Question

The frequency $n$ of vibration of stretched string depends upon its length $l$, its mass per unit length $m$ and Tension $t$ in the string Obtain dimensionally an expression for frequency $n$.

Answer 1

Let the relation be:

Frequency v = $K{L}^a{f}^b{m}^c$

where k is a dimensionless constant

The dimension of each quantity are:

$L=\left[M^0{L}^1{T}^o\right]$

$f=\left[M^1{L}^1{T}^{-2}\right]$

$m=\left[M^1{L}^{-1}{T}^0\right]$

$v=\left[M^0{L}^0{T}^{-1}\right]$

Thus now equating the dimension on both side of the equation:

$\left[M^0{L}^{0}T-1\right]=\left[M^0{L}^1{T}^0{]}^a\right[M^1{L}^1{T}^{-2}{]}^b[M^1{L}^{-1}{T}^0{]}^c$

thus, $b=\frac{1}{2}$

$c=-\frac{1}{2}$

$a=-1$

Thus the formula for frequency is:

$V=\frac{K}{L}$ $\sqrt{\frac{f}{m}}$