Assertion : In spring block system if length of spring and mass of block both are halved, then angular frequency of oscillations will remain unchanged.
Reason : Angular frequency is given by `omega = sqrt((k)/(m))` .

Options

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

Explanation

To solve the assertion and reason question regarding the spring-block system, we will break down the problem step by step.

### Step 1: Understand the Assertion and Reason
- **Assertion**: If the length of the spring and the mass of the block are both halved, then the angular frequency of oscillations will remain unchanged.
- **Reason**: Angular frequency is given by the formula \( \omega = \sqrt{\frac{k}{m}} \).

### Step 2: Analyze the Angular Frequency Formula
The angular frequency \( \omega \) is defined as:
\[ \omega = \sqrt{\frac{k}{m}} \]
where \( k \) is the spring constant and \( m \) is the mass of the block.

### Step 3: Determine the Effect of Halving the Length of the Spring
- The spring constant \( k \) is inversely proportional to the natural length of the spring. If the length of the spring is halved, the spring constant will double:
\[ k' = 2k \]
where \( k' \) is the new spring constant after halving the length.

### Step 4: Determine the Effect of Halving the Mass
- If the mass of the block is halved, the new mass will be:
\[ m' = \frac{m}{2} \]

### Step 5: Calculate the New Angular Frequency
Now, substituting the new values of \( k \) and \( m \) into the angular frequency formula:
\[ \omega' = \sqrt{\frac{k'}{m'}} = \sqrt{\frac{2k}{\frac{m}{2}}} = \sqrt{\frac{2k \cdot 2}{m}} = \sqrt{\frac{4k}{m}} = 2\sqrt{\frac{k}{m}} = 2\omega \]

### Step 6: Conclusion
From the calculations, we find that the new angular frequency \( \omega' \) is double the original angular frequency \( \omega \):
\[ \omega' = 2\omega \]
Thus, the assertion that the angular frequency remains unchanged is **false**. The reason that angular frequency is given by \( \omega = \sqrt{\frac{k}{m}} \) is **true**.

### Final Answer
- **Assertion**: False
- **Reason**: True
- Therefore, the correct option is that the assertion is false, and the reason is true.

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