A metal bar clamped at its centre resonates in its fundamental mode to produce longitudinal waves of frequency `4 kHz`. Now the clamp is moved to one end . If `f_(1)` and `f_(2)` be the frequencies of first overtone and second overtone respectively then ,

Options

A. `3 f_(2) = 5 f_(1)`
B. `3 f_(1) = 5 f_(2)`
C. `f_(2) = 2f_(1)`
D. `2f_(2) = f_(1)`

Answer

A

Explanation

With clamp at the centre `L = lambda//2` for the fundamental.
So , `f' = (v)/( 2 L) = 4 kHz`
When clamp is moved to one end then
`L = (lambda)/(4) , ( 3lambda)/(4) , ( 5 lambda)/(4) , … ( 2 n - 1) (lambda)/(4)`
For `n = 1 , 2 , …`
`f_(n) = ( 2 n - 1) (v)/( 4 L)`
`f_(0) = 2 kHz` ( `1 st` harmonic)
`f_(1) = 6 kHz` ( ` 2 nd` harmonic or `1 st` overtone)
`f_(2) = 10 kHz` ( ` 3rd` harmonic or ` 2 nd` overtone )