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Calculating strain from wave speed and tension in a wire

  1. Apr 23, 2015 #1
    1. The problem statement, all variables and given/known data

    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.

    2. Relevant equations

    v = √(τ/μ), F/A = E(ΔL/L)

    3. The attempt at a solution

    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?
     
  2. jcsd
  3. Apr 23, 2015 #2
    135 N ≠ F/A

    Check your units.
     
  4. Apr 23, 2015 #3
    Oh, right....F/A would be 135 N / 5.0 x 10-6 m2 = 2.7 x 107 N/m2?
     
  5. Apr 23, 2015 #4
    Correct.

    I always strongly recommend to include all your units in all your calculations. That way you can catch easy mistakes like this.
     
  6. Apr 23, 2015 #5
    So ΔL/L = 4.0 x 10-4?
     
  7. Apr 23, 2015 #6
    Looks about right.
     
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