A second harmonic has to be generated in a string of length l stretched between two rigid supports. The point where the string has to be plucked and touched are

Options

A. Plucked at `L/4` and touch at `L/2`
B. Plucked at `L/4` and touch at `(3L)/2`
C. Plucked at `L/2` and touch at `L/2`
D. Plucked at `L/2` and touch at `(3L)/4`

Explanation

To solve the problem of determining where to pluck and touch a string to generate the second harmonic, we can follow these steps:

### Step 1: Understand the Harmonics of a String
The second harmonic of a string fixed at both ends has a specific pattern of nodes and antinodes. In the second harmonic, there are two segments of the wave, resulting in one node at the center and two antinodes at the ends.

### Step 2: Identify the Nodes and Antinodes
For a string of length \( l \):
- The nodes (points of no displacement) are located at:
- \( x = 0 \) (left end)
- \( x = \frac{l}{2} \) (middle)
- \( x = l \) (right end)

- The antinodes (points of maximum displacement) are located at:
- \( x = \frac{l}{4} \)
- \( x = \frac{3l}{4} \)

### Step 3: Determine the Plucking and Touching Points
- **Plucking Point**: This is where the string should be plucked to create an antinode. Therefore, the plucking points can be at:
- \( x = \frac{l}{4} \) or \( x = \frac{3l}{4} \)

- **Touching Point**: This is where the string should be touched to create a node. The touching point can only be at:
- \( x = \frac{l}{2} \)

### Step 4: Conclusion
To generate the second harmonic in a string of length \( l \):
- The string should be plucked at \( x = \frac{l}{4} \) or \( x = \frac{3l}{4} \).
- The string should be touched at \( x = \frac{l}{2} \).

### Final Answer
- Plucking points: \( x = \frac{l}{4} \) and \( x = \frac{3l}{4} \)
- Touching point: \( x = \frac{l}{2} \)

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