Question

Frequency is the function of density ρ, length λ and tension T. The period of oscillation is proportional to

Answer 1

Dear Student,

Dimensionally

$$\begin{gathered} f=\left[T^{-1}\right] \\ density\rho =\left[M{L}^{-3}\right] \\ Length\lambda =\left[L\right] \\ Tension,Force:T=\left[ML{T}^{-2}\right] \\ forf=\rho \lambda Tbothsidehastodimensionallysame. \\ f=\frac{\sqrt{{\displaystyle \frac{T}{\rho }}}}{{\lambda }^2}\Rightarrow \frac{\sqrt{{\displaystyle \frac{ML{T}^{-2}}{M{L}^{-3}}}}}{L^2}\Rightarrow \frac{\sqrt{L^4{T}^{-2}}}{L^2}\Rightarrow T^{-1} \\ Timeperiodofoscillation:t=\left[T\right] \\ t=\frac{1}{f}\Rightarrow \frac{{\lambda }^2}{\sqrt{\frac{T}{\rho }}} \end{gathered}$$

Regards.