To solve the problem, we will follow these steps:
### Step 1: Understand the resonance conditions
We have two pipes:
1. A pipe closed at one end (let's call it Pipe 1).
2. A pipe open at both ends (let's call it Pipe 2).
Both pipes have the same length, \( L = 1.5 \, \text{m} \).
### Step 2: Determine the fundamental frequency for each pipe
- For a pipe closed at one end, the fundamental frequency is given by:
\[
f_1 = \frac{v_g}{4L}
\]
where \( v_g \) is the velocity of sound in the gas.
- For a pipe open at both ends, the fundamental frequency is given by:
\[
f_2 = \frac{v_a}{2L}
\]
where \( v_a \) is the velocity of sound in air.
### Step 3: Set the frequencies equal
Since both pipes resonate with the same tuning fork, we have:
\[
f_1 = f_2
\]
This implies:
\[
\frac{v_g}{4L} = \frac{v_a}{2L}
\]
### Step 4: Simplify the equation
We can cancel \( L \) from both sides:
\[
\frac{v_g}{4} = \frac{v_a}{2}
\]
Multiplying both sides by 4 gives:
\[
v_g = 2v_a
\]
### Step 5: Substitute the known value
We know that the velocity of sound in air at \( 30^\circ C \) is \( v_a = 360 \, \text{m/s} \):
\[
v_g = 2 \times 360 = 720 \, \text{m/s}
\]
### Step 6: Calculate the velocity of sound at \( 0^\circ C \)
The velocity of sound in gases is proportional to the square root of the temperature in Kelvin. Therefore, we can use the formula:
\[
\frac{v_1}{v_2} = \sqrt{\frac{T_1}{T_2}}
\]
where:
- \( v_1 \) is the velocity of sound at \( 0^\circ C \) (which we need to find),
- \( v_2 \) is the velocity of sound at \( 30^\circ C \) (which we found to be \( 720 \, \text{m/s} \)),
- \( T_1 = 273 \, \text{K} \) (temperature at \( 0^\circ C \)),
- \( T_2 = 303 \, \text{K} \) (temperature at \( 30^\circ C \)).
### Step 7: Substitute the values into the equation
\[
\frac{v_1}{720} = \sqrt{\frac{273}{303}}
\]
Now, solving for \( v_1 \):
\[
v_1 = 720 \times \sqrt{\frac{273}{303}}
\]
### Step 8: Calculate \( \sqrt{\frac{273}{303}} \)
Calculating the square root:
\[
\sqrt{\frac{273}{303}} \approx 0.952
\]
Thus,
\[
v_1 \approx 720 \times 0.952 \approx 685.44 \, \text{m/s}
\]
### Step 9: Final answer
Rounding to the nearest whole number, the velocity of sound at \( 0^\circ C \) in the gas is approximately:
\[
v_1 \approx 683 \, \text{m/s}
\]
### Conclusion
The velocity of sound at \( 0^\circ C \) in the gas is \( 683 \, \text{m/s} \).
---
