A particle executes S.H.M. with amplitude 'a' and time period T. The displacement of the particle when its speed is half of maximum speed is `(sqrt(x)a)/(2)`. The value of x is ………… .

Answer

To solve the problem, we need to determine the value of \( x \) when the displacement of a particle executing simple harmonic motion (SHM) is given as \( \frac{\sqrt{x} a}{2} \) while its speed is half of the maximum speed.

### Step-by-Step Solution:

1. **Understand the Maximum Speed in SHM**:
The maximum speed \( V_{\text{max}} \) of a particle in SHM is given by the formula:
\[ V_{\text{max}} = A \omega \]
where \( A \) is the amplitude and \( \omega \) is the angular frequency.

2. **Express the Speed in Terms of Displacement**:
The speed \( V \) of the particle at a displacement \( S \) is given by:
\[ V = \omega \sqrt{A^2 - S^2} \]

3. **Set the Condition for Half of Maximum Speed**:
We need to find the displacement when the speed is half of the maximum speed:
\[ V = \frac{V_{\text{max}}}{2} = \frac{A \omega}{2} \]

4. **Equate the Two Expressions for Speed**:
Setting the two expressions for speed equal gives:
\[ \omega \sqrt{A^2 - S^2} = \frac{A \omega}{2} \]
We can cancel \( \omega \) (assuming \( \omega \neq 0 \)):
\[ \sqrt{A^2 - S^2} = \frac{A}{2} \]

5. **Square Both Sides**:
Squaring both sides to eliminate the square root yields:
\[ A^2 - S^2 = \left(\frac{A}{2}\right)^2 \]
\[ A^2 - S^2 = \frac{A^2}{4} \]

6. **Rearranging the Equation**:
Rearranging gives:
\[ S^2 = A^2 - \frac{A^2}{4} \]
\[ S^2 = \frac{4A^2}{4} - \frac{A^2}{4} = \frac{3A^2}{4} \]

7. **Taking the Square Root**:
Taking the square root of both sides, we find:
\[ S = \pm \sqrt{\frac{3A^2}{4}} = \pm \frac{\sqrt{3} A}{2} \]

8. **Comparing with Given Displacement**:
We are given that the displacement when the speed is half of the maximum speed is:
\[ S = \frac{\sqrt{x} a}{2} \]
Setting this equal to our derived expression:
\[ \frac{\sqrt{x} a}{2} = \pm \frac{\sqrt{3} a}{2} \]

9. **Solving for \( x \)**:
Since \( a \) is the amplitude (and is positive), we can equate:
\[ \sqrt{x} = \sqrt{3} \]
Squaring both sides gives:
\[ x = 3 \]

### Final Answer:
The value of \( x \) is \( 3 \).