A particle performing simple harmonic motion undergoes unitial displacement of `(A)/(2)` (where A is the amplitude of simple harmonic motion) in 1 s. At `t=0`, the particle may be at he extreme position or mean position the time period of the simple harmonic motion can be

Options

A. `12s`
B. `2.4`
C. `6s`
D. `1.2s`

Explanation

To solve the problem step by step, we will analyze the information given and apply the principles of simple harmonic motion (SHM).

### Step-by-Step Solution:

1. **Understanding the Problem**:
The particle undergoes an initial displacement of \( \frac{A}{2} \) in 1 second. We need to find the time period \( T \) of the simple harmonic motion.

2. **Displacement in SHM**:
The displacement \( x \) of a particle in SHM can be expressed as:
\[ x = A \sin(\omega t) \]
where \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( t \) is the time.

3. **Substituting Given Values**:
At \( t = 1 \) second, the displacement \( x \) is given as \( \frac{A}{2} \). Therefore, we can write:
\[ \frac{A}{2} = A \sin(\omega \cdot 1) \]

4. **Simplifying the Equation**:
Dividing both sides by \( A \) (assuming \( A \neq 0 \)):
\[ \frac{1}{2} = \sin(\omega) \]

5. **Finding \( \omega \)**:
The equation \( \sin(\omega) = \frac{1}{2} \) implies:
\[ \omega = \sin^{-1}\left(\frac{1}{2}\right) \]
The principal value of \( \sin^{-1}\left(\frac{1}{2}\right) \) is \( \frac{\pi}{6} \) radians.

6. **Relating Angular Frequency to Time Period**:
The angular frequency \( \omega \) is related to the time period \( T \) by the formula:
\[ \omega = \frac{2\pi}{T} \]

7. **Substituting \( \omega \) into the Equation**:
We can substitute \( \omega = \frac{\pi}{6} \) into the equation:
\[ \frac{\pi}{6} = \frac{2\pi}{T} \]

8. **Solving for \( T \)**:
Rearranging the equation to solve for \( T \):
\[ T = \frac{2\pi}{\frac{\pi}{6}} = 2\pi \cdot \frac{6}{\pi} = 12 \text{ seconds} \]

9. **Conclusion**:
The time period of the simple harmonic motion is \( T = 12 \) seconds.

### Final Answer:
The time period of the particle performing simple harmonic motion is \( 12 \) seconds.

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