Table Of ContentOffbeat Physics
Offbeat Physics
Machines, Meditations and
Misconceptions
Edited by
P. I. C. Teixeira
First edition published 2022
by CRC Press
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Library of Congress Cataloging‑in‑Publication Data
Names: Teixeira, Paulo Ivo Cortez, 1965- editor.
Title: Offbeat physics : machines, meditations and misconceptions / edited
by P.I.C. Teixeira.
Description: First edition. | Boca Raton : CRC Press, 2022. | Includes
bibliographical references and index.
Identifiers: LCCN 2021049576 | ISBN 9781032021164 (hardback) | ISBN
9781032033938 (paperback) | ISBN 9781003187103 (ebook)
Subjects: LCSH: Physics. | Machinery, Dynamics of.
Classification: LCC QC23.2 .O44 2022 | DDC 530--dc23/eng/20211201
LC record available at https://lccn.loc.gov/2021049576
ISBN: 978-1-032-02116-4 (hbk)
ISBN: 978-1-032-03393-8 (pbk)
ISBN: 978-1-003-18710-3 (ebk)
DOI: 10.1201/9781003187103
Publisher’s note: This book has been prepared from camera-ready copy provided by the author.
Typeset in CMR10
by KnowledgeWorks Global Ltd.
This book is dedicated to Maria Helena Cardoso,
to whom the Physics Department at ISEL owes its
inception.
Contents
Preface xiii
Acknowledgments xv
Contributors xvii
I Machines 1
1 Dynamics of Braking Vehicles: From Coulomb Friction to
Anti-Lock Braking Systems 3
J. M. Tavares
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 The dynamics of braking using Coulomb friction . . . . . . . 5
1.2.1 Static friction force . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Kinetic friction force . . . . . . . . . . . . . . . . . . . 7
1.2.3 The two regimes for braking. . . . . . . . . . . . . . . 7
1.3 The advantage of the ABS . . . . . . . . . . . . . . . . . . . 9
1.4 Comparison with the model [3] and with real data . . . . . . 11
References 15
2 Simple Thermodynamics of Jet Engines 17
P. Patr´ıcio and J. M. Tavares
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Performances of jet engines . . . . . . . . . . . . . . . . . . . 19
2.3 The simplest model of a jet engine . . . . . . . . . . . . . . . 20
2.4 Jet engine with an ideal compressor and turbine . . . . . . . 22
2.5 Overall efficiency and thrust . . . . . . . . . . . . . . . . . . 25
2.6 Non-ideal compressor and turbine . . . . . . . . . . . . . . . 28
2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
References 33
3 Surprises of the Transformer as a Coupled Oscillator System 35
J. P. Silva and A. J. Silvestre
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
vii
viii Contents
3.2 Natural frequencies of a transformer . . . . . . . . . . . . . . 37
3.3 Resonant frequencies of a driven transformer . . . . . . . . . 39
3.3.1 Decoupled circuits . . . . . . . . . . . . . . . . . . . . 42
3.3.2 Maximum coupling . . . . . . . . . . . . . . . . . . . . 42
3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
References 45
4 MaximumThermodynamicPowerCoefficient ofaWind Tur-
bine 47
J. M. Tavares and P. Patr´ıcio
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 Power coefficient of a wind turbine . . . . . . . . . . . . . . . 49
4.3 One-dimensional reversible fluid flows . . . . . . . . . . . . . 50
4.3.1 Incompressible flow. . . . . . . . . . . . . . . . . . . . 52
4.3.2 Isentropic flow of an ideal gas . . . . . . . . . . . . . . 52
4.3.3 Isothermal flow of an ideal gas . . . . . . . . . . . . . 54
4.3.4 Power coefficient calculations . . . . . . . . . . . . . . 57
4.3.5 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.5 Supplementary material . . . . . . . . . . . . . . . . . . . . . 61
4.5.1 Generalized clausius inequality . . . . . . . . . . . . . 61
4.5.2 Linear mom‚entum equation . . . . . . . . . . . . . . . 62
4.5.3 Proof that pnˆ dS =0 for a compressible ideal flow 62
CV z
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
References 65
II Meditations 67
5 Mutual Inductance between Piecewise Linear Loops 69
A. C. Barroso and J. P. Silva
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.2 The vector potential . . . . . . . . . . . . . . . . . . . . . . . 71
5.3 Line integral along a straight path . . . . . . . . . . . . . . . 73
5.3.1 General case . . . . . . . . . . . . . . . . . . . . . . . 73
5.3.2 Planar wires . . . . . . . . . . . . . . . . . . . . . . . 77
5.4 The magnetic flux and mutual inductance . . . . . . . . . . . 77
5.5 First application: two square wires on the plane . . . . . . . 78
5.6 Second application: two square wires stacked . . . . . . . . . 80
5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
References 87
Contents ix
6 The Hertz Contact in Chain Elastic Collisions 89
P. Patr´ıcio
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.2 Independent collisions . . . . . . . . . . . . . . . . . . . . . . 90
6.3 Noninstantaneous collisions . . . . . . . . . . . . . . . . . . . 91
6.3.1 The Hertz contact . . . . . . . . . . . . . . . . . . . . 92
6.3.2 Dynamical equations . . . . . . . . . . . . . . . . . . . 93
6.3.3 Numerical resolution . . . . . . . . . . . . . . . . . . . 94
6.4 Discussion and conclusions . . . . . . . . . . . . . . . . . . . 98
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
References 101
7 Tilted Boxes on Inclined Planes 103
A. M. Nunes and J. P. Silva
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7.2 Boxes resting evenly on the plane . . . . . . . . . . . . . . . 105
7.2.1 Case 1: no sliding and no tumbling . . . . . . . . . . . 108
7.2.2 Case 2: no sliding and tumbling forward . . . . . . . . 108
7.2.3 Case 3: sliding down and no tumbling . . . . . . . . . 109
7.2.4 Case 4: sliding down and tumbling forward . . . . . . 110
7.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.3 Boxes tilted with respect to the plane . . . . . . . . . . . . . 112
7.3.1 The case where 0<ϕ≤β . . . . . . . . . . . . . . . . 114
7.3.2 The case where β <ϕ<π/2 . . . . . . . . . . . . . . 115
7.3.2.1 The case of β <ϕ<π/2 and a=0 . . . . . 116
7.3.2.2 The case of β <ϕ<π/2 and a>0 . . . . . 117
7.3.2.3 The case of β <ϕ<π/2 and a<0 . . . . . 118
7.3.2.4 Summary . . . . . . . . . . . . . . . . . . . . 119
7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
References 123
8 Magnetic Forces Acting on Rigid Current-Carrying Wires
Placed in a Uniform Magnetic Field 125
A. Casaca and J. P. Silva
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
References 131
9 Comparing a Current-Carrying Circular Wire with Polygons
of Equal Perimeter: Magnetic Field versus Magnetic Flux 133
J. P. Silva and A. J. Silvestre
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
9.2 Calculating the vector potential . . . . . . . . . . . . . . . . 136
