Physics for Geologists, Second edition

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Force 13 the Sun, and the Earth revolving about its own axis. It was Kepler and Newton, however, who put substance into these assertions. Johannes Kepler (1571-1630) used Tycho Brahe's database (as we would now call it) of planetary positions, but first he had to solve the problem of refraction if the true positions of the heavenly bodies were to be obtained from the observed positions. After this had been done, he stated that Planetary orbits are ellipses, with the sun at one of the two foci. The vector radius of each planet to the Sun sweeps out equal areas of the ellipse in equal intervals of time. The squares of the sidereal periods of revolution are proportional to the cubes of their mean distances. (The Earth's sidereal period of revolution is 365.26 days.) These laws are kinematic laws: they describe motion without reference to force. Isaac Newton (1643-1727) developed the three dynamical laws of motion. These are: If a body is at rest, or moving at a constant speed in a straight line, it will continue at rest or at the constant speed in a straight line unless acted on by a force. (This is also called the Law of Inertia; it was not a new law but a re-statement of Galilee's.) The acceleration, or rate of change of velocity with time, a, is directly proportional to the force F applied and inversely proportional to the mass, m : a = Flm. The actions of two bodies on each other are always equal and opposite (or, to each action there is an equal and opposite reaction). The concept of force is seen to lie in its effects, and a newton is defined as the force that gives to a mass of one kilogram an acceleration of one metre per second per second. Newton was able to derive his law of gravitation from his, and Kepler's third, laws of motion, as applied to the Moon. The argument runs like this: The Moon's motion is not in a straight line, therefore it is being acted on by a force. The action of this force has an equal and opposite reaction, so the Moon is attracting the Earth and the Earth is attracting the Moon equally. This attractive force gives rise to an acceleration of the Moon towards the Earth, and of the Earth towards the Moon. Newton found the kinematic relationship between the radius (r) of a circular orbit, the period (T), and the centripetal acceleration (a) to be a = 4 ~c~r/~~, and from that inferred the inverse square law (see Note 1 on page 140 if you cannot do that for yourself). The Moon's period is 27.3 days, or 2.36 x lo6 s; its distance is 384 x lo6 m (sufficiently well known in Newton's day) so the Moon's acceleration towards the Earth was found to be 2.7 x lop3 m sp2. Copyright 2002 by Richard E. Chapman
14 Force The ratio of the Moon's acceleration and the Earth's, 2 .7~ 10-~/9.8, is about 1/3600. The distance of the Moon from the Earth is about 60 Earth radii. Newton also showed that all orbits are conic sections (that is, the trace of a plane cutting a right cone). These are circles, ellipses, parabolas or hyperbolas, depending on the angle the plane makes with the axis of the cone, and the angle of the side of the cone. Parabolic and hyperbolic orbits are never completed. (For the mathematical details see Rosen 1992, pp. 61ff.) Weight, mass and density The distinction between weight and mass, weight density and mass density, concerns the force of gravity. Newton's Law of Gravitation states that every particle of mass ml attracts every particle of mass m2 with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres of mass (which we can take for the time being to be the centres of the objects); and this force acts along a straight line joining them: where r is the distance between their centres of mass, and G is the universal constant of gravitation. G is not a true constant because it has dimensions and the number depends on the units used: which are those of the force of attraction times the square of the distance between the bodies, divided by the product of their masses (N m2 kgp2). The force of attraction F between two bodies gives to each an acceleration towards the other. The amount of acceleration depends on the mass of the body because F = mlal = m2a2. For objects on the surface of the Earth, there is an observable acceleration owing largely to the gravitational attraction between the centres of mass of the Earth and the object. This is denoted by g and is known as the acceleration due to gravity or the acceleration of free fall. (This is strictly a vector, having direction and magnitude: we shall come to that.) So the weight W of the object is W = mg. (2.3) The observable acceleration due to gravity, g, has several components. The largest is indeed the acceleration due to gravity, G (M~~)/R;. This is reduced everywhere on Earth, except at the poles, by a centrifugal acceleration due to the Earth's rotation about its axis, which is greatest at the Equator, where it amounts to about 34 mm s-2 or 34 x lop3 m sP2, and decreases towards the Copyright 2002 by Richard E. Chapman
Page 1 and 2: Physics for Geologists Second editi
Page 3 and 4: Copyright 2002 by Richard E. Chapma
Page 5 and 6: ... vlii Contents Polarization 47 L
Page 7 and 8: Figures Copyright 2002 by Richard E
Page 9 and 10: Tables Copyright 2002 by Richard E.
