Review sheet
The review sheet has been updated with one change. The expression “amplitude” has been struck out in one place. To avoid any misunderstanding, let us restrict, for the purpose of this exam, the use of “amplitude” 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 “when waves mix, amplitudes add up, not intensities” (superposition principle).
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 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 discussed this problem.
I think it may be confusing to some students when I say “the sign is chosen by physics,” 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. \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.)
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 “amplitude.”