Question

A $160g$ rope $4m$ long is fixed at one end and tied to a light string of the same length at the other end. Its tension is $400N$. What are the frequencies of these standing waves ?

Answer 1

In such an arrangement, one fixed end acts as a node, and the other end attached to light string as antinode.

Speed of the wave=$\sqrt{\frac{T}{\mu }}=\sqrt{\frac{400}{0.16/4}}=100m/s$

Thus the fundamental frequency is $\frac{v}{2l}$

where $l$ is the length of string between two nodes=$8m$

Thus fundamental frequency =$\frac{100}{2\times 8}=6.25Hz$

In the first overtone, there are three half-waves in the string. Thus the frequency of first overtone=$\frac{3v}{2l}=18.75Hz$

In the second overtone, there are half half-waves in the string. Thus the frequency of second overtone=$\frac{5v}{2l}=31.25Hz$