Physics for Scientists and Engineers: Foundations and Connections
1st Edition
ISBN: 9781133939146
Author: Katz, Debora M.
Publisher: Cengage Learning
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Chapter 14, Problem 66PQ
A steel cable 2.00 m in length and with cross-sectional radius 0.350 mm is used to suspend from the ceiling a 10.0-kg model aircraft that is flying in a horizontal circle with an angular speed of 6.00 rad/s. What is the strain produced in the cable?
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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 14 Solutions
Physics for Scientists and Engineers: Foundations and Connections
Ch. 14.1 - A rubber duck floats in a bathtub. Imagine moving...Ch. 14.1 - Prob. 14.2CECh. 14.2 - CASE STUDY Hanging a Plane from a Single Point In...Ch. 14.2 - Prob. 14.4CECh. 14.4 - Imagine two vertical rods initially of equal...Ch. 14 - What Is Static Equilibrium? Problems 13 are...Ch. 14 - Prob. 2PQCh. 14 - Two identical balls are attached to a...Ch. 14 - While working on homework together, your friend...Ch. 14 - Consider the sketch of a portion of a...
Ch. 14 - Prob. 6PQCh. 14 - Prob. 7PQCh. 14 - Prob. 8PQCh. 14 - The keystone of an arch is the stone at the top...Ch. 14 - Prob. 10PQCh. 14 - Stand straight and comfortably with your feet...Ch. 14 - Prob. 12PQCh. 14 - Prob. 13PQCh. 14 - Prob. 14PQCh. 14 - Prob. 15PQCh. 14 - Prob. 16PQCh. 14 - Prob. 17PQCh. 14 - Prob. 18PQCh. 14 - Prob. 19PQCh. 14 - Prob. 20PQCh. 14 - Prob. 21PQCh. 14 - The inner planets of our solar system are...Ch. 14 - Two Boy Scouts, Bobby and Jimmy, are carrying a...Ch. 14 - Prob. 24PQCh. 14 - A painter of mass 87.8 kg is 1.45 m from the top...Ch. 14 - Consider the situation in Problem 25. Tests have...Ch. 14 - Children playing pirates have suspended a uniform...Ch. 14 - Prob. 28PQCh. 14 - Prob. 29PQCh. 14 - A 5.45-N beam of uniform density is 1.60 m long....Ch. 14 - A wooden door 2.1 m high and 0.90 m wide is hung...Ch. 14 - A 215-kg robotic arm at an assembly plant is...Ch. 14 - Problems 33 and 34 are paired. One end of a...Ch. 14 - For the uniform beam in Problem 33, find the...Ch. 14 - Prob. 35PQCh. 14 - A square plate with sides of length 4.0 m can...Ch. 14 - Prob. 37PQCh. 14 - At a museum, a 1300-kg model aircraft is hung from...Ch. 14 - A uniform wire (Y = 2.0 1011 N/m2) is subjected...Ch. 14 - A brass wire and a steel wire, both of the same...Ch. 14 - In Example 14.3, we found that one of the steel...Ch. 14 - A carbon nanotube is a nanometer-scale cylindrical...Ch. 14 - A nanotube with a Youngs modulus of 1.000 1012 Pa...Ch. 14 - Consider a nanotube with a Youngs modulus of 2.130...Ch. 14 - Prob. 45PQCh. 14 - Use the graph in Figure P14.46 to list the three...Ch. 14 - Prob. 47PQCh. 14 - A company is testing a new material made of...Ch. 14 - Prob. 49PQCh. 14 - Prob. 50PQCh. 14 - Prob. 51PQCh. 14 - Prob. 52PQCh. 14 - Prob. 53PQCh. 14 - Prob. 54PQCh. 14 - Prob. 55PQCh. 14 - Prob. 56PQCh. 14 - A copper rod with length 1.4 m and cross-sectional...Ch. 14 - Prob. 58PQCh. 14 - Prob. 59PQCh. 14 - Bruce Lee was famous for breaking concrete blocks...Ch. 14 - Prob. 61PQCh. 14 - Prob. 62PQCh. 14 - Prob. 63PQCh. 14 - A One end of a metal rod of weight Fg and length L...Ch. 14 - Prob. 65PQCh. 14 - A steel cable 2.00 m in length and with...Ch. 14 - Prob. 67PQCh. 14 - Prob. 68PQCh. 14 - Prob. 69PQCh. 14 - Prob. 70PQCh. 14 - Prob. 71PQCh. 14 - Prob. 72PQCh. 14 - Prob. 73PQCh. 14 - We know from studying friction forces that static...Ch. 14 - Ruby, with mass 55.0 kg, is trying to reach a box...Ch. 14 - An object is being weighed using an unequal-arm...Ch. 14 - Prob. 77PQCh. 14 - A massless, horizontal beam of length L and a...Ch. 14 - A rod of length 4.00 m with negligible mass is...Ch. 14 - A rod of length 4.00 m with negligible mass is...Ch. 14 - A horizontal, rigid bar of negligible weight is...Ch. 14 - Prob. 82PQCh. 14 - Prob. 83PQCh. 14 - Prob. 84PQCh. 14 - Prob. 85PQ
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