Simple Nature - Light and Matter

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Three functions with thesame curvature at x=0.c / The amplitude would usuallybe defined as the distancefrom equilibrium to one extremeof the motion, i.e., half the totaltravel.Physically, k is a measure of stiffness. For example, the heavy-dutysprings in a car’s shock absorbers would have a high value of k. Itis often referred to as the spring constant, but we’re only using aspring as an example here. As shown in figure b, any two functionsthat have U(0) = 0, dU/ dx = 0, and d 2 U/ dx 2 = k, with the samevalue of k, are virtually indistinguishable for small values of x, so ifwe want to analyze small oscillations, it doesn’t even matter whichfunction we assume. For simplicity, we’ll just use U(x) = (1/2)kx 2from now on.Now we’re ready to analyze the mass-on-a-spring system, whilekeeping in mind that it’s really only a representative example ofa whole class of similar oscillating systems. We expect that themotion is going to repeat itself over and over again, and since we’renot going to include frictional heating in our model, that repetitionshould go on forever without dying out. The most interesting thingto know about the motion would be the period, T , which is theamount of time required for one complete cycle of the motion. Wemight expect that the period would depend on the spring constant,k, the mass, m, and and the amplitude, A, defined in figure c. 11In examples like the brachistochrone and the Apollo 11 mission,it was generally necessary to use numerical techniques to determinethe amount of time required for a certain motion. Once again, let’sdust off the time3 function from page 93 and modify it for ourpurposes. For flexibility, we’ll define the function U(x) as a separatePython function. We really want to calculate the time required forthe mass to come back to its starting point, but that would beawkward to set up, since our function works by dividing up thedistance to be traveled into tiny segments. By symmetry, the timerequired to go from one end to the other equals the time requiredto come back to the start, so we’ll just calculate the time for half acycle and then double it when we return the result at the end of thefunction. The test at lines 16-19 is necessary because otherwise atthe very end of the motion we can end up trying to take the squareroot of a negative number due to rounding errors.11 Many kinds of oscillations are possible, so there is no standard definition ofthe amplitude. For a pendulum, the natural definition would be in terms of anangle. For a radio transmitter, we’d use some kind of electrical units.114 Chapter 2 Conservation of Energy
1 import math2 def u(k,x):3 return .5*k*x**245 def osc(m,k,a,n):6 x=a7 v=08 dx = -2.*a/n9 t=010 e = u(k,x)+.5*m*v**211 for i in range(n):12 x_old = x13 v_old = v14 x = x+dx15 kinetic = e-u(k,x)16 if kinetic>> print(osc(1.,1.,1.,100000))warning, K= -1.43707268307e-12 >> print(osc(1.,1.,2.,100000))warning, K= -5.7482907323e-12
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Fullerton, Californiawww.lightandma
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Contents0 Introduction and Review0.
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5.5 More About Heat Engines . . . .
