MCQ
A closed organ pipe and an open pipe of same length produce $4$ beats when they are set into vibrations simultaneously. If the length of each of them were twice their initial lengths, the number of beats produced will be
- ✓$2$
- B$4$
- C$1$
- D$8$
✓
Answer
Correct option: A.
$2$
a
Let the lengths of the open pipe and closed pipe be $L$ and $L^{\prime}$ and their frequencies be $\nu$ and $\nu^{\prime}$
Let the lengths of the open pipe and closed pipe be $L$ and $L^{\prime}$ and their frequencies be $\nu$ and $\nu^{\prime}$
Speed of sound in air is $v m / s$
Frequency in the closedpipe $\quad \nu^{\prime}=\frac{(2 n-1) v}{4 L^{\prime}}$
Frequency in the open pipe $\quad \nu=\frac{m v}{2 L}$
Beat frequency $b=\nu-\nu^{\prime} \Longrightarrow \frac{m v}{2 L}-\frac{(2 n-1) v}{4 L^{\prime}}=4$
Now the lengths of both the pipes become double, beat frequency
$b^{\prime}=\frac{m v}{2(2 L)}-\frac{(2 n-1) v}{4\left(2 L^{\prime}\right)}$
$\Longrightarrow b=\frac{4}{2}=2$
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