# Calculating strain from wave speed and tension in a wire

• hs764
Remember, you can always check your answer by plugging it back into the original equation and making sure everything balances out.
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?

135 N ≠ F/A

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

Correct.

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

So ΔL/L = 4.0 x 10-4?

## 1. What is the formula for calculating strain from wave speed and tension in a wire?

The formula for calculating strain from wave speed and tension in a wire is: strain = wave speed^2 / (tension x density)

## 2. What are the units for strain, wave speed, tension, and density in this formula?

The units for strain are dimensionless, as it is a ratio of two lengths. The units for wave speed are meters per second (m/s). The units for tension are Newtons (N). The units for density are kilograms per cubic meter (kg/m^3).

## 3. Can this formula be used for any type of wire or material?

Yes, this formula can be used for any type of wire or material as long as the tension and density are known.

## 4. How does strain affect the behavior of a wire?

Strain is a measure of the deformation of a material under stress. As strain increases, the wire will experience more stretching and may eventually reach its breaking point. This can also affect the wire's ability to transmit waves, as increased strain can alter the wire's physical properties.

## 5. What are some potential sources of error when using this formula to calculate strain?

Potential sources of error when using this formula include inaccuracies in measuring the wave speed and tension, variations in the wire's density, and external factors such as temperature or vibrations affecting the wire's behavior.

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