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Question

A tuning fork vibrating with a sonometer in fundamental mode having a wire of length 20 cm produces 5 beats per second. The beat frequency does not change if the length of the wire is changed to 21 cm. The frequency of the tuning fork is

A
200 Hz
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B
210 Hz
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C
205 Hz
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D
215 Hz
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Solution

The correct option is C 205 Hz
Let the frequency of the tuning fork is
f.
The frequency of vibrations of sonometer wire is,
f=nv2l [two ends fixed]
Fundamental frequency(n=1),
fo=v2l
Now, beat frequency,
fb=|fof|=|v2lf|
When l=20 cm=0.20 m
fb=v2×0.20f
fb=v2×0.20f=5 Hz .......(1) [given]
When l=21 cm=0.21 m
fb=fv2×0.21
[frequency of sonometer decreases as length increases]
So,
v2×0.20f=fv2×0.21
[given]
2f=v0.40+v0.42 .........(2)
2(v0.405)=v0.40+v0.42
[from (1)]
On solving we get, v=84 m/s
Putting this in (2),
2f=840.40+840.42
f=205 Hz

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