Two particles are in SHM along same line. Time period of each is `T` and amplitude is `A`. After how much time will they collide if at time `t = 0`. (a) first particle is at `x_(1) = + (A)/(2)` and moving towards positive x - axis and second particle is at `x_(2) = - (A)/(sqrt2)` and moving towards negative x - axis, (b) rest information are same as mentioned in part (a) except that particle first is also moving towards negative x - axis.

Explanation

To solve the problem, we need to analyze the motion of both particles in simple harmonic motion (SHM) and determine when they collide based on their initial positions and directions of motion.

### Part (a):
1. **Initial Positions and Directions**:
- Particle 1 is at \( x_1 = +\frac{A}{2} \) and moving towards the positive x-axis.
- Particle 2 is at \( x_2 = -\frac{A}{\sqrt{2}} \) and moving towards the negative x-axis.

2. **Angular Displacement Calculation**:
- For Particle 1, the angular displacement \( \theta_1 \) can be calculated using:
\[ \cos(\theta_1) = \frac{x_1}{A} = \frac{\frac{A}{2}}{A} = \frac{1}{2} \implies \theta_1 = 60^\circ = \frac{\pi}{3} \text{ radians} \]
- For Particle 2, the angular displacement \( \theta_2 \) is:
\[ \cos(\theta_2) = \frac{x_2}{A} = \frac{-\frac{A}{\sqrt{2}}}{A} = -\frac{1}{\sqrt{2}} \implies \theta_2 = 135^\circ = \frac{3\pi}{4} \text{ radians} \]

3. **Relative Motion**:
- Both particles will collide when the sum of the distances they cover equals the total distance between them.
- The total angular distance from \( \theta_1 \) to \( \theta_2 \) is:
\[ \theta = \theta_2 - \theta_1 = \frac{3\pi}{4} - \frac{\pi}{3} \]
- Finding a common denominator (12):
\[ \theta = \frac{9\pi}{12} - \frac{4\pi}{12} = \frac{5\pi}{12} \]

4. **Calculating Time to Collision**:
- The angular velocity \( \omega \) for both particles is given by:
\[ \omega = \frac{2\pi}{T} \]
- The time \( t \) taken for them to collide is given by:
\[ \omega t = \frac{5\pi}{12} \implies \frac{2\pi}{T} t = \frac{5\pi}{12} \]
- Solving for \( t \):
\[ t = \frac{5T}{24} \]

### Part (b):
1. **Initial Positions and Directions**:
- The initial positions remain the same, but now Particle 1 is also moving towards the negative x-axis.

2. **Angular Displacement**:
- The angular displacements remain the same as calculated before:
- Particle 1: \( \theta_1 = \frac{\pi}{3} \)
- Particle 2: \( \theta_2 = \frac{3\pi}{4} \)

3. **Relative Motion**:
- Since both particles are now moving towards negative x, we need to consider their combined angular motion:
- The total angular distance they need to cover to collide is still \( \theta = \theta_2 - \theta_1 = \frac{5\pi}{12} \).

4. **Calculating Time to Collision**:
- The equation for time remains:
\[ \omega t = \frac{5\pi}{12} \]
- However, since both are moving towards each other, they will cover the distance faster. The effective angular distance covered is doubled:
\[ 2\omega t = \frac{5\pi}{12} \]
- Solving for \( t \):
\[ t = \frac{5T}{48} \]

### Final Answers:
- (a) The time after which the particles collide is \( t = \frac{5T}{24} \).
- (b) The time after which the particles collide is \( t = \frac{5T}{48} \).