Question

A light string is tied to another light string of equal length $L$ and two strings are tied to the fixed supports at their other ends. The strings are set into vibration so that junction $A$ is a node. The wavelength of the loops formed to the left and to the right of the junction point $A$ is ${\lambda }_1$ and ${\lambda }_2$ respectively. Find the difference of the number of the loops formed to the right and to the left of the point $A$.

• A. $\frac{L({\lambda }_1-{\lambda }_2)}{2{\lambda }_1{\lambda }_2}$
• B. $\frac{2L({\lambda }_1-{\lambda }_2)}{{\lambda }_1{\lambda }_2}$
• C. $\frac{4L({\lambda }_1-{\lambda }_2)}{{\lambda }_1{\lambda }_2}$
• D. $\frac{4L({\lambda }_1+{\lambda }_2)}{{\lambda }_1({\lambda }_1-{\lambda }_2)}$

Answer 1

The correct option is B $\frac{2L({\lambda }_1-{\lambda }_2)}{{\lambda }_1{\lambda }_2}$

$n_1\frac{{\lambda }_1}{2}=L$

and $n_2\frac{{\lambda }_2}{2}=L$

The final answer can be found out using the above two equations as:
$n_2-n_1=\frac{2L}{{\lambda }_2}-\frac{2L}{{\lambda }_1}=$$\frac{2L({\lambda }_1-{\lambda }_2)}{{\lambda }_1{\lambda }_2}$