Amplitude of a vibrating string:

When a string is vibrating, it oscillates back and forth between two extreme points. The maximum displacement of the string from its equilibrium position is called the amplitude. In this question, we are given that the string is vibrating in its 3rd overtone with maximum amplitude a.

Understanding the string's vibration:

The frequency of a string's vibration depends on its length, tension, and mass per unit length. When a string is fixed at both ends, it can vibrate in different modes or harmonics. The fundamental frequency is the lowest frequency at which the string can vibrate, and it corresponds to the first harmonic. The string can also vibrate at higher frequencies, which correspond to higher harmonics or overtones.

In this question, we are considering the 3rd overtone, which means the string is vibrating at three times the frequency of the fundamental frequency. The 3rd overtone corresponds to the 3rd harmonic.

Understanding the amplitude at a specific distance:

To find the amplitude at a specific distance on the string, we need to understand the behavior of the string in different harmonics. In a vibrating string, the displacement of the string varies along its length. The string has nodes and antinodes. Nodes are points on the string where the displacement is always zero, while antinodes are points where the displacement is maximum.

In the case of a string fixed at both ends, the fundamental frequency has a single node at each end and one antinode at the center. As we move to higher harmonics, the number of nodes and antinodes increases.

In the 3rd harmonic (3rd overtone), the string has three nodes and two antinodes. The nodes are located at 1/4, 2/4, and 3/4 of the string's length, while the antinodes are located at 1/2 and the ends of the string.

In this question, we are asked to find the amplitude at a distance l/3 from one end. Since l/3 is between the first and second nodes (1/4 and 2/4), it corresponds to an antinode.

Key point: In a vibrating string, the amplitude is maximum at the antinodes.

Therefore, at a distance l/3 from one end, the amplitude is maximum, which is equal to a.

Hence, the correct answer is option C) 3a/2.