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Guess the expression for its frequency from dimensional analysis

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 $\mathrm{m}$. Guess the expression for its frequency from dimensional analysis.

Solution:

Let, frequency $\mathrm{v}=\mathrm{F}^{\mathrm{a}} \mathrm{L}^{\mathrm{b}} \mathrm{m}^{\mathrm{c}}$

or $[\mathrm{T}-1]=\left[\mathrm{ML}^{-2}\right]^{\mathrm{a}}\left[\mathrm{L}^{\mathrm{b}}\right]\left[\mathrm{M}^{\mathrm{c}}\right]$

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 $b=-1 / 2$.

$\text { So, } v=F^{-1 / 2} L^{-1 / 2} m^{-1 / 2}$

 

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