Problem

A cable has an unstretched length of $12 \, \mathrm{m}$ and stretches by $1.2 \times 10^{-4} \, \mathrm{m}$ when a stress of $6.4 \times 10^{8} \, \mathrm{Nm}^{-2}$ is applied. Calculate the strain energy per unit volume (strain energy density) in the cable.

Data

• Unstretched Length: $L = 12 \, \mathrm{m}$
• Change in Length: $\Delta L = 1.2 \times 10^{-4} \, \mathrm{m}$
• Stress: $\sigma = 6.4 \times 10^{8} \, \mathrm{Nm}^{-2}$

Prerequisite Concepts

1. Strain Energy Density: The strain energy per unit volume is given by:

$$\text{Strain Energy Density} = \frac{1}{2} \times \text{Stress} \times \text{Strain}$$

2. Strain: Strain is the ratio of the change in length to the original length:

$$\text{Strain} = \frac{\Delta L}{L}$$

Solution

Step 1: Calculate Strain

Using the formula for strain:

$$\text{Strain} = \frac{\Delta L}{L} = \frac{1.2 \times 10^{-4}}{12} = 1.0 \times 10^{-5}$$

Step 2: Calculate Strain Energy Density

Using the formula for strain energy density:

$$\text{Strain Energy Density} = \frac{1}{2} \times \sigma \times \text{Strain}$$

Substitute the values:

$$\text{Strain Energy Density} = \frac{1}{2} \times (6.4 \times 10^{8}) \times (1.0 \times 10^{-5})$$ $$\text{Strain Energy Density} = \frac{1}{2} \times 6.4 \times 10^{3} = 3.2 \times 10^{3} \, \mathrm{Jm}^{-3}$$

Answer

The strain energy per unit volume (strain energy density) in the cable is:

$$\text{Strain Energy Density} = 3.2 \times 10^{3} \, \mathrm{Jm}^{-3}$$