A vibrating string of certain length `l` under a tension `T` resonates with a mode corresponding to the first overtone (third harmonic ) of an air column of length `75 cm` inside a tube closed at one end. The string also generates `4 beats//s` with a tuning fork of frequency `n` . Now when the tension of the string is slightly increased the number of beats reduces to `2` per second. Assuming the velocity of sound in air to `340 m//s` , the frequency `n` the tuning fork in `H_(Z)` is
(a) `344`
(b) `336`
(c ) `117.3`
(d) `109.3`

Options

A. `344`
B. `336`
C. `117.3`
D. `109.3`

Answer

A

Explanation

As string and tube are in resonance `f_(1) = f_(2)`
`|f_(1) - n| = 4 Hz`.
When `T` increase, `f_(1)` also increases. It is given that beat frequency decreases to `2 Hz`.
`rArr n - f_(1) = 4`
`n = 4 + f_(1)`
as `f_(1) = f_(2)`
`n = 4 + f_(2)`

`n = (4 + f_(2))`
`f_(2) = (3V)/(4l) = (3 xx 340)/(4 xx (3//4)) = 340`
`n = 344`