Two particles A and B execute SHM on a straight line of path length 2 m starting from the two extreme points simultaneously. If their respective time periods are 1 s and 2 s, the minimum time in which they meet is

Options

A. `(1)/(4)s`
B. `(1)/(3)s`
C. `(1)/(2)s`
D. `(2)/(5)s`

Explanation

To solve the problem of finding the minimum time in which two particles A and B meet while executing simple harmonic motion (SHM) with different time periods, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Time Periods:**
- Let the time period of particle A (T_A) be 1 second.
- Let the time period of particle B (T_B) be 2 seconds.

2. **Calculate the Angular Frequencies:**
- The angular frequency (ω) is given by the formula:
\[ \omega = \frac{2\pi}{T} \]
- For particle A:
\[ \omega_A = \frac{2\pi}{T_A} = \frac{2\pi}{1} = 2\pi \, \text{rad/s} \]
- For particle B:
\[ \omega_B = \frac{2\pi}{T_B} = \frac{2\pi}{2} = \pi \, \text{rad/s} \]

3. **Set Up the Position Equations:**
- The position of particle A at time t can be described by:
\[ y_A = A \sin(\omega_A t + \frac{\pi}{2}) \]
- The position of particle B at time t can be described by:
\[ y_B = A \sin(\omega_B t - \frac{\pi}{2}) \]
- Here, A is the amplitude of the motion.

4. **Equate the Positions:**
- For the particles to meet, their positions must be equal:
\[ y_A = y_B \]
- This leads to the equation:
\[ A \sin(\omega_A t + \frac{\pi}{2}) = A \sin(\omega_B t - \frac{\pi}{2}) \]

5. **Simplify the Equation:**
- Since the amplitude A cancels out, we can simplify to:
\[ \sin(\omega_A t + \frac{\pi}{2}) = \sin(\omega_B t - \frac{\pi}{2}) \]
- Using the sine addition formula, we can express this as:
\[ \cos(\omega_A t) = -\cos(\omega_B t) \]

6. **Set Up the Condition for Meeting:**
- This leads to the condition:
\[ \omega_A t = \pi - \omega_B t + 2n\pi \quad (n \in \mathbb{Z}) \]
- Rearranging gives:
\[ (\omega_A + \omega_B)t = \pi + 2n\pi \]
- For the minimum time (n=0):
\[ t = \frac{\pi}{\omega_A + \omega_B} \]

7. **Calculate the Minimum Time:**
- Substitute the values of ω_A and ω_B:
\[ t = \frac{\pi}{2\pi + \pi} = \frac{\pi}{3\pi} = \frac{1}{3} \, \text{s} \]

### Final Answer:
The minimum time in which the two particles meet is \( \frac{1}{3} \) seconds.