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669.11.3 Magnetic Fields by Ampère
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The Mars Climate Orbiter is prepare
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temperatures and with many combinat
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idence that quarks have smaller par
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give clearer explanations.Finally,
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It can also be handy to have a rela
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The prefix centi-, meaning 10 −2
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goes like this:V = 1 3 Ah[1]A = πr
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the notation 10 0 to stand for one,
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calculation, 5.04 cm, was really no
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0.2 Scaling and Order-of-Magnitude
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to many times your own height. The
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g / 1. This plank is as long as it
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the front panels of the three violi
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Correct solution #4: The area of a
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here?” The scientific Mr. Spock w
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1. Don’t even attempt more than o
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Problem 10.mean, however, is define
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(d) Find the person’s acceleratio
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Albert Einstein, and his moustache,
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54 Chapter 0 Introduction and Revie
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a / Portrait of Monsieur Lavoisiera
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1.1.1 Problem-solving techniquesHow
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∆m = 0, where m is the total mass
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different substances will have diff
Page 64 and 65: c / Left: In a frame of referenceth
Page 66 and 67: f / Discussion question B.(discusse
Page 68 and 69: Self-check D.since the derivative o
Page 70 and 71: ProblemsThe symbols √ , , etc. ar
Page 72 and 73: difficult? ⊲ Solution, p. 932Key
Page 74 and 75: Heat energy can be convertedto ligh
Page 76 and 77: a new form of invisible “mystery
Page 78 and 79: f / A realistic drawing of Joule’
Page 80 and 81: numbers. With a purely numerical ap
Page 82 and 83: of gravity doesn’t change much if
Page 84 and 85: field. If the plane can start from
Page 86 and 87: m / Discussion question C.n / A hyd
Page 88 and 89: 88 Chapter 2 Conservation of Energy
Page 90 and 91: A car drives over a cliff.new frame
Page 92 and 93: How long does it take to move 1 met
Page 94 and 95: a / Approximations to thebrachistoc
Page 96 and 97: a / An ellipse is circle that hasbe
Page 98 and 99: to deduce the general equation for
Page 100 and 101: to the mass of the object that inte
Page 102 and 103: The minimum velocity required for t
Page 104 and 105: and since sin θ dθ occurs in the
Page 106 and 107: that is deeper than r. Under the as
Page 108 and 109: 2.3.6 ⋆ Evidence for repulsive gr
Page 110 and 111: a / A vivid demonstration thatheat
Page 112 and 113: ture. This is a very good question,
Page 116 and 117: This looks like a cosine function,
Page 118 and 119: ProblemsThe symbols √ , , etc. ar
Page 120 and 121: Problem 16.13 Anya and Ivan lean ov
Page 122 and 123: Problem 27.then we’d have U/m =
Page 124 and 125: 37 Two springs with spring constant
Page 126 and 127: ExercisesExercise 2A: Reasoning wit
Page 128 and 129: 128 Chapter 2 Conservation of Energ
Page 130 and 131: 3.1 Momentum In One Dimensiona / Sy
Page 132 and 133: eing to the right. The initial mome
Page 134 and 135: 3.1.3 Momentum compared to kinetic
Page 136 and 137: 3.1.4 Collisions in one dimensiong
Page 138 and 139: could never do that again in a mill
Page 140 and 141: i / The highjumper’s body passeso
Page 142 and 143: where x 1 is the mass of the first
Page 144 and 145: 3.1.6 The center of mass frame of r
Page 146 and 147: The airbag increases ∆tso as to r
Page 148 and 149: d / Two magnets exert forceson each
Page 150 and 151: x (m) t (s)10 1.8420 2.8630 3.8040
Page 152 and 153: 3.2.4 Forces between solidsConserva
Page 154 and 155: Maximum acceleration of a car examp
Page 156 and 157: Discussion QuestionA Criticize the
Page 158 and 159: C A pool ball is rebounding from th
Page 160 and 161: forces, as if the hand were a pair
Page 162 and 163: This may not sound like an impressi
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stroke are both executed in straigh
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Accelerating a cart example 35If yo
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w / A wedge.x / Archimedes’ screw
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As in the preceding example, we hav
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that have the same frequency is a s
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around until I got this result, sin
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quality factor, Q, is defined as Q
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ight direction to add energy to the
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i / Example 45: a viola withouta mu
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Collapse of the Nimitz Freeway exam
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end up canceling out, however:Q =
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186 Chapter 3 Conservation of Momen
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c / Bullets are dropped and shot at
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the asteroid’s energy and boostin
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coordinate axes. Even though ∆x,
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of the three components,∆p x = 0
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A useless vector operation example
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Solving for the unknowns gives∆x
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o / Example 64.p / Adding vectors g
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How to generalize one-dimensional e
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needed to support the object in fig
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The easiest method is the one demon
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Incorrect solution #4:(same notatio
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The magnitude of the acceleration i
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dropped out like a trap door, showi
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af / Breaking trail, by WalterE. Bo
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know how to integrate with respect
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ProblemsThe symbols √ , , etc. ar
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Problem 16.Problem 19.resistance.)1
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of gravity on Mars.(a) Find the tim
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partner force. (a) A swimmer speeds
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Problem 45the 0.4-gram masses would
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53 If you walk 35 km at an angle 25
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(b) Interpret this equation in the
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Problem 71.232 Chapter 3 Conservati
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77 A car accelerates from rest. At
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Problem 81.81 Complete example 71 o
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ExercisesExercise 3A: Force and Mot
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Exercise 3C: Worksheet on Resonance
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Exercise 3D: Vectors and MotionEach
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244 Chapter 3 Conservation of Momen
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An overhead view of apiece of putty
