Principles of Physics: A Calculus-Based Text
5th Edition
ISBN: 9781133104261
Author: Raymond A. Serway, John W. Jewett
Publisher: Cengage Learning
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Chapter 12, Problem 65P
A pendulum of length L and mass M has a spring of force constant k connected to it at a distance h below its point of suspension (Fig. P12.65). Find the frequency of vibration of the system for small values of the amplitude (small θ). Assume the vertical suspension rod of length L is rigid, but ignore its mass.
Figure P12.65
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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 12 Solutions
Principles of Physics: A Calculus-Based Text
Ch. 12.1 - A block on the end of a spring is pulled to...Ch. 12.2 - Consider a graphical representation (Fig. 12.3) of...Ch. 12.2 - Figure 12.4 shows two curves representing...Ch. 12.2 - An object of mass m is hung from a spring and set...Ch. 12.4 - A grandfather clock depends on the period of a...Ch. 12.5 - Prob. 12.6QQCh. 12 - Which of the following statements is not true...Ch. 12 - Prob. 2OQCh. 12 - Prob. 3OQCh. 12 - Prob. 4OQ
Ch. 12 - Prob. 5OQCh. 12 - Prob. 6OQCh. 12 - If a simple pendulum oscillates with small...Ch. 12 - Prob. 8OQCh. 12 - Prob. 9OQCh. 12 - Prob. 10OQCh. 12 - Prob. 11OQCh. 12 - Prob. 12OQCh. 12 - Prob. 13OQCh. 12 - You attach a block to the bottom end of a spring...Ch. 12 - Prob. 15OQCh. 12 - Prob. 1CQCh. 12 - The equations listed in Table 2.2 give position as...Ch. 12 - Prob. 3CQCh. 12 - Prob. 4CQCh. 12 - Prob. 5CQCh. 12 - Prob. 6CQCh. 12 - The mechanical energy of an undamped blockspring...Ch. 12 - Prob. 8CQCh. 12 - Prob. 9CQCh. 12 - Prob. 10CQCh. 12 - Prob. 11CQCh. 12 - Prob. 12CQCh. 12 - Consider the simplified single-piston engine in...Ch. 12 - A 0.60-kg block attached to a spring with force...Ch. 12 - When a 4.25-kg object is placed on top of a...Ch. 12 - The position of a particle is given by the...Ch. 12 - You attach an object to the bottom end of a...Ch. 12 - A 7.00-kg object is hung from the bottom end of a...Ch. 12 - Prob. 6PCh. 12 - Prob. 7PCh. 12 - Prob. 8PCh. 12 - Prob. 9PCh. 12 - A 1.00-kg glider attached to a spring with a force...Ch. 12 - Prob. 11PCh. 12 - Prob. 12PCh. 12 - A 500-kg object attached to a spring with a force...Ch. 12 - In an engine, a piston oscillates with simple...Ch. 12 - A vibration sensor, used in testing a washing...Ch. 12 - A blockspring system oscillates with an amplitude...Ch. 12 - A block of unknown mass is attached to a spring...Ch. 12 - Prob. 18PCh. 12 - Prob. 19PCh. 12 - A 200-g block is attached to a horizontal spring...Ch. 12 - A 50.0-g object connected to a spring with a force...Ch. 12 - Prob. 22PCh. 12 - Prob. 23PCh. 12 - Prob. 24PCh. 12 - Prob. 25PCh. 12 - Prob. 26PCh. 12 - Prob. 27PCh. 12 - Prob. 28PCh. 12 - The angular position of a pendulum is represented...Ch. 12 - A small object is attached to the end of a string...Ch. 12 - A very light rigid rod of length 0.500 m extends...Ch. 12 - A particle of mass m slides without friction...Ch. 12 - Review. A simple pendulum is 5.00 m long. What is...Ch. 12 - Prob. 34PCh. 12 - Prob. 35PCh. 12 - Show that the time rate of change of mechanical...Ch. 12 - Prob. 37PCh. 12 - Prob. 38PCh. 12 - Prob. 39PCh. 12 - Prob. 40PCh. 12 - Prob. 41PCh. 12 - Prob. 42PCh. 12 - Prob. 43PCh. 12 - Prob. 44PCh. 12 - Four people, each with a mass of 72.4 kg, are in a...Ch. 12 - Prob. 46PCh. 12 - Prob. 47PCh. 12 - Prob. 48PCh. 12 - Prob. 49PCh. 12 - Prob. 50PCh. 12 - Prob. 51PCh. 12 - Prob. 52PCh. 12 - Prob. 53PCh. 12 - Prob. 54PCh. 12 - Prob. 55PCh. 12 - A block of mass m is connected to two springs of...Ch. 12 - Review. One end of a light spring with force...Ch. 12 - Prob. 58PCh. 12 - A small ball of mass M is attached to the end of a...Ch. 12 - Prob. 60PCh. 12 - Prob. 61PCh. 12 - Prob. 62PCh. 12 - Prob. 63PCh. 12 - A smaller disk of radius r and mass m is attached...Ch. 12 - A pendulum of length L and mass M has a spring of...Ch. 12 - Consider the damped oscillator illustrated in...Ch. 12 - An object of mass m1 = 9.00 kg is in equilibrium...Ch. 12 - Prob. 68PCh. 12 - A block of mass M is connected to a spring of mass...
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