A long string carries a wave; an $8\mathrm{.}00-$m segment of the string contains six complete wavelengths and has a mass of $180$ g$.$ The string vibrates sinusoidally with a frequency of $57\mathrm{.}0$ Hz and a peak$-$to$-$valley displacement of $13\mathrm{.}0$ cm$.($The "peak$-$to$-$valley" 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 $=8\mathrm{.}00$ m

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 $=$ d sin

$2$n

L

x $2$ft

y $=$

d

$2$

sin

$2$n

L

x $$

$2$

T

t

correct

y $=$

d

$2$

sin

$2$n

L

x $2$ft

y $=$

d

$2$

sin

$2$

L

x $2$ft

none of the above

Correct: Your answer is correct.

Correct. This is the correct expression for the wave.

Part $4$ of $7-$ Analyze: $($cont$.)$

$(2)$ Based on the correct choice in question $(1),$ substitute numerical values in the following expression for the wave as a function of position x and time t$,$ with length units in meters and time in seconds. Do not perform any calculations in this step; simply substitute the numerical values in the correct positions.

y $=$

d

$2$

sin

$2$n

L

x $2$ft

$=$

$0\mathrm{.}13$

Correct: Your answer is correct.

seenKey $0\mathrm{.}13$ m

$2$

sin

$26$

Correct: Your answer is correct.

seenKey $6$

$8$

Correct: Your answer is correct.

seenKey $8\mathrm{.}00$ m

x $$

$2$

$57$

Correct: Your answer is correct.

seenKey $57\mathrm{.}0$

t

Part $5$ of $7-$ Analyze: $($cont$.)$

$(3)$ Now perform the calculations in question $(2)$ to determine the functional expression for the wave described in the problem. Substitute the results of those calculations in the expression below.

y $=$ A sin$($kx $$t$)$

$=$

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m

sin

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m$1$

x $$

Incorrect with feedback: Your answer is incorrect. Click to see feedback.

s$1$

t