A long tube contains air at a pressure of `1 atm` and a temperature of `107^(@) C`. The tube is open at one end and closed at the other by a movable piston. A tuning fork near the open end is vibrating with a frequency of `500 Hz`. Resonance is produced when the piston is at distance `19 , 58.5 and 98 cm` from the open end.
The speed of sound at `10^(@) C` is

Options

A. `330 m//s`
B. `340 m//s`
C. `395 m//s`
D. `495 m//s`

Explanation

To find the speed of sound at 10°C, we will follow these steps:

### Step 1: Understand the Resonance Condition
In a tube that is open at one end and closed at the other, the resonance occurs at specific lengths where the standing wave patterns form. The lengths at which resonance occurs can be expressed in terms of the wavelength (λ) of the sound wave.

### Step 2: Identify the Lengths of Resonance
Given the distances from the open end where resonance occurs:
- \( L_1 = 19 \, \text{cm} \)
- \( L_2 = 58.5 \, \text{cm} \)
- \( L_3 = 98 \, \text{cm} \)

### Step 3: Relate Lengths to Wavelength
For a tube closed at one end, the resonance lengths can be expressed as:
- \( L_1 + E = \frac{\lambda}{4} \) (first resonance)
- \( L_2 + E = \frac{3\lambda}{4} \) (second resonance)

Where \( E \) is the end correction.

### Step 4: Subtract the Two Equations
Subtract the first equation from the second to eliminate \( E \):
\[ (L_2 + E) - (L_1 + E) = \frac{3\lambda}{4} - \frac{\lambda}{4} \]
This simplifies to:
\[ L_2 - L_1 = \frac{2\lambda}{4} = \frac{\lambda}{2} \]
Thus, we can express the wavelength as:
\[ \lambda = 2(L_2 - L_1) \]

### Step 5: Calculate the Wavelength
Substituting the values of \( L_1 \) and \( L_2 \):
\[ \lambda = 2(58.5 \, \text{cm} - 19 \, \text{cm}) = 2(39.5 \, \text{cm}) = 79 \, \text{cm} \]
Convert this to meters:
\[ \lambda = 79 \, \text{cm} = 0.79 \, \text{m} \]

### Step 6: Calculate the Speed of Sound
Using the formula for the speed of sound:
\[ v = f \lambda \]
Where \( f = 500 \, \text{Hz} \) and \( \lambda = 0.79 \, \text{m} \):
\[ v = 500 \, \text{Hz} \times 0.79 \, \text{m} = 395 \, \text{m/s} \]

### Final Answer
The speed of sound at 10°C is \( 395 \, \text{m/s} \).

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