A vibrating tuning fork is held over a water column with one end closed and the other open. As the water level is allowed to fall, a loud sound is heard for water levels separated by 17 cm. If the speed of sound in air is 340 m/s, the frequency of the tuning fork is:

Options

A. 500 Hz
B. 1000 Hz
C. 2000 Hz
D. 5780 Hz

Answer

To solve the problem, we need to find the frequency of the tuning fork based on the information given about the resonance in a closed organ pipe formed by the water column. Here’s the step-by-step solution:

### Step 1: Understand the Resonance Conditions
In a closed organ pipe (one end closed and one end open), the resonance occurs at specific lengths of the air column. The first resonance occurs when the length of the air column (L1) is equal to \( \frac{\lambda}{4} \), and the second resonance occurs when the length of the air column (L2) is equal to \( \frac{3\lambda}{4} \).

### Step 2: Set Up the Equations
From the resonance conditions:
1. For the first resonance:
\[ L_1 = \frac{\lambda}{4} \quad \text{(Equation 1)} \]
2. For the second resonance:
\[ L_2 = \frac{3\lambda}{4} \quad \text{(Equation 2)} \]

### Step 3: Find the Difference Between the Two Lengths
The problem states that the difference between the two lengths (L2 - L1) is 17 cm. Therefore:
\[ L_2 - L_1 = 17 \text{ cm} = 0.17 \text{ m} \]

### Step 4: Substitute the Lengths into the Difference Equation
Substituting Equation 1 and Equation 2 into the difference:
\[ \frac{3\lambda}{4} - \frac{\lambda}{4} = 0.17 \]
This simplifies to:
\[ \frac{2\lambda}{4} = 0.17 \]
\[ \frac{\lambda}{2} = 0.17 \]
Multiplying both sides by 2 gives:
\[ \lambda = 0.34 \text{ m} \]

### Step 5: Use the Speed of Sound to Find Frequency
The speed of sound in air is given as 340 m/s. The frequency (f) can be calculated using the formula:
\[ f = \frac{v}{\lambda} \]
where \( v \) is the speed of sound and \( \lambda \) is the wavelength. Substituting the values:
\[ f = \frac{340 \text{ m/s}}{0.34 \text{ m}} = 1000 \text{ Hz} \]

### Conclusion
The frequency of the tuning fork is:
\[ \boxed{1000 \text{ Hz}} \]