(A): When a stretched string vibrates in two segments, then all the vibrating particles in first half of the string are in out of phase to that of in the remaining of the string
(R): In a stationary wave the phase difference between the vibrating particles in two consecutive loops is trad.

Options

A. Both 'A' and 'R' are true and 'R' is the correct, explanation of 'A'.
B. Both 'A' and 'R' are true and 'R' is not the correct explanation of 'A'
C. A' is true and 'R' is false
D. Both 'A' and 'R' are false

Answer

To analyze the statements given in the question, we need to evaluate both statements (A) and (R) separately.

### Step 1: Analyze Statement (A)
Statement (A) says: "When a stretched string vibrates in two segments, then all the vibrating particles in the first half of the string are out of phase with those in the remaining half of the string."

**Explanation:**
When a stretched string vibrates in two segments, it forms a stationary wave. In a stationary wave, the string is divided into segments called loops. The particles in one segment (loop) oscillate in the opposite direction to the particles in the adjacent segment. Therefore, the particles in the first half of the string will indeed be out of phase with the particles in the second half of the string.

**Conclusion for (A):** True

### Step 2: Analyze Statement (R)
Statement (R) says: "In a stationary wave, the phase difference between the vibrating particles in two consecutive loops is π radians (180 degrees)."

**Explanation:**
In a stationary wave, the phase difference between two consecutive nodes (or loops) is indeed π radians. This means that if one particle is at its maximum displacement in one loop, the particle in the next loop will be at its minimum displacement, indicating they are out of phase.

**Conclusion for (R):** True

### Step 3: Evaluate the Relationship between (A) and (R)
Now we need to determine if (A) is the correct reason for (R).

- Statement (A) correctly describes the behavior of particles in a vibrating string.
- Statement (R) correctly describes the phase difference in stationary waves.

Both statements are true, and (A) provides a valid explanation for (R).

### Final Conclusion:
Both statements (A) and (R) are true, and (A) is the correct explanation for (R).

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