RESONANCE IN A TUBE OF VARIABLE LENGTH GOAL Understand resonance in tubes and perform elementary calculations. PROBLEM Figure 14.25a shows a simple apparatus for demonstrating resonance in a tube. A long tube open at both ends is partially sub merged in a beaker of water, and a vibrating tuning

Question

RESONANCE IN A TUBE OF VARIABLE LENGTH

GOAL Understand resonance in tubes and perform elementary calculations.

PROBLEM Figure 14.25a shows a simple apparatus for demonstrating resonance in a tube. A long tube open at both ends is partially sub merged in a beaker of water, and a vibrating tuning fork of unknown frequency is placed near the top of the tube. The length of the air column, [latex]L[/latex], is adjusted by moving the tube vertically. The sound waves gen erated by the fork are reinforced when the length of the air column corresponds to one of the resonant frequencies of the tube. Suppose the smallest value of [latex]L[/latex] for which a peak occurs in the sound intensity is [latex]9.00 \mathrm{~cm}[/latex]. (a) With this measurement, determine the frequency of the tuning fork. (b) Find the wavelength and the next two air-column lengths giving resonance. Take the speed of sound to be [latex]343 \mathrm{~m} / \mathrm{s}[/latex].

STRATEGY Once the tube is in the water, the setup is the same as a pipe closed at one end. For part (a), substitute values for [latex]v[/latex] and [latex]L[/latex] into Equation [latex]14.19[/latex]

[latex]f_n = n \frac{v}{4L} = nf_1 \qquad n = 1,3,5, . . .[/latex] [14.19]

with [latex]n=1[/latex], and find the frequency of the tuning fork. (b) The next resonance maximum occurs when the water level is low enough in the straw to allow a second node (see Fig. 14.25b), which is another half wavelength in distance. The third resonance occurs when the third node is reached, requiring yet another half-wavelength of distance. The frequency in each case is the same because it's generated by the tuning fork.

Accepted Answer

(a) Find the frequency of the tuning fork.

Substitute [latex]n=1, v=343 \mathrm{~m} / \mathrm{s}[/latex], and [latex]L_{1}=9.00 imes 10^{-2} \mathrm{~m}[/latex] into Equation 14.19:

[latex]f_{1}=\frac{v}{4 L_{1}}=\frac{343 \mathrm{~m} / \mathrm{s}}{4\left(900 imes 10^{-2} \mathrm{~m} ight)}=953 \mathrm{~Hz}[/latex]

(b) Find the wavelength and the next two water levels giving resonance.

Calculate the wavelength, using the fact that, for a tube open at one end, [latex]\lambda=4 L[/latex] for the fundamental.

[latex]\lambda=4 L_{1}=4\left(9.00 imes 10^{-2} \mathrm{~m} ight)=0.360 \mathrm{~m}[/latex]

Add a half-wavelength of distance to [latex]L_{1}[/latex] to get the next resonance position:

[latex]\begin{aligned}& L_{2}=L_{1}+\lambda / 2=0.090 0 \mathrm{~m}+0.180 \mathrm{~m}=0.270 \mathrm{~m}\end{aligned}[/latex]

Add another half-wavelength to [latex]L_{2}[/latex] to obtain the third resonance position:

[latex]L_{3}=L_{2}+\lambda / 2=0.270 \mathrm{~m}+0.180 \mathrm{~m}=0.450 \mathrm{~m}[/latex]

REMARKS This experimental arrangement is often used to measure the speed of sound, in which case the frequency of the tuning fork must be known in advance.

Question 14.10: RESONANCE IN A TUBE OF VARIABLE LENGTH GOAL Understand reson...

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