Assertion: The fundamental frequency of an open organ pipe increases as the temperature is increased.
Reason: As the temperature increases, the velocity of sound increases more rapidly than length of the pipe.

Options

A. If both assertion and reason are true and reason is the correct explanation of assertion.
B. If both assertion and reason are true but reason is not the correct explanation of assertion.
C. If assertion is true but reason is false.
D. If both assertion and reason are false.

Explanation

To solve the question, we need to analyze both the assertion and the reason provided.

### Step 1: Understanding the Assertion
The assertion states that the fundamental frequency of an open organ pipe increases as the temperature is increased.

- **Fundamental Frequency Formula**: The fundamental frequency (f) of an open organ pipe is given by the formula:
\[ f = \frac{V}{2L} \]
where \( V \) is the speed of sound in air and \( L \) is the length of the pipe.

### Step 2: Analyzing the Effect of Temperature on Speed of Sound
As temperature increases, the speed of sound in air also increases. The speed of sound in a gas is given by:
\[ V = \sqrt{\frac{\gamma P}{\rho}} \]
where \( \gamma \) is the adiabatic index, \( P \) is the pressure, and \( \rho \) is the density of the gas.

- **Relation to Temperature**: The speed of sound can also be expressed as:
\[ V \propto \sqrt{T} \]
This means that as temperature (T) increases, the speed of sound (V) increases.

### Step 3: Analyzing the Effect of Temperature on Length of the Pipe
The length of the air column (L) in the pipe can also change with temperature due to thermal expansion, but the increase in length is generally less significant compared to the increase in speed of sound.

### Step 4: Conclusion on the Assertion
Since the speed of sound increases with temperature and the fundamental frequency is directly proportional to the speed of sound, we can conclude that:
- As the temperature increases, the fundamental frequency of the open organ pipe increases.

### Step 5: Understanding the Reason
The reason states that as the temperature increases, the velocity of sound increases more rapidly than the length of the pipe.

- This statement is true because while both speed and length may increase with temperature, the increase in speed is more pronounced due to the square root relationship with temperature, while the increase in length is linear.

### Final Conclusion
Both the assertion and the reason are true, and the reason correctly explains the assertion. Therefore, the correct answer is that both the assertion and reason are true, and the reason explains the assertion.

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