== Review sheet ==

The <<ln("/Homework+?action=AttachFile&do=get&target=EX01-Review.pdf", review sheet)>> has been updated with one change.  The expression &ldquo;amplitude&rdquo; has been struck out in one place.  To avoid any misunderstanding, let us restrict, for the purpose of ''this'' exam, the use of &ldquo;amplitude&rdquo; in the following two cases (discussed in class) only.

  * The amplitude of a SHM.
  * The amplitude referring to $D(x,t)$ itself, when we say &ldquo;when waves mix, amplitudes add up, not intensities&rdquo; (superposition principle).

{{{#!wiki comment
However, the struck-out use of the word amplitude is a valid one, in a general context, if one takes into account the sign change, when appropriate.  In general, if you have a sinusoidal function, $A_x \sin (kx + \text{const})$ or $A_t \sin (\omega t + \text{const})$, then either $A_x$ or $A_t$ (both assumed to be $\ge 0$) is called the &ldquo;amplitude.&rdquo;
}}}

== Things that we discussed ==

 * Parallel axis theorem: $I = I_{cm} + M d^2$, $M$ is the total mass, $d$ is the distance between the pivot point and the center of mass, and $I$ ($I_{cm}$) is the rotational inertia around the pivot point (the center of mass).
 * We <<la(1.png,discussed this problem)>>.  One thing to note about finding extremum.  If $f(x) > 0$ and if you are given a task to find extrema for $\sqrt{f(x)}$.  Then, finding the extremum value of $x$ for $f(x)$ is equivalent to finding the extremum values of $\sqrt{f(x)}$ (or any positive power of $f(x)$, for that matter).  You can prove this by using differential calculus (chain rule), but the essential thing to note is that the function $G(Y) = Y^n$ for $n > 0$ and $Y > 0$ is a strictly increasing function of $Y$.  Therefore, the extremum for $Y$ is also the extremum for $G(Y)$ ($Y \equiv f(x)$, in our case).
 * We also <<la(2.png,discussed this problem)>>.
 * I think it may be confusing to some students when I say &ldquo;the sign is chosen by physics,&rdquo; in regards to the Doppler effect formula (see the review sheet to learn my notation here): $$ f^\prime = \frac{v \pm v_o}{v \pm v_s}\, f. $$  For those students, the following may be helpful.
 {{{#!mathjax
  \begin{align}
    +v_o & \text{ if observer is approaching source (higher pitch),} \\\\
    -v_o & \text{ if observer is moving away from source (lower pitch),} \\\\
    +v_s & \text{ if source is moving away from observer (lower pitch),} \\\\
    -v_s & \text{ if source is approaching observer (higher pitch)}. 
  \end{align}
 }}}
 Another important thing to note: when you determine the sign for $v_o$, you should temporarily put $v_s = 0$, and also when you determine the sign for $v_s$, you should temporarily put $v_o = 0$.
 * The area symbol $S$ should be taken as the cross sectional area of the string, when one considers the string wave and its energy and so on.  One overarching principle: $x, y, z$ (spatial coordinates) or $S$ (a derived quantity in terms of the shape of the string in the $yz$ plane) all refer to the string (or general the medium) '''in equilibrium''' when there is no wave.  They remain static variables even when waves occur, since $D$ takes care of the dynamical displacement that happens when waves occur.
 * $\omega = v k$ is completely equivalent to $v = f\lambda$ (not discussed during the review session, but I thought this might be helpful to some students.)