Question

A metal string is fixed between rigid supports. It is initially at negligible tension. Its Young's modulus is Y, density is $\rho$ and coefficient of linear expansion is $\alpha$. It is now cooled through a temperature t, transverse waves will move along it with a speed of :

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

Answer 1

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

By the definition of Young's modulus, we have
$y=\left(\frac{F}{A}\right)/\left(\frac{△L}{l}\right)\Rightarrow F=\frac{YAl\left(\alpha t\right)}{l}$
$△l=l\alpha t$
$\therefore$ F = YA $\alpha$ t
The velocity $V=\sqrt{\frac{T}{\mu }}=\sqrt{\frac{YA\alpha t}{\rho \left(A\right)}}$
$\mu =\rho \left(A\right)$
$\Rightarrow V=\sqrt{\frac{Y\alpha t}{\rho }}$