Question 229: Waves

Question

An open organ pipe and closed organ pipe of same length produce 2 beats per second, when they are set into vibrations together, in fundamental mode. The length of open pipe is made half and that of closed pipe is doubled.
The number of beats produced per second will be (neglect end correction)

Accepted Answer

Step 1: Find the original frequencies
The fundamental frequency of a closed pipe is given by:

$ \mathrm{n}_{\mathrm{c}} = \frac{v}{4L} $
The fundamental frequency of an open pipe is:

$ \mathrm{n}_{\mathrm{o}} = \frac{v}{2L} $
The question says they produce 2 beats per second, so:

$ \mathrm{n}_{\mathrm{o}} - \mathrm{n}_{\mathrm{c}} = 2 $
Substituting the values:

$ \frac{v}{2L} - \frac{v}{4L} = 2 $
Simplify:

$ \frac{v}{4L} = 2 $

So,

$ \frac{v}{L} = 8 $
Step 2: Change the lengths of the pipes
If the length of the open pipe is halved, its new length is $ \frac{L}{2} $, so the new frequency is:

$ \mathrm{n}_o' = \frac{v}{2 \left( \frac{L}{2} ight)} = \frac{v}{L} $
If the closed pipe is doubled in length (now $2L$), its new frequency is:

$ \mathrm{n}_c' = \frac{v}{4 	imes 2L} = \frac{v}{8L} $
Step 3: Calculate the new beat frequency
The new number of beats per second is the difference between the new frequencies:

$ \mathrm{Beat\ frequency} = \mathrm{n}_o' - \mathrm{n}_c' = \frac{v}{L} - \frac{v}{8L} $
Find a common denominator to subtract:

$ \frac{v}{L} - \frac{v}{8L} = \frac{8v - v}{8L} = \frac{7v}{8L} $
Recall from earlier that $ \frac{v}{L} = 8 $. So:

$ \frac{7v}{8L} = 7 $
Final Answer
: The number of beats produced per second will be 7.

Answer

4

Answer

6

Answer

7

Answer

8

Waves

An open organ pipe and closed organ pipe of same length produce 2 beats per second, when they are set into vibrations together, in fundamental mode. The length of open pipe is made half and that of closed pipe is doubled. The number of beats produced per second will be (neglect end correction)

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