Question 273: Waves

Question

The closed and open organ pipe have same length and when they are vibrating simultaneously in first overtone produce 3 beats. The length of open pipe is made $\left(\frac{1}{3}\right)^{\mathrm{rd}}$ and that of closed pipe is made 3 times the original, the number of beats produced will be (Neglect end correction)

Accepted Answer

For open pipe first overtone, $n_1=\frac{v}{L}$
For closed pipe first overtone, $\mathrm{n}_1^{\prime}=\frac{3 \mathrm{v}}{4 \mathrm{~L}}$
$	herefore \quad$ Number of beats produced are,
$$ \begin{array}{ll}  & \mathrm{n}_1-\mathrm{n}_1^{\prime}=\frac{\mathrm{v}}{\mathrm{~L}}-\frac{3 \mathrm{v}}{4 \mathrm{~L}}=3 \ 	herefore & \frac{\mathrm{v}}{4 \mathrm{~L}}=3 \ 	herefore & \frac{\mathrm{v}}{\mathrm{~L}}=12 \ldots .(\mathrm{i}) \end{array} $$
When length of open pipe is made $\frac{\mathrm{L}}{3}$, the fundamental frequency becomes,
$$ \mathrm{n}=\frac{\mathrm{v}}{2\left(\frac{\mathrm{~L}}{3}ight)}=\frac{3 \mathrm{v}}{2 \mathrm{~L}} $$
When length of closed pipe is made 3 times, the fundamental frequency becomes,
$$ \mathrm{n}^{\prime}=\frac{\mathrm{v}}{4(3 \mathrm{~L})}=\frac{\mathrm{v}}{12 \mathrm{~L}} $$
$$ \begin{array}{ll} 	herefore \quad & 	ext { Beats produced }=n-n^{\prime} \ & =\frac{3 v}{2 L}-\frac{v}{12 L} \ & =\frac{17}{12} 	imes \frac{v}{L}=\frac{17}{12} 	imes 12 \,\,\,\,\,\,\,	ext { ....[From (i)] }\ & =17 \end{array} $$

Answer

14

Answer

17

Answer

18

Answer

12

Waves

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