One string of a certain musical instrument is 75.0 cm long and has a mass of 8.75 g. It is bei

One string of a certain musical instrument is 75.0 cm long...

One string of a certain musical instrument is $75.0 \mathrm{~cm}$ long and has a mass of $8.75 \mathrm{~g}$. It is being played in a room where the speed of sound is $344 \mathrm{~m} / \mathrm{s}$. (a) To what tension must you adjust the string so that, when vibrating in its second overtone, it produces sound of wavelength $0.765 \mathrm{~m} ?$ (Assume that the breaking stress of the wire is very large and isn't exceeded.) (b) What frequency sound does this string produce in its fundamental mode of vibration?

Key Concepts

Mass per Unit Length

This concept relates to the density of the string material as distributed along its length. It is obtained by dividing the string's total mass by its length and is a key parameter in determining the wave speed on the string when it vibrates. A lower mass per unit length results in a higher wave speed for a given tension, and vice versa.

Wave Speed on a String

Wave speed on a string is determined by the square root of the tension divided by the mass per unit length. This relationship, v = ?(T/?), is crucial for understanding how changes in the tension of the string affect the frequency of the vibrations. It forms the basis for analyzing the vibrational modes of the string.

Standing Waves on Strings

When a string is fixed at both ends, only certain wavelengths and frequencies can form standing waves. These standing waves, or modes, occur when the string supports an integer number of half-wavelengths along its length. This principle is essential for determining the specific vibration patterns, including the fundamental mode and its overtones.

Harmonics and Overtones

Harmonics are the integer multiples of the fundamental frequency that occur when a string vibrates in its various modes. The fundamental mode (first harmonic) has the lowest frequency, and the overtones (or higher harmonics) correspond to the higher modes of vibration. The second overtone, for example, is typically the third harmonic mode in a string fixed at both ends.

Relationship Between Sound Frequency and Wavelength

In the context of musical instruments, the frequency of the sound produced by a vibrating string is related to the speed of sound in the surrounding medium, typically air, and the wavelength of the sound. This relationship, defined as f = v/?, connects the vibrational properties of the string to the acoustic output that is heard, thereby linking the physical vibrations to the perceived pitch.

Transcript

00:01 So this question is asking us to find the ah tension such that the string in the second overtone actually vibrates at a wavelength of point seven six five meters.

00:16 So we can say that l i's going.

00:20 We know that alice.

00:20 Seventy five centimeters or point two seven five meters and we know that the velocity of sound within this room is going to be three hundred forty four meters per second.

00:33 And we know the mass is going to be eight point five seven, eight point seven five grams or point zero zero eight seven five kilograms.

00:42 Always ah, right in s i u nits and then we know that the seconds overtone is the third harmonic.

00:53 So we have also ate wavelength of point seven six five meters.

01:00 So we should first find the linear density.

01:02 So lonely air density is going to be the master divided by the length point zero zero eight seven five kilograms divided by point seven five meters.

01:17 This is going to give us point zero one one seven kilograms per meter.

01:27 Then we need to find the wavelength in the third overtone.

01:31 Sorry, second overtone or the third harmonic.

01:34 This is going to be? ah, the wavelength of the wavelength of any harmonic is just going to be too well over.

01:39 And so that means that the wavelength of the third harmonic or the second overtone again will be too well over three.

01:47 And this will be two times point seven five, divided by three.

01:51 This is going to give us point five meters and at this point, we and say ok, what's the frequency? and the third in the second overtone? the third harmonic is going to be the propagation speed divided by the wavelength, the desired wavelength.

02:10 So this is actually going to be this wavelength here divided by point seven six five because i want to have a frequency such that it was be vibrating at in that have a frequency such that it will be vibrating and producing a wavelength of point seven six five meters...

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