Assertion: In a simple harmonic motion the kinetic and potential energy becomes equal when the displacement is `(1)/(sqrt(2))` time the amplitude
Reason: is `SHM` kinetic energy is zero when potential energy is maximum

Options

A. If both assertion and reason are ture 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 reson are false.

Explanation

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

### Step 1: Understand the Assertion
The assertion states that in simple harmonic motion (SHM), the kinetic energy (KE) and potential energy (PE) become equal when the displacement (y) is \( \frac{1}{\sqrt{2}} \) times the amplitude (A).

### Step 2: Write the Expressions for Kinetic and Potential Energy
In SHM, the expressions for kinetic energy and potential energy are given by:
- Kinetic Energy (KE):
\[ KE = \frac{1}{2} m \omega^2 (A^2 - y^2) \]
- Potential Energy (PE):
\[ PE = \frac{1}{2} m \omega^2 y^2 \]

### Step 3: Set Kinetic Energy Equal to Potential Energy
According to the assertion, we set KE equal to PE:
\[ \frac{1}{2} m \omega^2 (A^2 - y^2) = \frac{1}{2} m \omega^2 y^2 \]

### Step 4: Simplify the Equation
We can cancel \( \frac{1}{2} m \omega^2 \) from both sides (assuming it is not zero):
\[ A^2 - y^2 = y^2 \]
This simplifies to:
\[ A^2 = 2y^2 \]

### Step 5: Solve for y
Rearranging gives:
\[ y^2 = \frac{A^2}{2} \]
Taking the square root:
\[ y = \pm \frac{A}{\sqrt{2}} \]
This confirms that the displacement \( y \) is indeed \( \frac{1}{\sqrt{2}} \) times the amplitude \( A \).

### Conclusion for Assertion
Thus, the assertion is correct.

### Step 6: Analyze the Reason
The reason states that in SHM, kinetic energy is zero when potential energy is maximum.

### Step 7: Understand the Energy Relationship in SHM
In SHM, the total mechanical energy (E) is constant and is given by:
\[ E = KE + PE \]
When the displacement is at its maximum (i.e., at amplitude A), the potential energy is maximum and the kinetic energy is zero. This is indeed true.

### Conclusion for Reason
Thus, the reason is also correct. However, it does not provide a direct explanation for the assertion.

### Final Answer
Both the assertion and reason are correct, but the reason is not the correct explanation for the assertion.