The fundamental frequencies of a closed pipe and an open pipe of different lengths are 300 Hz and 400 Hz respectively. If they are joined to form a longer pipe, the fundamental frequency of the long pipe so formed is

Options

A. 350 Hz
B. 50 Hz
C. 120 Hz
D. 100 Hz

Answer

To solve the problem, we need to find the fundamental frequency of a longer pipe formed by joining a closed pipe and an open pipe. Let's break it down step by step.

### Step 1: Understand the frequencies of the individual pipes
- The fundamental frequency of the closed pipe (f2) is given as 300 Hz.
- The fundamental frequency of the open pipe (f1) is given as 400 Hz.

### Step 2: Relate frequency to the speed of sound and length of the pipes
For an open pipe, the fundamental frequency is given by the formula:
\[ f_1 = \frac{v}{2L_1} \]
For a closed pipe, the fundamental frequency is given by:
\[ f_2 = \frac{v}{4L_2} \]

Where:
- \( v \) = speed of sound in air
- \( L_1 \) = length of the open pipe
- \( L_2 \) = length of the closed pipe

### Step 3: Express lengths in terms of speed of sound and frequencies
From the open pipe frequency:
\[ L_1 = \frac{v}{2f_1} = \frac{v}{2 \times 400} = \frac{v}{800} \]

From the closed pipe frequency:
\[ L_2 = \frac{v}{4f_2} = \frac{v}{4 \times 300} = \frac{v}{1200} \]

### Step 4: Calculate the total length of the combined pipe
When the two pipes are joined, the total length \( L \) of the longer pipe is:
\[ L = L_1 + L_2 = \frac{v}{800} + \frac{v}{1200} \]

To add these fractions, we need a common denominator:
The least common multiple of 800 and 1200 is 2400.

Converting each term:
\[ L_1 = \frac{v}{800} = \frac{3v}{2400} \]
\[ L_2 = \frac{v}{1200} = \frac{2v}{2400} \]

Now, adding them:
\[ L = \frac{3v}{2400} + \frac{2v}{2400} = \frac{5v}{2400} = \frac{v}{480} \]

### Step 5: Calculate the fundamental frequency of the longer pipe
For the longer pipe (which behaves like an open pipe), the fundamental frequency \( f_0 \) is given by:
\[ f_0 = \frac{v}{2L} \]

Substituting \( L \):
\[ f_0 = \frac{v}{2 \times \frac{v}{480}} = \frac{v}{\frac{2v}{480}} = \frac{480}{2} = 240 \text{ Hz} \]

### Conclusion
The fundamental frequency of the longer pipe formed by joining the closed pipe and the open pipe is **240 Hz**.