Physics for Geologists, Second edition

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Notes 1 The centripetal force required to keep a body in circular orbit is mv2/r, as we found in the dimensional analysis on page 11. The velocity of the body is 2nr/T where T is the period, the time taken for one orbit. So, mV2 m4n2r F,=ma=--- - T2 4n2r and a = - Y T2 ' Kepler's Third Law states that the square of the period is proportional to the cube of the mean distance, so Similarly, given the inverse square law, you can infer Kepler's third law of planetary motion, that r3 cc T2. 2 Work is a weight multiplied by the distance moved in the direction of a force. So if you drop a mass m from a height h above the ground, its velocity on impact is v = gt. Draw a graph now of v against t, and you have a triangle (you don't need to put numbers to the graph) the area of which is the distance fallen. Thus the distance fallen is The potential energy before dropping is converted to kinetic energy on impact (ignoring friction) so Ep = force x height = mg x igt2 = imgZt2 = LmU2 2 = E k . 3 If you take the Earth's radius as the unit of distance, then, as stated in the text, the attraction of the Moon is roughly proportional to 1/602 at the Earth's centre, 1159~ at the sublunar point, and 1/612 at the antipode. If x is the number of Earth's radii between the centre of the Earth and the centre of the Moon, the force at the sublunar point is proportional to l/(x- at the centre of the Earth, to 1/x2; and at the antipode, to l/(x+ I ) ~ Two . approximations are used. (x - 1)2 = x2 -2x+ 1 Copyright 2002 by Richard E. Chapman
Notes 141 and since x is very large relative to 1,x2 - 2x + 1 is very nearly equal to x2 - 2x which is equal to x2[1 - (2/x)]. The second approximation is that (1 - a)-' is very nearly equal to (1 +a). This will be demonstrated below. So, the difference between the force at the sublunar point and the force at the centre of the Earth is proportional to Similarly, the difference between the forces at the antipode and at the Earth's centre is proportional to -2/x3. The second approximation can be demonstrated as follows: Set up an equation: 1/(1 - a) = 1 + b. So, If a and b are small relative to 1, the product ab can be neglected, and a is very nearly equal to b, and 1/(1 - a) is very nearly equal to 1 +a. The approximation is also the first two terms of the binomial expansion (1 - a)-' = 1 + a + a2 + . . . for a2 < 1, in this case, very much less than one. Returning to the quantities in the text, the Earth's radius RE and the distance to the moon dM, and the difference is 4 Let the velocities of two parallel wave-trains travelling in the same direction in deep water differ by dV, and their wavelengths by dh - the faster waves having the longer wavelength. Each time the faster, longer wave passes the shorter and slower by dh, the coincidence of waves will fall back by one wavelength in the time dX/dV. So the group velocity will be V - h(dv/dh). In deep water, V = (g~/2n)1/2, and h(dV/dh) = (h/2)(g/2nh)'/2 = i(g~/2n)'/2 = V/2. See, for example, Tricker (1984) for a fuller discussion of water waves, and John (1984) for their motion. 5 The meaning of the term degrees of freedom is only superficially simple. In the matter of correlation, any two points will lie on a straight line. Even two points on the circumference of a circle will lie on a straight line, but not three. Two points provide no information about the relationship between the two. If you have ten paired measurements and you wish to assess their degree of correlation, any test of significance must be carried out for (10 - 2) degrees of freedom. More generally, if there are n pairs, there are (n - 2) degrees of freedom in regression analysis. If you make 10 independent measurements of, for example, the ages and heights of some students in a school, then there are 10 degrees of freedom. But if you continue the measurements until you have 1000, and record the numbers within Copyright 2002 by Richard E. Chapman
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Physics for Geologists Second editi
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Copyright 2002 by Richard E. Chapma
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... vlii Contents Polarization 47 L
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Figures Copyright 2002 by Richard E
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Tables Copyright 2002 by Richard E.
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xiv Preface waves diminishes with d
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xvi Preface provides. This book is
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xviii Acknowledgements to K. Whaler
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xx Symbols mass refractive index bu
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2 Basic concepts Table 1.1 Dimensio
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4 Basic concepts Table 1.2 Basic SI
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6 Basic concepts pascal (Pa) and si
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8 Basic concepts all the relevant d
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10 Basic concepts relating them by
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2 Force The dimensions of force are
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14 Force The ratio of the Moon's ac
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16 Force Figure 2.1 The sum of the
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18 Force When a motorcycle takes a
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20 Force Figure 2.3 Torque is the p
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22 Force How does momentum differ f
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24 Force the weights - that is, the
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26 Force Figure 2.6 Wall-sticking i
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3 Gravity Universal gravity When di
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30 Gravity are significant to geolo
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32 Gravity Figure 3.2 The tractive,
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34 Gravity Mean sea level It might
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36 Gravity The benefits of space fl
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38 Gravity Figure 3.6 Profile of th
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40 Gravity Figure 3.8 The siphon at
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4 Optics Light is an electromagneti
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44 Optics Figure 4.2 Refraction. (a
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46 Optics The coeficient of reflect
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48 Optics only the light oscillatin
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50 Optics called Iceland Spar. Two
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52 Optics Diffraction If you shine
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54 Optics Figure 4.7 Geometrical op
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56 Optics Figure 4.9 The mirrors in
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5 Atomic structure and age-dating P
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60 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
Page 105 and 106: These three elastic constants are r
Page 107 and 108: Stress and strain 91 Figure 9.1 New
Page 109 and 110: Stress and strain 93 Figure 9.2 Com
Page 111 and 112: Stress and strain 95 Figure 9.3 Ide
Page 113 and 114: Fracture Stress and strain 97 We fo
Page 115 and 116: Stress and strain 99 from which we
Page 117 and 118: 10 Sea waves The nature of water wa
Page 119 and 120: Sea waves 103 Figure 10.1 Wave moti
Page 121 and 122: Sea waves 105 refraction by islands
Page 123 and 124: Acoustics: sound and other waves 10
Page 125 and 126: Acoustics: sound and other waves 10
Page 127 and 128: 12 Fluids and fluid flow Introducti
Page 129 and 130: Fluids and fluid flow 1 13 Figure 1
Page 131 and 132: Fluids and fluid flow 115 Figure 22
Page 133 and 134: Fluids and fluid flow 117 and the f
Page 135 and 136: water above a vertical thickness h
Page 137 and 138: Fluids and fluid flow 121 where V i
Page 139 and 140: Fluids and fluid flow 123 ardentes,
Page 141 and 142: Fluids and fluid flow 125 Figure 12
Page 143 and 144: - 14 $ 12 E 10 K .- 3 8 r cll + 6 0
Page 145 and 146: Fluids and fluid flow 129 the liqui
Page 147 and 148: Some dangers of mathematical statis
Page 155: Appendix 139 What would happen if y
Page 159 and 160: References Allum, J. A. E. (1966) P
Page 161 and 162: References 145 -(1950) 'Mechanism o
Page 163: Further reading 147 Trigg, G. L. an
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Physics for Geologists, Second edition

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