A sounding fork whose frequency is `256 Hz` is held over an empty measuring cylinder. The sound is faint , but if just the right amount of water is poured into the cyclinder , it becomes loud. If the optimal amount of water produce an air column of length `0.31 m`, then the speed of sound in air to a first approximation is

Options

A. `317 m//s`
B. `371 m//s`
C. `340 m//s`
D. `332 m//s`

Explanation

To find the speed of sound in air using the given information, we can follow these steps:

### Step 1: Understand the relationship between the air column and the wavelength
When the tuning fork is sounded over the measuring cylinder, it creates a standing wave in the air column. The length of the air column corresponds to a quarter of the wavelength (λ/4) because the open end of the cylinder is a displacement node and the water surface is a pressure node.

### Step 2: Set up the equation for the length of the air column
Given that the length of the air column (L) is 0.31 m, we can express this relationship mathematically:
\[ L = \frac{\lambda}{4} \]
Thus, we can rearrange this to find the wavelength (λ):
\[ \lambda = 4L \]

### Step 3: Calculate the wavelength
Substituting the value of L:
\[ \lambda = 4 \times 0.31 \]
\[ \lambda = 1.24 \, \text{m} \]

### Step 4: Use the speed of sound formula
The speed of sound (v) is related to frequency (f) and wavelength (λ) by the formula:
\[ v = f \times \lambda \]
Given that the frequency (f) of the tuning fork is 256 Hz, we can substitute the values:
\[ v = 256 \, \text{Hz} \times 1.24 \, \text{m} \]

### Step 5: Calculate the speed of sound
Now, performing the multiplication:
\[ v = 256 \times 1.24 \]
\[ v \approx 317.44 \, \text{m/s} \]

### Step 6: Round to the nearest whole number
To the first approximation, we round this value:
\[ v \approx 317 \, \text{m/s} \]

### Conclusion
Thus, the speed of sound in air to a first approximation is approximately **317 m/s**.

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