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inward toward the hinge will have n
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special case, we can choose to visu
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j / Two asteroids collide.k / Every
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Note that although the factors of 2
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Relationship between force and torq
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is straight down, which is perpendi
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⊲ All three objects in the figure
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aa / Example 10.to either side of e
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momentum tells us that L = mrv sin
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In the absence of any torque, a rig
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Radial acceleration at the surface
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The parallel axis theorem example 1
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differential. The result isarea ===
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of inertia as if the object was smo
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h / Moments of inertia of somegeome
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4.3 Angular Momentum In Three Dimen
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14 deg, and it points along an axis
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manner like this, then the definiti
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k / Example 27.torque to the left w
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We can also generalize the plane-ro
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Problem 1.Problem 6.Problem 8.Probl
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14 (a) The bar of mass m is attache
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Problem 22.22 The sun turns on its
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35 The nucleus 168 Er (erbium-168)
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ExercisesExercise 4A: TorqueEquipme
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pesky set of constraints on heat en
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mit more air into the cavity behind
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Pressure of lava underneath a volca
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h / A simplified version of anideal
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for us to construct a simple connec
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For the first time we have an inter
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a / 1. The temperature differencebe
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Carnot engines operating between a
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from A to B, he lets it by, but whe
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B When we run the Carnot engine in
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f / A phase space for a singleatom
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time, the energy sharing is very un
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will the the one that maximizes the
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If the gas is monoatomic, then we k
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5.4.4 The arrow of time, or “this
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5.4.6 Summary of the laws of thermo
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may be in the form of microscopic d
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significant amount of heat to flow
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while the smaller area under the bo
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7 (a) Determine the ratio between t
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338 Chapter 5 Thermodynamics
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end of this book, we’ll even see
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c / As the wave pattern passes the
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the wave move ahead faster and get
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k / Hitting a key on a pianocauses
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Our final result for the speed of t
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care, because the delay is the same
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An easy way to visualize this is in
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can be constructed as a superpositi
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⊲ Looking up the speed of light i
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scientists, to speak of the Big Ban
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6.2 Bounded WavesSpeech is what sep
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d / An uninverted reflection. There
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our intuitive expectation of strong
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valid solutions. In the following s
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j / In the mirror image, theareas o
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m / A pulse bounces backand forth.i
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q / Graphs of loudness versusfreque
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s / Standing waves on a rope. (PSSC
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“cavity” and “neck” parts o
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Problem 5.Problem 8.5 The figure sh
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Problem 16.16 A Fabry-Perot interfe
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c / Newton’s laws do not distingu
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f / The correspondence principlereq
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d / A Galilean version of therelati
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g / Three types of transformations
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system, velocities are always unitl
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p / Apparatus used for the testof r
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sumption, the assumption that it ma
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at rest relative to the water. But
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7.2.4 No action at a distanceThe Ne
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so that Alice can complete her moti
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time into different regions accordi
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position relative to the sun at exa
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7.2.7 ⋆ Four-vectors and the inne
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would depend on the relative veloci
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7.3 DynamicsSo far we have said not
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In this frame, as expected, the sma
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But this whole argument was based o
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meters per second, so converting to
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Expressing γ as ( 1 − v 2 /c 2)
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7.3.4 ⋆ ProofsThis optional secti
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these lines. For example, a car sit
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An Einstein’s ring. Thedistant ob
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two-dimensional universe as if it w
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h / 1. A ray of light is emittedupw
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object that that isn’t influenced
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it impossible for any observer to b
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a variant on the Penrose singularit
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in it. Instead, it is currently spe
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2 Astronauts in three different spa
