Question

The frequency of vibration of a depends upon the length of the string, tension in the string & mass per unit length of the string $\left(m\right)$. Using dimension obtain the expression for the same.

Answer 1

According to the question , frequency $f=F(l,T,m)=k{l}^a{T}^b{m}^c$, here k=any dimensionless constant, l=length of string , T= tension in string

using property of dimension, i.e, dimension of LHS=RHS

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

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

comparing both sides we get :

(1) $b+c=0\Rightarrow b=-c$

(2) $-2b-c=-1\Rightarrow b=1,c=-1$

(3) $a+b-c=0\Rightarrow a=-2$

so, we get the following formula ,

$f=k{l}^{-2}T{m}^{-1}=\frac{kT}{m{l}^2}$