Question

Assuming that the frequency $\gamma$ of a vibrating string may depend upon $i)$ applied force $\left(F\right)ii)$ length $\left(\ell \right)iii)$ mass per unit length $\left(m\right)$, prove that $\gamma \alpha \frac{1}{l}\sqrt{\frac{F}{m}}$ using dimensional analysis.

Answer 1

dimension of frequency=$\left[T^{-1}\right]$

Dimension of applied force,F=$\left[ML{T}^{-2}\right]$

dimension of length,l=[L]

dimension of mass per unit length,m=$\left[M{L}^{-1}\right]$

Now suppose,frequency =F^x L^y m^z

so,$\left[T^{-1}\right]$= $\left[ML{T}^{-2}\right]$^$\times$[L]^y $\left[M{L}^{-1}\right]$^z

$\left[T^{-1}\right]=\left[M\right]$^(X+Z)[L]^(X+Y-Z)[T]^(-2X)

Compare both sides,

X+Z=0

X+Y-Z=0

-2X=-1 $\Rightarrow$X=1/2

So,Z=-1/2 and Y=-1

Hence, frequency is directly proportional to $1/l\sqrt{\left(F/m\right)}$