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(d) Simplify your answer to part c
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Problem 25b. Redrawn fromVan Baak,
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ExercisesExercise 7A: The Michelson
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Exercise 7B: Sports in Slowlightlan
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448 Chapter 7 Relativity
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Exercise 7D: Misconceptions about R
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452 Chapter 7 Relativity
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sun, moon, stars, and planets were
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Two types of chargeWe can easily co
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the other acquires an equal amount
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to get the beam up to speed in the
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telling us that we know about matte
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the muddle. The row-and-column sche
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the force of air friction canceled
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ectly evaluate the implications of
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that they were indeed electrically
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C Thomson found that the m/q of an
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the object with a net positive char
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thickness of material the radioacti
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It was already known that although
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particle. It turned out that it was
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For example they could easily strip
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cording to Mosely, the atomic numbe
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that fly off to see what was inside
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solution was found by measuring the
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o / A nuclear power plant at Catten
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neutrons. In a nuclear fission bomb
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We can now list all four of the kno
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1. Our sun’s source of energy is
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a / The known nuclei, represented o
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killed, because the DNA becomes una
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e / Wild Przewalski’s horsesprosp
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Problem 1. Top: A realisticpicture
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12 The subatomic particles called m
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ExercisesExercise 8A: Nuclear decay
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saxophone, every technological tool
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Example 1Ions moving across a cell
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ut the original meaning was to trav
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Force depends only on position. Sin
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Here are a few questions and answer
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flow through it.For many substances
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superconductivity in metals that wo
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j / Example 9. In 1 and 2,charges t
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look like the usual resistor. The f
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9.1.5 Current-conducting properties
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attery acid becomes depleted of hyd
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9.2 Parallel and Series CircuitsIn
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to each resistance, resulting inI t
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where “...” means that the sum
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the two resistors in figure h/3.We
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Choice of high voltage for power li
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A complicated circuit example 20⊲
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Problem 2.ProblemsThe symbols √ ,
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Problem 16.only count that as one u
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knob turned all the way clockwise,
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A printed circuit board, likethe ki
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ExercisesExercise 9A: Voltage and C
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Exercise 9C: Reasoning About Circui
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556 Chapter 9 Circuits
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a / A bar magnet’s atoms are(part
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defined. The ship’s captain can m
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Reduction in gravity on Io due to J
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j / Example 3.self-check AFind an e
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a dipole moment — they are define
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10.2.1 One dimensionVoltage is elec
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The interpretation is that if you b
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Figure c shows some examples of way
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For large values of d, this express
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where Q is the total charge of the
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g / Example 12: variation of the fi
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corners to the disk and transform i
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10.4 Energy In Fieldsa / Two opposi
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self-check DWe can think of the qua
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of the inward field contributed by
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space, 2 while charge doesn’t, we
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a / The symbol for a capacitor.b /
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ounding each capacitor will be half
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in time:x ↔ qv ↔ Iself-check GH
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due to its own momentum. It perform
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or|V | =∣ LdIdt ∣ ,which in man
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the inductor resists such a sudden
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Example 26.Death by solenoid; spark
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ordering.( 1 √2+√ i ) 2 = √ 1
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x / The complex number e iφlies on
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Figure aa shows a useful way to vis
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Inductors tend to be big, heavy, ex
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The reason for using the trig ident
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a purely inductive or capacitive lo
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Resonance with damping example 33
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orE outward, on side 1 A + E outwar
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|E| = kq totalr 2 ,where r is the r
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y considering its point charges ind
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Discussion Questionsg / Discussion
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with charge, change the Coulomb con
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The symmetry between the two sides
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where dv is the volume of the cube.
