A pipe of 30 cm long open at both the ends produces harmonics. Which harmonic mode of pipe resonates at 1.1…
A pipe of 30 cm long open at both the ends produces harmonics. Which harmonic mode of pipe resonates at 1.1 kHz source?
(Take, speed of sound in air $= 330\text{ ms}^{-1}$)
Fifth harmonic
Fourth harmonic
Third harmonic
Second harmonic
Solution
According to the mode of vibration of air column in an open organ pipe,
First mode of vibration, $\nu_1 = \frac{v}{2l}$ ...(i)
where, $\nu_1 =$ frequency of vibration,
$v =$ speed of sound and $l =$ length of organ pipe.
The frequency of $n$th mode of vibration,
$\nu_n = n \times \nu_1 = n \times \left(\frac{v}{2l}\right)$ ...(ii)
Given, $\nu_n = 1.1\text{ kHz} = 1100\text{ Hz}$, $v = 330\text{ ms}^{-1}$ and
$l = 30\text{ cm} = 0.30\text{ m}$
Substituting these values in Eq. (ii), we get
$1100 = n \times \left(\frac{330}{2 \times 0.30}\right)$
$\Rightarrow n = \frac{1100 \times 2 \times 0.30}{330} \Rightarrow n = \frac{660}{330} = 2$
So, second harmonic mode of pipe resonates.