Question:
The frequency of vibration of a string depends on the length $L$ between the nodes, the tension $F$ in the string and its mass per unit length $m$. Guess the expression for its frequency from dimensional analysis.
Solution:
Let, frequency $v=F^{a} L^{b} m^{c}$
or $[T-1]=\left[M L T^{-2}\right]^{a}\left[L^{b}\right]\left[M^{c}\right]$
Equating the terms, we get $-2 a=-1$, or $a=1 / 2$, and $c+a=0$, so $c=-1 / 2$
and $a+b=0$, so $b=-1 / 2$.
So, $v=F^{-1 / 2} L^{-1 / 2} m^{-1 / 2}$