Two particles start oscillating together in shm along the same straight line. Their periods are 2 s and 4 s. What is their phase difference afterls from the start?

Explanation

To solve the problem of finding the phase difference between two particles oscillating in simple harmonic motion (SHM) after 1 second, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Periods of the Particles:**
- Let the period of Particle 1 (T1) be 2 seconds.
- Let the period of Particle 2 (T2) be 4 seconds.

2. **Calculate the Angular Frequencies:**
- The angular frequency (ω) is given by the formula:
\[ \omega = \frac{2\pi}{T} \]
- For Particle 1:
\[ \omega_1 = \frac{2\pi}{2} = \pi \text{ rad/s} \]
- For Particle 2:
\[ \omega_2 = \frac{2\pi}{4} = \frac{\pi}{2} \text{ rad/s} \]

3. **Determine the Phase Positions After 1 Second:**
- The phase (φ) of a particle in SHM can be calculated using:
\[ \phi = \omega t \]
- For Particle 1 after 1 second:
\[ \phi_1 = \omega_1 \cdot 1 = \pi \cdot 1 = \pi \text{ rad} \]
- For Particle 2 after 1 second:
\[ \phi_2 = \omega_2 \cdot 1 = \frac{\pi}{2} \cdot 1 = \frac{\pi}{2} \text{ rad} \]

4. **Calculate the Phase Difference:**
- The phase difference (Δφ) is given by:
\[ \Delta \phi = |\phi_1 - \phi_2| \]
- Substituting the values:
\[ \Delta \phi = |\pi - \frac{\pi}{2}| = |\frac{2\pi}{2} - \frac{\pi}{2}| = |\frac{\pi}{2}| = \frac{\pi}{2} \text{ rad} \]

5. **Conclusion:**
- The phase difference between the two particles after 1 second is \(\frac{\pi}{2}\) radians.