suppose you set your textbook sliding across a...Ch. 10 - (a) What is the angular speed of the second hand...Ch. 10 - One blade of a pair of scissors rotates...Ch. 10 - If you see an object rotating, is there...Ch. 10 - If a small sphere of mass M were placed at the end...Ch. 10 - Three objects of uniform densitya solid sphere, a...Ch. 10 - Which of the entries in Table 10.2 applies to...Ch. 10 - Figure CQ10.15 shows a side view of a childs...Ch. 10 - A person balances a meterstick in a horizontal...Ch. 10 - (a) Find the angular speed of the Earths rotation...Ch. 10 - A potters wheel moves uniformly from rest to an...Ch. 10 - During a certain time interval, the angular...Ch. 10 - A bar on a hinge starts from rest and rotates with...Ch. 10 - A wheel starts from rest and rotates with constant...Ch. 10 - A centrifuge in a medical laboratory rotates at an...Ch. 10 - An electric motor rotating a workshop grinding...Ch. 10 - A machine part rotates at an angular speed of...Ch. 10 - A dentists drill starts from rest. After 3.20 s of...Ch. 10 - Why is the following situation impossible?...Ch. 10 - A rotating wheel requires 3.00 s to rotate through...Ch. 10 - The tub of a washer goes into its spin cycle,...Ch. 10 - A spinning wheel is slowed down by a brake, giving...Ch. 10 - Review. Consider a tall building located on the...Ch. 10 - A racing car travels on a circular track of radius...Ch. 10 - Make an order-of-magnitude estimate of the number...Ch. 10 - A discus thrower (Fig. P10.9) accelerates a discus...Ch. 10 - Figure P10.18 shows the drive train of a bicycle...Ch. 10 - A wheel 2.00 m in diameter lies in a vertical...Ch. 10 - A car accelerates uniformly from rest and reaches...Ch. 10 - A disk 8.00 cm in radius rotates at a constant...Ch. 10 - Prob. 10.22PCh. 10 - A car traveling on a flat (unbanked), circular...Ch. 10 - A car traveling on a flat (unbanked), circular...Ch. 10 - In a manufacturing process, a large, cylindrical...Ch. 10 - Review. A small object with mass 4.00 kg moves...Ch. 10 - Find the net torque on the wheel in Figure P10.14...Ch. 10 - The fishing pole in Figure P10.28 makes an angle...Ch. 10 - An electric motor turns a flywheel through a drive...Ch. 10 - A grinding wheel is in the form of a uniform solid...Ch. 10 - A 150-kg merry-go-round in the shape of a uniform,...Ch. 10 - Review. A block of mass m1 = 2.00 kg and a block...Ch. 10 - A model airplane with mass 0.750 kg is tethered to...Ch. 10 - A disk having moment of inertia 100 kg m2 is free...Ch. 10 - The combination of an applied force and a friction...Ch. 10 - Review. Consider the system shown in Figure P10.36...Ch. 10 - A potters wheela thick stone disk of radius 0.500...Ch. 10 - Imagine that you stand tall and turn about a...Ch. 10 - A uniform, thin, solid door has height 2.20 m,...Ch. 10 - Two balls with masses M and m are connected by a...Ch. 10 - Figure P10.41 shows a side view of a car tire...Ch. 10 - Following the procedure used in Example 10.7,...Ch. 10 - Three identical thin rods, each of length L and...Ch. 10 - Rigid rods of negligible mass lying along the y...Ch. 10 - The four particles in Figure P10.45 are connected...Ch. 10 - Many machines employ cams for various purposes,...Ch. 10 - A war-wolf or trebuchet is a device used during...Ch. 10 - A horizontal 800-N merry-go-round is a solid disk...Ch. 10 - Big Ben, the nickname for the clock in Elizabeth...Ch. 10 - Consider two objects with m1 m2 connected by a...Ch. 10 - The top in Figure P10.51 has a moment of inertia...Ch. 10 - Why is the following situation impossible? In a...Ch. 10 - In Figure P10.53, the hanging object has a mass of...Ch. 10 - Review. A thin, cylindrical rod = 24.0 cm long...Ch. 10 - Review. An object with a mass of m = 5.10 kg is...Ch. 10 - This problem describes one experimental method for...Ch. 10 - A uniform solid disk of radius R and mass M is...Ch. 10 - The head of a grass string trimmer has 100 g of...Ch. 10 - A cylinder of mass 10.0 kg rolls without slipping...Ch. 10 - A solid sphere is released from height h from the...Ch. 10 - (a) Determine the acceleration of the center of...Ch. 10 - A smooth cube of mass m and edge length r slides...Ch. 10 - A uniform solid disk and a uniform hoop are placed...Ch. 10 - A tennis ball is a hollow sphere with a thin wall....Ch. 10 - A metal can containing condensed mushroom soup has...Ch. 10 - As shown in Figure 10.13 on page 306, toppling...Ch. 10 - Review. A 4.00-m length of light nylon cord is...Ch. 10 - An elevator system in a tall building consists of...Ch. 10 - A shaft is turning at 65.0 rad/s at time t = 0....Ch. 10 - A shaft is turning at angular speed at time t =...Ch. 10 - Review. A mixing beater consists of three thin...Ch. 10 - The hour hand and the minute hand of Big Ben, the...Ch. 10 - A long, uniform rod of length L and mass M is...Ch. 10 - A bicycle is turned upside down while its owner...Ch. 10 - A bicycle is turned upside down while its owner...Ch. 10 - Prob. 10.76APCh. 10 - Review. As shown in Figure P10.77, two blocks are...Ch. 10 - Review. A string is wound around a uniform disk of...Ch. 10 - The reel shown in Figure P10.79 has radius R and...Ch. 10 - A common demonstration, illustrated in Figure...Ch. 10 - A uniform solid sphere of radius r is placed on...Ch. 10 - Review. A spool of wire of mass M and radius R is...Ch. 10 - A solid sphere of mass m and radius r rolls...Ch. 10 - A thin rod of mass 0.630 kg and length 1.24 m is...Ch. 10 - Prob. 10.85APCh. 10 - Review. A clown balances a small spherical grape...Ch. 10 - A plank with a mass M = 6.00 kg rests on top of...Ch. 10 - As a gasoline engine operates, a flywheel turning...Ch. 10 - As a result of friction, the angular speed of a...Ch. 10 - To find the total angular displacement during the...Ch. 10 - A spool of thread consists of a cylinder of radius...Ch. 10 - A cord is wrapped around a pulley that is shaped...Ch. 10 - A merry-go-round is stationary. A clog is running...Ch. 10 - A uniform, hollow, cylindrical spool has inside...
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