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 ___

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

Q. 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 ____.

(A) 3

(B) 2

(D) 1

Correct Answer: 2

Explanation

For a closed organ pipe, only odd harmonics are present. The frequency of the nth harmonic of a closed pipe is given by:

f_n = n v / (4L_c) (n = 1, 3, 5, ...)

Here, the fifth harmonic of the closed pipe corresponds to n = 5.

Frequency of the fifth harmonic of closed pipe:

f = 5v / (4L_c)

For an open organ pipe, all harmonics are present and the frequency of the first harmonic is:

f = v / (2L_o)

Since the two frequencies are in unison, they are equal:

5v / (4L_c) = v / (2L_o)

Cancel v from both sides:

5 / (4L_c) = 1 / (2L_o)

Cross-multiplying:

10L_o = 4L_c

L_c / L_o = 10 / 4 = 5 / 2

Given ratio = 5 / x

So,

5 / x = 5 / 2

x = 2

Hence, the value of x = 2.

Explanation

Related JEE Main Questions