Question 285: Waves

Question

The frequency of fourth overtone of a closed pipe is in unison with the fifth overtone of an open pipe. The ratio of length of closed pipe to that of open pipe is

Accepted Answer

Let the length of the closed pipe be $L_1$ and the length of the open pipe be $L_2$.
Step 1: Frequencies in closed and open pipes
For a
closed pipe
, only odd harmonics are present.
$n$-th overtone $=$ $(2n+1)$-th harmonic.
So,
Fourth overtone
$(n=4)$ is:
Harmonic number $= 2	imes4 + 1 = 9$
Frequency:
$$     f_{closed} = \frac{9v}{4L_1}     $$
(where $v$ is speed of sound)
For an
open pipe
, all harmonics are present.
$n$-th overtone $=$ $(n+1)$-th harmonic.
So,
Fifth overtone
$(n=5)$ is:
Harmonic number $= 5+1 = 6$
Frequency:
$$     f_{open} = \frac{6v}{2L_2} = \frac{3v}{L_2}     $$
Step 2: Set frequencies equal (in unison)
$$ \frac{9v}{4L_1} = \frac{3v}{L_2} $$
Cancel $v$ from both sides:
$$ \frac{9}{4L_1} = \frac{3}{L_2} $$
Cross-multiplied,
$$ 9L_2 = 12L_1 $$
$$ \frac{L_1}{L_2} = \frac{9}{12} = \frac{3}{4} $$
Step 3: Final Answer
The ratio of length of closed pipe to open pipe is
$$ \boxed{3:4} $$
So, the correct option is
Option B
.

Answer

$2: 3$

Answer

$3: 4$

Answer

$4: 5$

Answer

$5: 6$

Waves

The frequency of fourth overtone of a closed pipe is in unison with the fifth overtone of an open pipe. The ratio of length of closed pipe to that of open pipe is

Choose the correct answer: