How long after the beginning of motion is the displacement of a oscillating particle equal to one half its amplitude if the period is `24s` and particle starts from rest from extreme position.

Options

A. `12s`
B. `2s`
C. `4s`
D. `6s`

Explanation

To solve the problem step by step, we will follow the outlined approach in the video transcript.

### Step-by-Step Solution:

1. **Understand the Motion**: The particle starts from rest at the extreme position, which means it starts at maximum displacement (A). The equation of motion for a particle in simple harmonic motion (SHM) is given by:
\[ x(t) = A \cos(\omega t) \]
where \( x(t) \) is the displacement at time \( t \), \( A \) is the amplitude, and \( \omega \) is the angular frequency.

2. **Determine the Angular Frequency**: The period \( T \) of the motion is given as 24 seconds. The relationship between the period and angular frequency is:
\[ T = \frac{2\pi}{\omega} \]
Rearranging this gives:
\[ \omega = \frac{2\pi}{T} = \frac{2\pi}{24} = \frac{\pi}{12} \text{ rad/s} \]

3. **Set Up the Displacement Condition**: We need to find the time \( t_1 \) when the displacement \( x \) is equal to half the amplitude, i.e., \( x = \frac{A}{2} \). Substituting this into the equation of motion gives:
\[ \frac{A}{2} = A \cos(\omega t_1) \]

4. **Simplify the Equation**: Dividing both sides by \( A \) (assuming \( A \neq 0 \)):
\[ \frac{1}{2} = \cos(\omega t_1) \]

5. **Find the Angle**: We know that \( \cos(\theta) = \frac{1}{2} \) corresponds to angles:
\[ \theta = \frac{\pi}{3} + 2n\pi \quad \text{or} \quad \theta = -\frac{\pi}{3} + 2n\pi \quad (n \in \mathbb{Z}) \]
For the first instance (smallest positive time), we take:
\[ \omega t_1 = \frac{\pi}{3} \]

6. **Substitute the Value of \( \omega \)**: Now substituting \( \omega = \frac{\pi}{12} \):
\[ \frac{\pi}{12} t_1 = \frac{\pi}{3} \]

7. **Solve for \( t_1 \)**: Canceling \( \pi \) from both sides:
\[ \frac{t_1}{12} = \frac{1}{3} \]
Multiplying both sides by 12 gives:
\[ t_1 = 12 \times \frac{1}{3} = 4 \text{ seconds} \]

### Final Answer:
The time after the beginning of motion when the displacement of the oscillating particle is equal to one half of its amplitude is **4 seconds**.