If in an experiment for determination of velocity of sound by resonance tube method using a tuning fork of `512 Hz`, first resonance was observed at` 30.7 cm` and second was obtained at `63.2 cm`, then maximum possible error in velocity of sound is (consider actual speed of sound in air is `332m//s

Options

A. `204cm//sec`
B. `110cm//sec`
C. `58cm//sec`
D. `80cm//sec`

Explanation

To solve the problem of determining the maximum possible error in the velocity of sound using the resonance tube method, we can follow these steps:

### Step 1: Understand the relationship between resonance and wavelength
In the resonance tube method, the distance between the first and second resonance points corresponds to half the wavelength of the sound wave. Therefore, we can express this relationship as:
\[ \frac{\lambda}{2} = X_2 - X_1 \]
where \(X_1\) is the first resonance point and \(X_2\) is the second resonance point.

### Step 2: Calculate the wavelength
Given:
- \(X_1 = 30.7 \, \text{cm}\)
- \(X_2 = 63.2 \, \text{cm}\)

We can calculate the difference:
\[ X_2 - X_1 = 63.2 \, \text{cm} - 30.7 \, \text{cm} = 32.5 \, \text{cm} \]

Now, substituting this into the equation for wavelength:
\[ \frac{\lambda}{2} = 32.5 \, \text{cm} \implies \lambda = 2 \times 32.5 \, \text{cm} = 65 \, \text{cm} \]

### Step 3: Calculate the velocity of sound
The velocity \(V\) of sound can be calculated using the formula:
\[ V = f \cdot \lambda \]
where \(f\) is the frequency of the tuning fork. Given that \(f = 512 \, \text{Hz}\) and converting the wavelength into meters:
\[ \lambda = 65 \, \text{cm} = 0.65 \, \text{m} \]

Now substituting the values:
\[ V = 512 \, \text{Hz} \times 0.65 \, \text{m} = 332.8 \, \text{m/s} \]

### Step 4: Calculate the error in the velocity of sound
The actual speed of sound in air is given as \(332 \, \text{m/s}\). To find the maximum possible error:
\[ \text{Error} = |V_{\text{calculated}} - V_{\text{actual}}| = |332.8 \, \text{m/s} - 332 \, \text{m/s}| = 0.8 \, \text{m/s} \]

### Final Answer
The maximum possible error in the velocity of sound is \(0.8 \, \text{m/s}\).

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