Calculating strain from wave speed and tension in a wire

[hs764]

Homework Statement

Just wanted to check my work on this one.

An aluminum wire is clamped at each end under zero tension at room temperature. The tension in the wire is increased by reducing the temperature which results in a decrease in the wire's equilibrium length. What strain (ΔL/L) will result in a transverse wave speed of 100 m/s? Take the cross-sectional area of the wire to be 5.0 x 10^-6 m². The density of aluminum is ρ = 2.7 x 10³ kg/m³ and Young's modulus is Y = 6.8 x 10¹⁰ N/m².

Homework Equations

[/B]
v = √(τ/μ), F/A = E(ΔL/L)

The Attempt at a Solution

[/B]
100 m/s = √(τ/μ) μ = ρ x A = 2.7 x 10³ kg/m³ x 5.0 x 10^-6 m² = 1.35 x 10^-2 kg/m. τ = v²μ = 1 x 10⁴ m²/s² x 1.35 x 10^-2 kg/m = 135 N = F/A. F/A = E(ΔL/L), 135 N = 6.8 x 10¹⁰(ΔL/L). ΔL/L = 2.0 x 10^-9.

Does this look correct?

 

[paisiello2]

135 N ≠ F/A

Check your units.

 

[hs764]

Oh, right...F/A would be 135 N / 5.0 x 10^-6 m² = 2.7 x 10⁷ N/m²?

 

[paisiello2]

Correct.

I always strongly recommend to include all your units in all your calculations. That way you can catch easy mistakes like this.

 

[hs764]

So ΔL/L = 4.0 x 10^-4?

 

[paisiello2]

Looks about right.