A tuning fork of frequency 480 Hz is in unison with the first overtone of a pipe closed at one end. What is the fundamental frequency of the closed pipe ?

Options

A. 120 Hz
B. 140 Hz
C. 150 Hz
D. 160 Hz

Explanation

To find the fundamental frequency of a closed pipe when given the frequency of the first overtone, we can follow these steps:

### Step 1: Understand the relationship between harmonics in a closed pipe
In a closed pipe, the harmonics are produced as follows:
- The fundamental frequency (1st harmonic) is given by \( f_1 \).
- The first overtone (3rd harmonic) is given by \( f_3 \).

The relationship between these frequencies is:
\[ f_3 = 3f_1 \]

### Step 2: Given information
We know from the problem that the tuning fork frequency \( f_3 \) is 480 Hz, which is in unison with the first overtone of the closed pipe.

### Step 3: Set up the equation
Using the relationship from Step 1, we can write:
\[ f_3 = 3f_1 \]
Substituting the known value of \( f_3 \):
\[ 480 \, \text{Hz} = 3f_1 \]

### Step 4: Solve for the fundamental frequency \( f_1 \)
To find \( f_1 \), we can rearrange the equation:
\[ f_1 = \frac{480 \, \text{Hz}}{3} \]

### Step 5: Calculate \( f_1 \)
Now, perform the calculation:
\[ f_1 = \frac{480}{3} = 160 \, \text{Hz} \]

### Conclusion
The fundamental frequency of the closed pipe is \( 160 \, \text{Hz} \).

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