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Question

A string fixed at both ends has consecutive standing wave modes for which the distances between any two adjacent nodes are 9 cm and 8 cm respectively. If the tension is 20 N and the linear mass density of the string is 8 g/m, then:

A
The length of the string is 72 cm
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B
Fundamental frequency is 35 Hz
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C
Speed of wave on the string is 100 m/s
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D
Speed of wave on the string is 50 m/s
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Solution

The correct options are
A The length of the string is 72 cm
B Fundamental frequency is 35 Hz
D Speed of wave on the string is 50 m/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=72 cm
For fundamental frequency,



λ=2lλ=2×72=144 cm=1.44 m

wave speed on the stretched string is given by,
v=Tμ=208×103=1044=1002=50 m/s
Fundamental frequency.
f=vλ=501.44=34.7235 Hz

Thus, options (a) ,(b) and (d) are the correct answers.

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