How Close is a Copper Bar to Breaking Under High Tensile Stress and Sound Waves?

[lizzyb]

The tensile stress in a thick copper bar is 99.5% of its elastic breaking point of 13.0 X 10^10 N/m^2. A 500 Hz sound wave is transmitted through the material. (a) What displacement amplitude will cause the bar to break? (b) What is the maximum speed of the particle at this moment?

Comments:

(b) is easy because one we know the maximum displacement (or amplitude), we may use $$v_{max} = \omega A$$

For (a), though, it seems like the frequency isn't related to this part of the question. If the copper is stretched so far, beyond it's elastic breaking point, then it will break, but how do I determine this? Thank you.

 

[lizzyb]

Well, we know that $$\Delta P = 13.0 \times 10^{10} - .995 \times 13.0 \times 10^{10} = 6.5 \times 10^8$$
and we can assume that this is $$\Delta P_{max}$$ and we may use the equation $$\Delta P_{max} = \rho v \omega s_{max}$$ where rho and omega are easily determined.

What about v? The book says $$v = \sqrt{\frac{B}{\rho}}$$ so how would I determine B?

 

[Ahmed-Mahmoud]

B is the bulk's modulus