A closed pipe of length 1 m emits the fundamental note. What is the percentage change in the frequency when the temperature is changed from `30^@C" to "45^@C.`

Answer

To solve the problem, we need to determine the percentage change in frequency of a closed pipe when the temperature changes from 30°C to 45°C. Here’s a step-by-step solution:

### Step 1: Convert Celsius to Kelvin
To work with temperature in calculations related to frequency, we first convert the temperatures from Celsius to Kelvin.

- **Temperature 1 (T1)**:
\[ T1 = 30°C + 273 = 303 \, K \]

- **Temperature 2 (T2)**:
\[ T2 = 45°C + 273 = 318 \, K \]

### Step 2: Understand the relationship between frequency and temperature
The frequency of a closed pipe is directly proportional to the square root of the absolute temperature (in Kelvin). Therefore, we can express the frequencies at the two temperatures as follows:

- **Fundamental frequency at T1 (F1)**:
\[ F1 \propto \sqrt{T1} = \sqrt{303} \]

- **Fundamental frequency at T2 (F2)**:
\[ F2 \propto \sqrt{T2} = \sqrt{318} \]

### Step 3: Calculate the frequencies
Let’s express the frequencies in terms of a constant \( K \):
- \( F1 = K \sqrt{303} \)
- \( F2 = K \sqrt{318} \)

### Step 4: Calculate the change in frequency
Now, we can find the change in frequency \( \Delta F \):
\[ \Delta F = F2 - F1 = K \sqrt{318} - K \sqrt{303} \]
Factoring out \( K \):
\[ \Delta F = K (\sqrt{318} - \sqrt{303}) \]

### Step 5: Calculate the percentage change in frequency
The percentage change in frequency is given by:
\[ \text{Percentage Change} = \frac{\Delta F}{F1} \times 100 \]
Substituting the expressions for \( \Delta F \) and \( F1 \):
\[ \text{Percentage Change} = \frac{K (\sqrt{318} - \sqrt{303})}{K \sqrt{303}} \times 100 \]
The \( K \) cancels out:
\[ \text{Percentage Change} = \frac{\sqrt{318} - \sqrt{303}}{\sqrt{303}} \times 100 \]

### Step 6: Calculate the numerical values
Now we can calculate the square roots:
- \( \sqrt{318} \approx 17.83 \)
- \( \sqrt{303} \approx 17.4 \)

Substituting these values:
\[ \text{Percentage Change} = \frac{17.83 - 17.4}{17.4} \times 100 \]
Calculating the difference:
\[ \text{Percentage Change} = \frac{0.43}{17.4} \times 100 \approx 2.47\% \]

### Final Answer
The percentage change in frequency when the temperature is changed from 30°C to 45°C is approximately **2.47%**.

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