Question

The frequency of the fourth harmonic of an organ pipe is $33\text{Hz}$ less than the frequency of the fifth harmonic of the same organ pipe. Find the difference in wavelengths for the given two cases. Neglect end corrections.
[speed of sound in air $=330\text{m/s}$]

• A. $60\text{cm}$
• B. $50\text{cm}$
• C. $40\text{cm}$
• D. $30\text{cm}$

Answer 1

The correct option is B $50\text{cm}$

As there is fourth and fifth harmonics, therefore, the organ pipe is open because the closed organ pipe doesn't allow even harmonics.
According to question,
$f_4=f_5-33$
$\Rightarrow \frac{4v}{2l}=\frac{5v}{2l}-33$
$\Rightarrow \frac{v}{2l}=33$
$\Rightarrow l=\frac{v}{2\times 33}=\frac{330}{66}=5\text{m}$
Now, wavelength for fourth harmonic,
$\lambda =\frac{2l}{n}$
$[f=\frac{nv}{2l}$ and $v=\lambda f]$
$\Rightarrow {\lambda }_4=\frac{2\times 5}{4}=2.5\text{m}$
Similarly, for fifth harmonic,
${\lambda }_5=2\text{m}$
So,
$\Delta \lambda ={\lambda }_4-{\lambda }_5=2.5-2=0.5\text{m}=50\text{cm}$