Question

What must be the stress (F//A) in a stretched wire of a material whose Young's modulus is Y for the speed of lonitudinal waves equal to 30 times the speed of transverese waves?

Answer 1

Given ;

Young's modulus of material = γ

Speed of longitudinal wave = 30(Speed of transverse wave)

$V_L = 30\times V_T$                                                 (equation 1)

To find ;

Stress in a streched wire = $\frac{F}{A}$

Solution ;

We know that Young's modulus

γ = $\frac{\textbf{Stress}}{\textbf{Strain}}$                                                         (equation 2)

Stress = $\frac{\textbf{F}}{\textbf{A}}$                                                       (equation 3)

Longitudinal  speed   $V_L = \sqrt{\frac{\gamma}{\rho}}$

where  γ = Young's modulus

          ρ = Density

transverse speed $V_T = \sqrt{\frac{F}{\mu}}$

where F = Force

          μ = Mass per unit length = ρA                        (equation 4)

Given that

$V_L = 30\times V_T$

so   $\sqrt{\frac{\gamma}{\rho}} = 30\times \sqrt{\frac{F}{\mu}}$                                             (Squaring both sides)

$\frac{\gamma}{\rho}}= 900\times \frac{F}{\mu}$                                                          (putting equation 4 here)

$\frac{\gamma}{\rho}}= 900\times \frac{F}{\rho \times A}$                                                      (cancelling ρ on both sides)

γ = 900 $\times \frac{F}{A}$                                                               (by equation 3)

So,

Stress = $\frac{\gamma}{900}$