A string has a mass of 12.1g. The string is stretched with a force of 89.1 N, giving it length of 18.7 m. Then, the string vibrates transversely at precisely the frequency that corresponds to its fourth normal mode; that is at its fourth harmonic. What is the wavelength of the standing wave created in the string and what is the frequency in the standing wave
Question
A string has a mass of 12.1g. The string is stretched with a force of 89.1 N, giving it length of 18.7 m. Then, the string vibrates transversely at precisely the frequency that corresponds to its fourth normal mode; that is at its fourth harmonic. What is the wavelength of the standing wave created in the string and what is the frequency in the standing wave
Solution
To solve this problem, we need to use the formulas for the speed of a wave on a stretched string, the frequency of a harmonic on a string, and the relationship between speed, frequency, and wavelength.
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First, we calculate the speed of the wave on the string. The speed (v) of a wave on a stretched string depends on the tension force (F) and the mass per unit length (mu) of the string. The formula is:
v = sqrt(F/mu)
The mass per unit length of the string is the total mass divided by the total length, or mu = 12.1g / 18.7m = 0.647g/m = 0.000647 kg/m.
So, the speed of the wave on the string is:
v = sqrt(89.1N / 0.000647 kg/m) = 350.5 m/s.
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Next, we calculate the frequency of the fourth harmonic. The frequency (f) of the nth harmonic on a string of length L that is fixed at both ends is given by:
f = n*v / 2L
For the fourth harmonic (n=4), the frequency is:
f = 4350.5 m/s / (218.7m) = 74.8 Hz.
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Finally, we calculate the wavelength of the standing wave. The wavelength (lambda) is related to the speed and frequency of the wave by:
lambda = v / f
So, the wavelength of the standing wave is:
lambda = 350.5 m/s / 74.8 Hz = 4.69 m.
So, the wavelength of the standing wave created in the string is 4.69 m and the frequency of the standing wave is 74.8 Hz.
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