A tube, open at only one end, is cut into two shorter (non equal) lengths. The piece that is open at both ends has a fundamental frequency of 425 Hz, while the piece open only at one end has a fundamental frequency of 675 Hz. What is the fundamental frequency of the original tube?

Options

A. 127 Hz
B. 162 Hz
C. 209 Hz
D. 148 Hz

Answer

To find the fundamental frequency of the original tube, we will use the information given about the two shorter tubes formed after cutting the original tube. Let's denote the following:

- \( f_1 = 675 \, \text{Hz} \) (fundamental frequency of the closed tube)
- \( f_2 = 425 \, \text{Hz} \) (fundamental frequency of the open tube)
- \( L_1 \) = length of the closed tube
- \( L_2 \) = length of the open tube
- \( f \) = fundamental frequency of the original tube

### Step 1: Write the equations for the lengths of the tubes
For the closed tube (one end closed, one end open):
- The fundamental frequency is given by the equation:
\[ L_1 = \frac{v}{4f_1} \]
where \( v \) is the speed of sound in air.

For the open tube (both ends open):
- The fundamental frequency is given by the equation:
\[ L_2 = \frac{v}{2f_2} \]

### Step 2: Write the equation for the original tube
The original tube's length \( L \) is the sum of the lengths of the two shorter tubes:
\[ L = L_1 + L_2 \]
Substituting the equations from Step 1:
\[ L = \frac{v}{4f_1} + \frac{v}{2f_2} \]

### Step 3: Simplify the equation
Factor out \( v \):
\[ L = v \left( \frac{1}{4f_1} + \frac{1}{2f_2} \right) \]

### Step 4: Find the fundamental frequency of the original tube
For the original tube, the fundamental frequency \( f \) is given by:
\[ L = \frac{v}{4f} \]
Setting the two expressions for \( L \) equal:
\[ \frac{v}{4f} = v \left( \frac{1}{4f_1} + \frac{1}{2f_2} \right) \]

### Step 5: Cancel \( v \) and solve for \( f \)
Cancel \( v \) from both sides:
\[ \frac{1}{4f} = \frac{1}{4f_1} + \frac{1}{2f_2} \]

### Step 6: Multiply through by 4f to clear the fractions
\[ 1 = \frac{f}{f_1} + \frac{2f}{f_2} \]
Rearranging gives:
\[ 1 = \frac{f}{675} + \frac{2f}{425} \]

### Step 7: Combine the fractions
To combine the fractions, we need a common denominator:
\[ 1 = \frac{425f + 2 \cdot 675f}{675 \cdot 425} \]
This simplifies to:
\[ 1 = \frac{425f + 1350f}{675 \cdot 425} \]
\[ 1 = \frac{1775f}{675 \cdot 425} \]

### Step 8: Solve for \( f \)
Cross-multiplying gives:
\[ 675 \cdot 425 = 1775f \]
Thus,
\[ f = \frac{675 \cdot 425}{1775} \]

### Step 9: Calculate the value
Calculating this value:
\[ f = \frac{286875}{1775} \approx 162 \, \text{Hz} \]

### Final Answer:
The fundamental frequency of the original tube is approximately **162 Hz**.

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