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The three terms in the divergence a
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ProblemsThe symbols √ , , etc. ar
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Problem 19.Problem 20.proton, for e
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the lightning strike.(b) Based on y
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units.(b) Verify that RC has units
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lating freely (without any driving
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ExercisesExercise 10A: Field Vector
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5. Now hook up the two solenoids in
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A large current is createdby shorti
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time I used it implicitly was in fi
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f / A standard dipole madefrom a sq
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and substituting q = λh and v = m/
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o / Magnetic forces cause abeam of
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electric field, a magnetic one, can
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u / In this scene from SwanLake, th
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so the total field in the z directi
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For the y component, we havee / A s
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We can pin down the result even mor
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i / The field of a dipole.where r i
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circulates around the y axis, so at
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to be, not where it is now. Coulomb
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We have found one specific example
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to a flagpole, we can cancel out a
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11.4 Ampère’s Law In Differentia
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y evaluating the field at the midpo
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i.e.,F = axˆx + byˆx + cˆx + dx
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k / A summary of the derivative, gr
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c / Detail from Ascending andDescen
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the magnet. Are these atomic curren
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field in her region of space has be
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A changing magnetic flux makes a cu
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nearly zero. By Faraday’s law, th
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c / An Ampèrian surface superimpos
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and Maxwell’s equations becomeΦ
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k / Red and blue light travelat the
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second, the zero-point is located a
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is the magnitude of the momentum ve
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Discussion question A.Discussion qu
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only penetrates to a very small dep
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elationship D = ɛE would actually
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esulting in cancellation.The opposi
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a low permeability, while the other
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n / A fluxgate compass.is externall
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6 Two parallel wires of length L ca
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an instant at the upper right, but
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Problem 25.20 Four long wires are a
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Problem 35.Problem 37.32 Verify Amp
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Problem 42.beam of light usually co
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(c) Discuss the relationship betwee
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(f) Use conservation of energy to r
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5. Now position yourself with your
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12.1.1 The nature of lightThe cause
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An image of Jupiter andits moon Io
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d / Two self-portraits of theauthor
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ultimate truth about light, but the
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• There is a tendency to conceptu
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self-check AEach of these diagrams
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m / Discussion question B.Discussio
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12.2 Images by ReflectionInfants ar
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Discussion QuestionA The figure sho
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down magnification is AB/DE. A repe
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i / A Newtonian telescopebeing used
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B Locate the images formed by two p
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12.3 Images, QuantitativelyIt sound
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⊲ The object and image angles are
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12.3.2 Other cases with curved mirr
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signs also have to be memorized for
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h / A diverging mirror in the shape
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j / Spherical mirrors are thecheape
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general type of eye that we share w
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shown on this graph and then attemp
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the mechanical model would predict
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that your calculator will flash an
