Question

A metal string is fixed between rigid supports. It is initially at negligible tension. It Young’s modulus is Y, density is ρ and coefficient of thermal expansion is $\alpha$. If it is now cooled through a temperature = t, transverse wave will move along it with speed

• A. $Y\frac{\alpha t}{\rho }$
  
• B. $\alpha t\sqrt{\frac{Y}{\rho }}$
  
• C. $\sqrt{\frac{Y\alpha t}{\rho }}$
  
• D. $t\sqrt{\frac{Y\alpha }{\rho }}$

Answer 1

The correct option is C $\sqrt{\frac{Y\alpha t}{\rho }}$


From, $Y=\frac{\frac{F}{A}}{\Delta \frac{L}{L}}=\frac{TL}{A\Delta L}$
where, F = T = tension.
On cooling, $\Delta L=\alpha L\left(\Delta \theta \right)$
$=\alpha Lt\therefore Y=\frac{TL}{A\alpha Lt}T=YA\alpha t$
Also, mass per unit length of string
$m=A\rho$
As wave velocity =$\sqrt{\left(\frac{T}{m}\right)}$
$\therefore v=\sqrt{\frac{YA\alpha t}{A\rho }}=\sqrt{\left(\frac{Y\alpha t}{\rho }\right)}$