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Cell[CellGroupData[{
Cell["Period, Frequency, Amplitude", "Section",
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Cell[TextData[{
  StyleBox["Things that Repeat",
    FontFamily->"Trebuchet MS",
    FontSize->12,
    FontWeight->"Bold"],
  "\n",
  "Today we will discuss phenomena that ",
  StyleBox["repeat in time",
    FontSlant->"Italic"],
  ".  There are some terms and models that apply to any system -- circular \
motion, stock market cycles, the displacement of your vocal chords while \
singing a C note -- which may be said to ",
  StyleBox["repeat in time",
    FontSlant->"Italic"],
  ". Name another system which repeats in time."
}], "Text"],

Cell[TextData[{
  StyleBox["Period",
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    FontWeight->"Bold"],
  "\nAny system that repeats in time in a predictable fashion can be \
characterized by the ",
  StyleBox["period",
    FontSlant->"Italic"],
  " - the amount of time between repetitions.\n",
  StyleBox["Period (",
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  StyleBox["T ",
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  StyleBox[" is the Length of Time between repetitions.  It is expressed in \
Seconds.   ",
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  StyleBox["\n",
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  StyleBox["Frequency",
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  "\nEquivalently, you can say how ",
  StyleBox["many ",
    FontSlant->"Italic"],
  "(a number) fit in a certain amount of time -- the ",
  StyleBox["frequency",
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  ".\n",
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  StyleBox["f",
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  StyleBox["    ",
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  "\nIf a system takes 0.1 seconds to repeat, it must repeat 10 times a \
second - 10 Hz.\n\nEx. A 33 rpm turntable rotates at a frequency of   ",
  Cell[BoxData[
      \(TraditionalForm\`33\ rev\/min\[CenterDot]\(1\ min\)\/\(60\ s\) = \
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Is the frequency higher or lower than before? Is the period higher or lower \
than before?\n\nPlease note: a Hertz is just a convenient name for the unit \
\"inverse seconds\", ",
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  StyleBox["Amplitude",
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  "\nWe can average a repeating phenomenon over time. The ups will generally \
cancel the downs, and we can describe the behavior as an oscillation around \
an average, or equilibrium, position. The maximum excursion from equilibrium \
is called the amplitude.  For the common systems remember that\n",
  StyleBox["   For Spring Problems, Amplitude is a distance (meters).   ",
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  "\nFor an alternating current electrical system, the amplitude is a current \
(",
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  Cell[BoxData[
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Cell[TextData[{
  "The law for a spring (Hooke's Law) states that a spring extended from \
equilibrium imposes a restoring force equal to the extension:\n",
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  "\nIf we stretch the spring to the right, there is a force, proportional \
and opposite to the extension, pulling it leftwards to equilibrium.  If we \
release it, it will zip back to the center -- but it will overshoot, as it \
now has momentum leftwards.  The momentum will carrry it to a point on the \
left, opposite its initial release point, where it will pause (velocity \
zero). It will then zip rightwards, back through the equilibrium. As it \
passes through the equilibrium point the acceleration is zero, but now its \
rightward momentum carries it through to the point at which it was released.  \
This will repeat indefinitely for a high quality spring (one with no drag or \
dissipation).\n\nPosition is left max\t\t\tPosition is zero\t\t\t\tPosition \
is right max\nVelocity is zero  \t\t\tVelocity is max (l/r)\t\t\t       \
Velocity is zero\nAccel is max, to right\t\t\tAccel is zero\t\t\t\t Accel is \
max, to left\n\
|------------------------------------------------------|----------------------\
-----------------------------------------|\n\n",
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Cell[TextData[{
  "For a pendulum near equilibrium,\n",
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  "\nIf we tip the pendulum to the right, there is a force, proportional and \
opposite to the extension, pulling it leftwards to equilibrium.  If we \
release it, it will zip back to the center -- but it will overshoot, as it \
now has momentum leftwards.  The momentum will carrry it to a point on the \
left, opposite its initial release point, where it will pause (velocity \
zero). It will then zip rightwards, back through the equilibrium. As it \
passes through the equilibrium point the acceleration is zero, but now its \
rightward momentum carries it through to the point at which it was released.  \
This will repeat indefinitely for a high quality pendulum, with no drag or \
dissipation.\n\nAngle is left max\t\t\tAngle is zero\t\t\t\t\t   Angle is \
right max\nVelocity is zero  \t\t\tVelocity is max (l/r)\t\t\t\t       \
Velocity is zero\nAccel is max, to right\t\t\tAccel is zero\t\t\t\t\t Accel \
is max, to left\n\
|------------------------------------------------------|----------------------\