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D Classify the examples shown in fi
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12.4.6 ⋆ Microscopic description
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12.5 Wave OpticsElectron microscope
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of a crystal? Sound waves are used
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j / Thomas Youngk / Double-slit dif
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object that diffracts it, so the tr
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Although the equation λ/d = sin θ
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things we’ve learned about diffra
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apidly changing distances; on reuni
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6 The figure on the next page shows
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14 Here’s a game my kids like to
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27 Suppose a converging lens is con
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same depth, but not quite. [Check:
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Problem 42.42 Panel 1 of the figure
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46 The figure below shows two diffr
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53 The beam of a laser passes throu
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Problem 59.the ancient problem of i
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3. Now imagine the following new si
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Exercise 12B: Object and Image Dist
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Exercise 12D: Double-Source Interfe
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Exercise 12E: Single-slit diffracti
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Exercise 12F: Diffraction of LightE
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energy, instead of being spread out
Page 824 and 825:
correlated. If they have been playi
Page 826 and 827:
small.The statement that the rule f
Page 828 and 829:
But the y axis can no longer be a u
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for a long time once it gets there,
Page 832 and 833:
j / Calibration of the 14 C dating
Page 834 and 835:
given by(probability of a ≤ x ≤
Page 836 and 837:
k / In recent decades, a huge hole
Page 838 and 839:
A wave is partially absorbed.c / A
Page 840 and 841:
f / The hamster in her hamsterball
Page 842 and 843:
How would you extract h from the gr
Page 844 and 845:
F Does E = hf imply that a photon c
Page 846 and 847:
of approaching this issue.)j / Bull
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We assume v is small enough so that
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the probability distribution will b
Page 852 and 853:
trons that we have already used for
Page 854 and 855:
equations of general validity are t
Page 856 and 857:
wave carries high probability and w
Page 858 and 859:
An infinite sine wave can only tell
Page 860 and 861:
momentum implies a definite kinetic
Page 862 and 863:
of snapshots would amount to a desc
Page 864 and 865:
close are the electrons to the limi
Page 866 and 867:
dimensional particle in a box, and
Page 868 and 869:
Applying this to conservation of en
Page 870 and 871:
the probability of making it throug
Page 872 and 873:
Three dimensionsFor simplicity, we
Page 874 and 875:
1. Oscillations can go backand fort
Page 876 and 877:
C The figure shows a skateboarder t
Page 878 and 879:
a / Eight wavelengths fit aroundthi
Page 880 and 881:
As shown by these examples, the unc
Page 882 and 883:
e / The three states of the hydroge
Page 884 and 885:
f / The energy levels of a particle
Page 886 and 887:
∂r/∂x = x/r comes in handy. Com
Page 888 and 889:
50% of its time in each atom. It’
Page 890 and 891:
tus to the z axis) and more memorab
Page 892 and 893:
ut why does that have anything to d
Page 894 and 895:
the data are only inaccurate due to
Page 896 and 897:
14 The photoelectric effect can occ
Page 898 and 899:
Problem 25.(b) Sketch a graph showi
Page 900 and 901:
conservation of energy and momentum
Page 902 and 903:
Problem 43.43 On pp. 884-885 of sub
Page 904 and 905:
ExercisesExercise 13A: Quantum Vers
Page 906 and 907:
⊲ First we convert the equation i
Page 908 and 909:
Programming With PythonThe purpose
Page 910 and 911:
Appendix 2: MiscellanyUnphysical
Page 912 and 913:
18 bestt = t19 c1 = bestc120 c2 = b
Page 914 and 915:
p i + ∑ iThe spin theoremTheorem:
Page 916 and 917:
Appendix 3: Photo CreditsExcept as
Page 918 and 919:
Appendix 4: Hints and SolutionsHint
Page 920 and 921:
Hints for Chapter 6Page 377, proble
Page 922 and 923:
Answers to Self-Checks for Chapter
Page 924 and 925:
Page 301: (1) Not valid. The equati
Page 926 and 927:
Page 584:N −1 m −2 C 2 V 2 m
Page 928 and 929:
the dashed ray.Page 748: You should
Page 930 and 931:
direction it was initially going (i
Page 932 and 933:
Page 52, problem 38: The cone of mi
Page 934 and 935:
Page 224, problem 37: (a) Spring co
Page 936 and 937:
object, we have static friction, wh
Page 938 and 939:
Page 549, problem 29:(a) Conservati
Page 940 and 941:
Page 798, problem 19: (a) The objec
Page 942 and 943:
.0.8 Notation and unitsquantity uni
Page 944 and 945:
surfaces. For comparison, a typical
Page 946 and 947:
elations:a = ∆v∆tx = 1 2 at2 +
Page 948 and 949:
we are moving along with the projec
Page 950 and 951:
the number of oscillations required
Page 952 and 953:
In general, the cross product of ve
Page 954 and 955:
Sound waves consist of increases an
Page 956 and 957:
signs for charge is that with this
Page 958 and 959:
know that there is a delay in time
Page 960 and 961:
is related byΦ = 4πkq into the ch
Page 962 and 963:
to use sophisticated models such as
Page 964 and 965:
Chapter 13, Quantum Physics, page 8
Page 966 and 967:
oth position and momentum, the Heis
Page 968 and 969:
Schrödinger’s, 864cathode rays,
Page 970 and 971:
unstable, 87equipartition theorem,
Page 972 and 973:
Joule, James, 73paddlewheel experim
Page 974 and 975:
Einstein’s early theory, 838energ
Page 976 and 977:
Nikola, 178tesla (unit), 652thermal
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Simple Nature - Light and Matter

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