A string fixed at both ends has consecutive standing wave modes for which the distances between any two adjacent nodes are 9cm and 8cm respectively. If the tension is 20N and the linear mass density of the string is 8g/m, then:
A
The length of the string is 72cm
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B
Fundamental frequency is 35Hz
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C
Speed of wave on the string is 100m/s
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D
Speed of wave on the string is 50m/s
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Solution
The correct options are A The length of the string is 72cm B Fundamental frequency is 35Hz D Speed of wave on the string is 50m/s Let l be the length of the string and x be the seperation between any two consecutive nodes. Then For n loops on the string: 9n=l.........(1) For n+1 loops on the string: 8(n+1)=l.........(2)
From (1) and (2), we get n=8 so, l=9×8=72cm For fundamental frequency,
λ=2l⇒λ=2×72=144cm=1.44m
wave speed on the stretched string is given by, v=√Tμ=√208×10−3=√1044=1002=50m/s ∴ Fundamental frequency. f=vλ=501.44=34.72≈35Hz
Thus, options (a) ,(b) and (d) are the correct answers.