In a resonance tube with tuning fork of frequency `512 Hz`, first resonance occurs at water level equal to `30.3cm` and second resonance ocuurs at `63.7cm`. The maximum possible error in the speed of sound is

Options

A. `5.12 cm//s`
B. `102.4 cm//s`
C. `204.8 cm//s`
D. `153.6 cm//s`

Explanation

To solve the problem, we need to determine the maximum possible error in the speed of sound based on the given resonance tube measurements. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Resonance Condition
In a resonance tube, the first resonance occurs when the length of the air column corresponds to a quarter of the wavelength (λ/4), and the second resonance occurs when the length corresponds to three-quarters of the wavelength (3λ/4).

### Step 2: Set Up the Equations
Let:
- \( L_1 = 30.3 \, \text{cm} \) (first resonance)
- \( L_2 = 63.7 \, \text{cm} \) (second resonance)
- \( e \) = end correction

From the first resonance condition:
\[ \frac{\lambda}{4} = L_1 + e \]
(Equation 1)

From the second resonance condition:
\[ \frac{3\lambda}{4} = L_2 + e \]
(Equation 2)

### Step 3: Subtract the Equations
Subtract Equation 1 from Equation 2:
\[ \frac{3\lambda}{4} - \frac{\lambda}{4} = (L_2 + e) - (L_1 + e) \]
This simplifies to:
\[ \frac{2\lambda}{4} = L_2 - L_1 \]
Thus:
\[ \frac{\lambda}{2} = L_2 - L_1 \]

### Step 4: Relate Wavelength to Speed of Sound
The wavelength is related to the speed of sound (V) and frequency (f) by:
\[ \lambda = \frac{V}{f} \]
Substituting this into our equation gives:
\[ \frac{V}{2f} = L_2 - L_1 \]

### Step 5: Solve for Speed of Sound
Rearranging gives:
\[ V = 2f(L_2 - L_1) \]

### Step 6: Calculate the Maximum Possible Error
To find the maximum possible error in the speed of sound, we need to consider the errors in \( L_1 \) and \( L_2 \). The least count (error) in both measurements is \( 0.1 \, \text{cm} \).

Let:
- \( \Delta L_1 = 0.1 \, \text{cm} \)
- \( \Delta L_2 = 0.1 \, \text{cm} \)

The total error in the length difference \( L_2 - L_1 \) is:
\[ \Delta(L_2 - L_1) = \Delta L_2 + \Delta L_1 = 0.1 + 0.1 = 0.2 \, \text{cm} \]

### Step 7: Substitute into the Error Formula
The change in speed of sound \( \Delta V \) can be expressed as:
\[ \Delta V = 2f \cdot \Delta(L_2 - L_1) \]
Substituting the values:
- \( f = 512 \, \text{Hz} \)
- \( \Delta(L_2 - L_1) = 0.2 \, \text{cm} = 0.002 \, \text{m} \)

Thus:
\[ \Delta V = 2 \times 512 \times 0.002 = 2.048 \, \text{m/s} = 204.8 \, \text{cm/s} \]

### Final Answer
The maximum possible error in the speed of sound is:
\[ \Delta V = 204.8 \, \text{cm/s} \]

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