Question 84: Waves

Question

The fifth harmonic of a closed organ pipe is found to be in unison with the first harmonic of an open pipe. The ratio of lengths of closed pipe to that of the open pipe is $5 / x$. The value of $x$ is $\_\_\_\_$

Accepted Answer

For a closed organ pipe
, only odd harmonics are present. The frequencies are:
$$f_n = \frac{nv}{4L_c}, \quad n = 1, 3, 5, 7, ...$$
The
fifth harmonic
(n = 5) of the closed pipe:
$$f_{closed} = \frac{5v}{4L_c}$$
For an open organ pipe
, all harmonics are present. The frequencies are:
$$f_n = \frac{nv}{2L_o}, \quad n = 1, 2, 3, ...$$
The
first harmonic
(n = 1) of the open pipe:
$$f_{open} = \frac{v}{2L_o}$$
Setting Frequencies Equal (Unison)
Since they are in unison:
$$\frac{5v}{4L_c} = \frac{v}{2L_o}$$
Solving for the Ratio
Cancelling $v$ from both sides:
$$\frac{5}{4L_c} = \frac{1}{2L_o}$$
Cross-multiplying:
$$5 	imes 2L_o = 4L_c 	imes 1$$
$$10L_o = 4L_c$$
$$\frac{L_c}{L_o} = \frac{10}{4} = \frac{5}{2}$$
Finding x
Given that the ratio is $\frac{5}{x}$:
$$\frac{5}{x} = \frac{5}{2}$$
$$\boxed{x = 2}$$
The answer is
Option B: 2
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Answer

3

Answer

2

Answer

4

Answer

1

Waves

The fifth harmonic of a closed organ pipe is found to be in unison with the first harmonic of an open pipe. The ratio of lengths of closed pipe to that of the open pipe is 5 / x. The value of x is \_\_\_\_

Choose the correct answer: