Question

a hollow metallic tube closed at one end produces resonance with a tuning fork of frequency n at temperature $T_0$. The temperature of the tube is increased by dT. If the coefficient of thermal expansion of the tube metal is $\alpha$, the beat frequency will be

Answer 1

velocity of sound $\left(v\right)=\sqrt{\frac{VRT}{M}}$

$\Rightarrow v\propto \sqrt{T}$

let $v=\beta \sqrt{T}$..........(1)

let $LB$ the length of rod at $0^{o}C$

$L^{\prime }=L(1+\alpha T)\{L^{\prime }=$ at $TC$

$\{\begin{array}{c}{L}_0=L(1+\alpha T_0)\\ v_0=\beta \sqrt{T_0}\end{array}$

now, for resonance

$\frac{k{\lambda }_0}{4}=L_0$

$f_0=\frac{k}{4}\frac{v_0}{L_0}$

$\Rightarrow n=\frac{k}{4}\frac{\beta \sqrt{T_0}}{L(1+\alpha T_0)}$.......(iii)

$L_1=L_0(1+\alpha dT),{\alpha }_1=\beta \sqrt{T_0+dt}$

$\Rightarrow \{L_1=L(1+\alpha T_0\left)\right(1-\left(\alpha dT\right)\}$

$f_1=\frac{k}{4}\frac{v_1}{L_1}=\frac{k}{4}\left(\frac{\beta \sqrt{T_0+dt}}{L(1+\alpha T_0)(1+\alpha dt)}\right)$

$f_1=\frac{k\beta \sqrt{T_0}}{4L(1+\alpha T_0)}\left(\sqrt{1+\frac{dt}{T_0}}\right(1-\alpha dT\left)\right)$

using binomial approximation

$f_1=n(1+\frac{1}{2}\frac{dT}{T_0-\alpha dT})$......(iii)

$\Rightarrow beatfrequency\Rightarrow \left(iii\right)-\left(ii\right)=n(\frac{1}{2}\frac{dT}{T_0}-\alpha dT)$