Assertion : For a given simple harmonic motion displacement (from the mean position) and acceleration have a constant ratio.
Reason : `T = 2pi sqrt(|("displacement")/("acceleration")|)`.

Options

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

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 "For a given simple harmonic motion, displacement (from the mean position) and acceleration have a constant ratio."

In simple harmonic motion (SHM), the acceleration \( a \) is directly proportional to the displacement \( x \) from the mean position and is given by the formula:
\[ a = -\omega^2 x \]
where \( \omega \) is the angular frequency.

From this equation, we can express the ratio of acceleration to displacement:
\[ \frac{a}{x} = -\omega^2 \]
This shows that the ratio \( \frac{a}{x} \) is constant, as \( \omega \) is a constant for a given SHM. Therefore, the assertion is true.

### Step 2: Understanding the Reason
The reason states that \( T = 2\pi \sqrt{\frac{\text{displacement}}{\text{acceleration}}} \).

In SHM, the period \( T \) is related to the angular frequency \( \omega \) by the formula:
\[ T = \frac{2\pi}{\omega} \]
We can also relate \( \omega \) to displacement and acceleration. From the earlier relationship, we have:
\[ \omega^2 = \frac{a}{x} \]
Thus, we can write:
\[ \omega = \sqrt{\frac{a}{x}} \]
Substituting this into the period formula gives:
\[ T = 2\pi \sqrt{\frac{x}{a}} \]
This shows that the reason is also true and correctly explains the assertion.

### Conclusion
Both the assertion and the reason are true, and the reason is a correct explanation of the assertion.

### Final Answer
- Assertion: True
- Reason: True
- The reason is a correct explanation of the assertion.

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