In a SHM with time period 2s. Time taken for displacement x to reach `A/2` from mean position is `1/a` find `a`?

Answer

To solve the problem, we need to find the value of \( a \) given that a particle in simple harmonic motion (SHM) has a time period of 2 seconds and takes \( \frac{1}{a} \) seconds to reach a displacement of \( \frac{A}{2} \) from the mean position.

### Step-by-Step Solution:

1. **Identify the Time Period and Calculate Angular Frequency**:
The time period \( T \) is given as 2 seconds. The angular frequency \( \omega \) can be calculated using the formula:
\[ \omega = \frac{2\pi}{T} \]
Substituting \( T = 2 \):
\[ \omega = \frac{2\pi}{2} = \pi \, \text{rad/s} \]

**Hint**: Remember that angular frequency is related to the time period by this formula.

2. **Set Up the Displacement Equation**:
The displacement \( x \) in SHM can be expressed as:
\[ x = A \sin(\omega t) \]
We need to find the time \( t \) when \( x = \frac{A}{2} \).

**Hint**: The sine function relates the displacement to time in SHM.

3. **Substitute the Displacement**:
Set \( x = \frac{A}{2} \):
\[ \frac{A}{2} = A \sin(\omega t) \]
Dividing both sides by \( A \) (assuming \( A \neq 0 \)):
\[ \frac{1}{2} = \sin(\omega t) \]

**Hint**: Simplifying the equation helps isolate the sine function.

4. **Find the Angle**:
From the equation \( \sin(\omega t) = \frac{1}{2} \), we know that:
\[ \omega t = \sin^{-1}\left(\frac{1}{2}\right) \]
The angle whose sine is \( \frac{1}{2} \) is \( \frac{\pi}{6} \) radians.

**Hint**: Recall the values of sine for standard angles.

5. **Solve for Time \( t \)**:
Substitute \( \omega = \pi \):
\[ \pi t = \frac{\pi}{6} \]
Dividing both sides by \( \pi \):
\[ t = \frac{1}{6} \, \text{seconds} \]

**Hint**: Isolate \( t \) to find the time taken to reach the specified displacement.

6. **Relate Time to \( a \)**:
According to the problem, the time taken to reach \( \frac{A}{2} \) is given as \( \frac{1}{a} \):
\[ \frac{1}{a} = \frac{1}{6} \]
Therefore, we can equate and solve for \( a \):
\[ a = 6 \]

**Hint**: Equating the two expressions for time allows you to find \( a \).

### Final Answer:
The value of \( a \) is \( 6 \).