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Question

A 1 m long horizontal rope, having a mass of 40 g, is fixed at one end and is tied to a light string at the other end. The tension in the rope is 400 N. What will be the wavelengths (in metres) in the first and second overtone ?

A
34 , 34
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B
43 , 45
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C
54 , 53
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D
45 , 43
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Solution

The correct option is B 43 , 45
Given,
Length of rope, L=1 m
Mass, M=40 g or 0.04 kg
Mass per unit length, μ=ML=0.04 kg/m
Tension in the rope, T=400 N

It behaves like a string fixed at one end.
wave speed on the stretched string, v=Tμ=4000.04=100 m/sec

Fundamental frequency, f0 on a string fixed at one end is given by
f0=v4L
f0=1004=25 Hz

Frequency of the first overtone on the string fixed at one end
f1=3v4L
Wavelength of first overtone λ1=vf1=4L3
From the given data,
λ1=4×13=43 m

Frequency of the second overtone, f2 on the string,
f2=5v4L
Wavelength of second overtone λ2=vf2=4L5
From the given data,
λ2=4×15=45 m

Thus, option (b) is the correct answer.

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