Assertion : Acceleration is proportional to the displacement. This condition is not sufficient for motion in simple harmonic.
Reason : In simple harmonic motion direction of displacement is also considered.

Options

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

Answer

To solve the question regarding the assertion and reason related to simple harmonic motion (SHM), we will analyze both statements step by step.

### Step 1: Understanding the Assertion
The assertion states that "Acceleration is proportional to the displacement," and that this condition is not sufficient for motion in simple harmonic motion.

- In SHM, the acceleration (a) of an object is given by the equation:
\[ a = -\omega^2 x \]
where \(x\) is the displacement from the equilibrium position and \(\omega\) is the angular frequency.

### Step 2: Analyzing the Proportionality
The assertion implies that if acceleration is proportional to displacement, it does not necessarily mean that the motion is simple harmonic.

- For SHM, the relationship is specifically that acceleration is proportional to the **negative** of the displacement. This means:
\[ a \propto -x \]
This negative sign indicates that the acceleration acts in the opposite direction to the displacement, which is a key characteristic of SHM.

### Step 3: Understanding the Reason
The reason provided states, "In simple harmonic motion, the direction of displacement is also considered."

- This is true because the direction of the displacement determines whether the acceleration will be positive or negative. In SHM, when the object is displaced to one side (positive displacement), the acceleration acts in the opposite direction (negative), pulling it back towards the equilibrium position.

### Step 4: Conclusion
Based on the analysis:
- The assertion is true because simply stating that acceleration is proportional to displacement does not capture the necessary condition of SHM, which requires that acceleration is proportional to the negative of displacement.
- The reason is also true because the direction of displacement is crucial in determining the nature of the acceleration in SHM.

Thus, both the assertion and reason are true, and the reason correctly explains the assertion.

### Final Answer
Both the assertion and reason are true, and the reason is the correct explanation of the assertion. Therefore, the answer is option A.

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