Two SHM particles `P_(1)` and `p_(2)` start from `+ (A)/(2)` and `-sqrt(3A)/(2)`, both in negative directions. Find the time (in terms of T) when they collide. Both particles have same omega, `A` and `T` and the execute SHM along the same line.

Explanation

To find the time when the two SHM particles \( P_1 \) and \( P_2 \) collide, we can follow these steps:

### Step 1: Understand the Initial Positions
The initial positions of the particles are given as:
- Particle \( P_1 \) starts from \( +\frac{A}{2} \)
- Particle \( P_2 \) starts from \( -\frac{\sqrt{3}A}{2} \)

Both particles are moving in the negative direction.

### Step 2: Determine the Angles for Each Particle
Using the cosine function, we can find the angles corresponding to the initial positions of both particles.

For \( P_1 \):
\[ \cos(\alpha) = \frac{\text{Position of } P_1}{A} = \frac{\frac{A}{2}}{A} = \frac{1}{2} \]
Thus,
\[ \alpha = \cos^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3} \text{ (or 60 degrees)} \]

For \( P_2 \):
\[ \cos(\beta) = \frac{\text{Position of } P_2}{A} = \frac{-\frac{\sqrt{3}A}{2}}{A} = -\frac{\sqrt{3}}{2} \]
Thus,
\[ \beta = \cos^{-1}\left(-\frac{\sqrt{3}}{2}\right) = \frac{5\pi}{6} \text{ (or 120 degrees)} \]

### Step 3: Find the Condition for Collision
The two particles will collide when they reach the same position. Since they are moving in the same direction with the same angular frequency \( \omega \), the angle between them will be \( 90^\circ \) (or \( \frac{\pi}{2} \)) at the time of collision.

### Step 4: Calculate the Total Angle Rotated
At the time of collision, the angle \( P_1 \) has rotated from \( \frac{\pi}{3} \) to \( \frac{\pi}{2} \):
\[ \text{Angle rotated by } P_1 = \frac{\pi}{2} - \frac{\pi}{3} = \frac{3\pi}{6} - \frac{2\pi}{6} = \frac{\pi}{6} \]

### Step 5: Calculate the Time Taken
Using the relationship between angle and time:
\[ \theta = \omega t \]
Where \( \omega = \frac{2\pi}{T} \). Thus,
\[ \frac{\pi}{6} = \frac{2\pi}{T} \cdot t \]
Rearranging gives:
\[ t = \frac{\pi/6 \cdot T}{2\pi} = \frac{T}{12} \]

### Final Answer
The time \( t \) when the two particles collide is:
\[ t = \frac{T}{12} \]