Question

The frequency of vibration (v) of a string may depend upon length (l) of the string, tension (T) in the string and mass per unit length (m) of the string. Use method of dimensions for establishing the formula for frequency v

Answer 1

Here, $v\propto l^a{T}^b{m}^c=k{l}^a{T}^b{m}^c$ where k is proportionality constant.

Now substituting dimension of each quantities,

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

Comparing the both sides, $b+c=0,a+b=0$ and $-1=-2b$

Thus, $b=1/2,c=-b=-1/2,a=-b=-1/2$

So, $v=k\sqrt{\frac{T}{ml}}$