A pipe of length 1.5m closed at one end is filled with gas and resonates in its fundamental mode with a tuning fork . Another pipe of same dimension filled with air resonates in its fundamental mode with same tuning fork. If experiment is performed at `30^@ C` (speed of sound in air is 360 m/sec) the velocity of sound at `0^@C` in gas is

Options

A. 580 m/sec
B. 680 m/s
C. 880 m/sec
D. none

Answer

To find the velocity of sound in gas at 0°C, we can follow these steps:

### Step 1: Understand the fundamental frequency of a closed pipe
For a pipe closed at one end, the fundamental frequency (f₀) is given by the formula:
\[ f₀ = \frac{V}{4L} \]
where:
- \( V \) is the speed of sound in the medium (gas or air),
- \( L \) is the length of the pipe.

### Step 2: Calculate the fundamental frequency in air
Given:
- Length of the pipe, \( L = 1.5 \, \text{m} \)
- Speed of sound in air at 30°C, \( V_a = 360 \, \text{m/s} \)

Using the formula for fundamental frequency in air:
\[ f₀ = \frac{360}{4 \times 1.5} \]
Calculating this gives:
\[ f₀ = \frac{360}{6} = 60 \, \text{Hz} \]

### Step 3: Relate the fundamental frequency in gas
Since both pipes resonate with the same tuning fork, they have the same fundamental frequency:
\[ f₀ = \frac{V_g}{4L} \]
where \( V_g \) is the speed of sound in the gas.

### Step 4: Set the frequencies equal and solve for \( V_g \)
From the previous step:
\[ 60 = \frac{V_g}{4 \times 1.5} \]
Rearranging gives:
\[ V_g = 60 \times 6 = 360 \, \text{m/s} \]

### Step 5: Adjust for temperature to find \( V_0 \) at 0°C
The speed of sound in gas changes with temperature. The formula to relate the speed of sound at different temperatures is:
\[ V_t = V_0 \left(1 + \frac{\Delta T}{546}\right) \]
where:
- \( V_t \) is the speed of sound at temperature \( T \) (30°C),
- \( V_0 \) is the speed of sound at 0°C,
- \( \Delta T = T - 0 = 30 \, \text{°C} \).

Substituting the known values:
\[ 360 = V_0 \left(1 + \frac{30}{546}\right) \]

### Step 6: Solve for \( V_0 \)
Calculating the term in parentheses:
\[ 1 + \frac{30}{546} = 1 + 0.0548 \approx 1.0548 \]
Now substituting back:
\[ 360 = V_0 \times 1.0548 \]
Solving for \( V_0 \):
\[ V_0 = \frac{360}{1.0548} \approx 341.25 \, \text{m/s} \]

### Final Answer
The velocity of sound in gas at 0°C is approximately:
\[ V_0 \approx 341.25 \, \text{m/s} \]

---