-----------------------------------------|\n\n",
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   \n       Velocity is large when Angle & Acceleration are small.    \n      \
 Velocity is small when Angle & Acceleration are large.  ",
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Cell[TextData[{
  "The force laws for spring motion and pendulum motion are suspiciously \
similar: ",
  Cell[BoxData[
      \(TraditionalForm\`F\  = \ Junk\ \[CenterDot]\ Amplitude\)]],
  ". This is the simplest model we have in physics for a repeating system, \
and it is called a Simple Harmonic Oscillator. The spring system has a \
solution in terms of the cosine of the amplitude:\n\t",
  Cell[BoxData[
      \(TraditionalForm\`x\  = \ A\ \(cos(2\ \[Pi]\ \(\(f\ t\)\&\ \))\)\)]],
  "\nThis is a repeating function of time, with amplitude ",
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  " and frequency ",
  Cell[BoxData[
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  ".  The frequency of the oscillation stems directly from the force law; it \
is the frequency of the system if disturbed and then left to cope.  That ",
  StyleBox["natural frequency",
    FontSlant->"Italic"],
  "  is given by Kim's Equation:\n",
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  StyleBox[" ",
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  StyleBox[" ",
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  "\t\"Gil frequently drops his two pies\"\n",
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  StyleBox["Natural Frequency for Simple Harmonic Motion is ",
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  "Notice that the frequency does NOT depend on the amplitude.\n\n\t\"The \
more things change, the more they stay the same\""
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Cell["\<\
Humans have a leg length of about 1 meter, and a stride of about 0.5 meter.   \
What is the natural frequency of the human stride? The period? The amplitude? \
How does the natural frequency of a short person compare to that of a tall \
person?
Adults tend to get carsick from frequencies below one Hz. Assume that \
oscillations at or above our natural walking frequency are comfortable but \
that frequencies below our natural walking frequency induce carsickness. \
Should children get carsick more or less easily?\
\>", "Text",
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Cell["Kinetic and Potential Energy", "Subsection",
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Cell[TextData[{
  "The law for a spring (Hooke's Law) states that a spring extended from \
equilibrium imposes a restoring force equal to the extension:\n",
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      FormBox[
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  "\n\nIf an object compresses the spring, that makes the system mad; \
therefore it\n\tCompressing:\tIncreases PE; decreases KE; system does \
negative work on object.\nAs the spring relaxes back to equilibrium (string \
uncompresses to its default position), the system is happy:\n\tRelaxing:\t\
Decreases PE; increases KE; system does positive work on object.\t\nAs the \
system passes equilibrium, the object still has KE; thus, it overshoots, \
making the system mad again:\n\tStretching:\tIncreases PE; decreases KE; \
system does negative work on object.\n\nPosition is left max\t\t\tPosition is \
zero\t\t\t\tPosition is right max\nVelocity is zero  \t\t\tVelocity is max \
(l/r)\t\t\t       Velocity is zero\nAccel is max, to right\t\t\tAccel is zero\
\t\t\t\t Accel is max, to left\nPE is maximum\t\t\tPE is zero\t\t\t\t       \
PE is maximum\nKE is zero\t\t\t\tKE is maximum\t\t\t\t   KE is zero\n\
|------------------------------------------------------|----------------------\
-----------------------------------------|\n\nFor Simple Harmonic Motion,\n",
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 ",
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  "Extremely Important Wave Formula:\n\t",
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  ", frequency: \"per seconds\"; gives ",
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  ", meters per second.\nWhat is the wavelength of radio station FM 99.1? \
What frequency is the size of an atom? of a cell?\nA rule of thumb is that a \
speaker cannot reproduce sound whose wavelength is larger than its diameter. \
What is the frequency cutoff for a 10\" (0.25 m) speaker."
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Cell["\<\
Standing wave
node / anti-node
fundamental\
\>", "Text",
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Cell[TextData[{
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  "\n\t\t\t",
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  "\nThe velocity of a standing wave on a stretched string is \n\t",
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  ",\tvelocity = ",
  Cell[BoxData[
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  "\nEx. A stand-up bass has 1.5m strings which weigh 50 g, tensioned to 3000 \
N.  What is its fundamental frequency? If the string is plucked at the \
midpoint, what will the frequency and wavelength of the harmonic be?"
}], "Text",
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Cell["In-Class Compendium, Passages 4 and 5 (time: 18:00)", "Subsubsection",
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Cell["If time -- In-Class Compendium, Passage 15 (time: 8:00)", \
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