Question

What must be the stress (F/A) in a stretched wire of material whose Young's modulus is Y for the speed of longitudinal waves to equal 3o times the speed of transverse waves?

Answer 1

We know the formula for the speed of the longitudinal waveis

$$ v_{l}=\sqrt{\frac{Y}{p}} $$

We know the formula for the speed of the transverse waveis

$$ v_{t}=\sqrt{\frac{F}{\mu}} $$

From the problem given that

$$ \begin{aligned} v_{l} &=30 v_{t} \\ \sqrt{\frac{\gamma}{\rho}} &=30 \sqrt{\frac{F}{\mu}} \\ \frac{y}{\rho} &=(900)\left(\frac{F}{\mu}\right) \end{aligned} $$

Since we have \(\mu=\rho A\)

Then

$$ \frac{\gamma}{\rho}=(900)\left(\frac{F}{p A}\right) $$

$$ \begin{array}{l} Y=(900)\left(\frac{F}{A}\right) \\ \frac{F}{A}=\frac{Y}{900} \end{array} $$