Question

A wire of density $\rho$ is stretched between the clamps at a distance $L$ apart, while being subjected to an extension $l(l<<L),Y$ is the Young's modulus of the wire. The lowest resonant frequency of transverse vibration of the wire is approximately given by :

• A. $f=\frac{1}{2L}\sqrt{\frac{YL}{l\rho }}$
• B. $f=\frac{1}{2L}\sqrt{\frac{Y\rho L}{l^2}}$
• C. $f=\frac{1}{2L}\sqrt{\frac{Yl}{L\rho }}$
• D. $f=\frac{1}{2L}\sqrt{\frac{L\rho }{Yl}}$

Answer 1

Answer: (C)

$f=\frac{1}{2L}\sqrt{\frac{Yl}{L\rho }}$

$Y=\frac{Stress}{Strain}=\frac{TL}{Al}$

$T=\frac{YAl}{L}$

Mass per unit length $=\rho A$

$f=\frac{1}{2L}\sqrt{\frac{T}{m}}$

$m\to$ mass per unit length
$f=\frac{1}{2L}\sqrt{\frac{Yl}{L\rho }}$