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Analysis Model Tutorial

A long string carries a wave; an $8\mathrm{.}00-m$ segment of the string contains six complete wavelengths and has a mass of $180g.$ The string vibrates sinusoidally with a frequency of $57\mathrm{.}0Hz$ and a peak$-$to$-$valley displacement of $13\mathrm{.}0cm.($The "peak$-$tovalle $y^n$ distance is the vertical distance from the farthest positive position to the farthest negative position.$)$

$($a$)$ Write the function that describes this wave traveling in the positive $x$ direction.

$($b$)$ Determine the power being supplied to the string.

Part $1$ of $7-$ Conceptualize:

Notice what information we are given in this problem. We have a representative length $L=8\mathrm{.}00m$ of the string and the number $n=6$ of wavelengths in it$,$ allowing us to find the wavelength of the wave. The mass given in the problem statement for that segment is $m.$ The length $L$ of the segment of the string is also the length $L$ of a segment of the wave that contains $n$ wavelengths. The segment of string stays fixed in its horizontal position while the segment of the wave moves along the string with a constant speed. We are also given the frequency $f$ of the wave and the peak$-$to$-$valley distance, which we will call $d.$

Part $2$ of $7-$ Categorize:

There is no particle, rigid object, or system described in this problem on which to base an analysis model. There is a wave, and we apply the only wave model we have seen so far, the traveling wave analysis model from this chapter.

Part $3$ of $7-$ Analyze:

$($a$)$ Write the function that describes this wave traveling in the positive $x$ direction.

$(1)$ In terms of the quantities given in the problem statement and denoted by the symbols indicated in the Conceptualize step, write a symbolic expression as a function of position $x$ and time $t$ for the wave described in the problem.

$y=$dsin$(\frac{2n}{L}x-2ft)$

$y=\frac{d}{2}sin(\frac{2n}{L}x-\frac{2}{T}t)$

$y=\frac{d}{2}sin(\frac{2n}{L}x-2ft)$

$y=\frac{d}{2}sin(\frac{2}{L}x-2ft)$

none of the above