Page 11 and 12: xiv Preface waves diminishes with d
Page 13 and 14: xvi Preface provides. This book is
Page 15 and 16: xviii Acknowledgements to K. Whaler
Page 17 and 18: xx Symbols mass refractive index bu
Page 19 and 20: 2 Basic concepts Table 1.1 Dimensio
Page 21 and 22: 4 Basic concepts Table 1.2 Basic SI
Page 23 and 24: 6 Basic concepts pascal (Pa) and si
Page 25 and 26: 8 Basic concepts all the relevant d
Page 27 and 28: 10 Basic concepts relating them by
Page 29: 2 Force The dimensions of force are
Page 33 and 34: 16 Force Figure 2.1 The sum of the
Page 35 and 36: 18 Force When a motorcycle takes a
Page 37 and 38: 20 Force Figure 2.3 Torque is the p
Page 39 and 40: 22 Force How does momentum differ f
Page 41 and 42: 24 Force the weights - that is, the
Page 43 and 44: 26 Force Figure 2.6 Wall-sticking i
Page 45 and 46: 3 Gravity Universal gravity When di
Page 47 and 48: 30 Gravity are significant to geolo
Page 49 and 50: 32 Gravity Figure 3.2 The tractive,
Page 51 and 52: 34 Gravity Mean sea level It might
Page 53 and 54: 36 Gravity The benefits of space fl
Page 55 and 56: 38 Gravity Figure 3.6 Profile of th
Page 57 and 58: 40 Gravity Figure 3.8 The siphon at
Page 59 and 60: 4 Optics Light is an electromagneti
Page 61 and 62: 44 Optics Figure 4.2 Refraction. (a
Page 63 and 64: 46 Optics The coeficient of reflect
Page 65 and 66: 48 Optics only the light oscillatin
Page 67 and 68: 50 Optics called Iceland Spar. Two
Page 69 and 70: 52 Optics Diffraction If you shine
Page 71 and 72: 54 Optics Figure 4.7 Geometrical op
Page 73 and 74: 56 Optics Figure 4.9 The mirrors in
Page 75 and 76: 5 Atomic structure and age-dating P
Page 77 and 78: 60 Atomic structure and age-dating
Page 79 and 80: 62 Atomic structure and age-dating
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64 Atomic structure and age-dating
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66 Electromagnetic radiation radiat
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68 Electromagnetic radiation Source
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Heat and heat flow The noun heat ha
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72 Heat and heat flow Second Law: A
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8 Electricity and magnetism Electri
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Electricity and magnetism 77 Figure
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Electricity and magnetism 79 rivett
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Electricity and magnetism 8 1 defin
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Electricity and magnetism 83 is kno
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9 Stress and strain Pressure, p, is
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Stress and strain 87 than some natu
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These three elastic constants are r
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Stress and strain 91 Figure 9.1 New
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Stress and strain 93 Figure 9.2 Com
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Stress and strain 95 Figure 9.3 Ide
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Fracture Stress and strain 97 We fo
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Stress and strain 99 from which we
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10 Sea waves The nature of water wa
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Sea waves 103 Figure 10.1 Wave moti
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Sea waves 105 refraction by islands
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Acoustics: sound and other waves 10
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Acoustics: sound and other waves 10
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12 Fluids and fluid flow Introducti
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Fluids and fluid flow 1 13 Figure 1
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Fluids and fluid flow 115 Figure 22
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Fluids and fluid flow 117 and the f
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water above a vertical thickness h
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Fluids and fluid flow 121 where V i
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Fluids and fluid flow 123 ardentes,
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Fluids and fluid flow 125 Figure 12
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- 14 $ 12 E 10 K .- 3 8 r cll + 6 0
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Fluids and fluid flow 129 the liqui
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Some dangers of mathematical statis
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Appendix 139 What would happen if y
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Notes 141 and since x is very large
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References Allum, J. A. E. (1966) P
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References 145 -(1950) 'Mechanism o
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Further reading 147 Trigg, G. L. an
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