A 2⋅00 m-long rope, having a mass of 80 g, is fixed at one end and is tied to a light string at the other end. The tension in the string is 256 N. (a) Find the frequencies of the fundamental and the first two overtones. (b) Find the wavelength in the fundamental and the first two overtones.
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Explanation:
- In the question, it is given:
- Long rope has the length (L) = 2.00 m
- Rope has the mass = 0.08 kg
- Tension (T) = 256 N
- Linear mass density, m = 0.04 kg/m
- Tension, T=256 N
- Wave velocity, v = Tm
- ⇒v = 1602
- ⇒v = 80 m/s
- For fundamental frequency:
- L = λ4 ⇒f = vλ = 10 Hz
(a) The overtones of frequency are shown below:
- 1st overtone = 30 Hz
- 2nd overtone = 50 Hz
(b) The wavelength in the fundamental and the first two overtones
- Λ = 4l = 8 m
- Therefore, λ1 = 2.67 m
- λ2 =1.6 m
- So, 8 m, 2.67 m and 1.6 m are the respective wavelengths.
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