Introduction:
In this problem, we are given a string of length l, fixed at both ends, and vibrating in second harmonics. We need to find the amplitude of the particle at a distance (1/8)l from the fixed end.

Understanding the problem:
To solve this problem, we need to understand the concept of standing waves. In a standing wave, the particles of the medium vibrate with maximum amplitude at certain points called antinodes. The distance between two adjacent antinodes is equal to half of the wavelength of the wave. The amplitude of the particles at any point on the string depends on the distance of that point from the nearest antinode.

Solution:
Given, the string is vibrating in second harmonics. This means that the wavelength of the wave is equal to twice the length of the string. Thus, the wavelength of the wave is given by λ = 2l.

We know that the distance between two adjacent antinodes is equal to half of the wavelength. Thus, the distance between two adjacent antinodes is given by λ/2 = l.

The particle whose amplitude we need to find is located at a distance (1/8)l from the fixed end. We can calculate the distance of this point from the nearest antinode as follows:

Distance from the fixed end to the nearest antinode = (1/4)l

Distance from the fixed end to the given point = (1/8)l

Thus, the distance between the given point and the nearest antinode is:

Distance between the given point and the nearest antinode = (1/4)l - (1/8)l = (1/8)l

The amplitude of the particle at the given point is equal to half the amplitude of the antinode. Thus, the amplitude of the particle at the given point is:

Amplitude of the particle at the given point = (1/2) x 5mm = 2.5mm

Conclusion:
The amplitude of the particle at a distance (1/8)l from the fixed end is 2.5mm.

Can you please explain me this how u got