Problem

A cable has a length of $12 \, \mathrm{m}$ and is stretched by $1.2 \times 10^{-4} \, \mathrm{m}$ when a stress of $8.0 \times 10^{8} \, \mathrm{Nm}^{-2}$ is applied. What is the strain energy per unit volume in the cable when the stress is applied?

Data

• Length of cable, $L = 12 \, \mathrm{m}$
• Change in length, $\Delta L = 1.2 \times 10^{-4} \, \mathrm{m}$
• Stress, $\sigma = 8.0 \times 10^{8} \, \mathrm{Nm}^{-2}$

To find:

• Strain energy per unit volume (strain energy density), $U \, \mathrm{Jm^{-3}}$

Prerequisite Concepts

• Strain energy density is defined as:

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

• Strain is given by:

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

Solution

1. Calculate the strain:

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

2. Substitute the values into the formula for strain energy density:

$$U = \frac{1}{2} \times \text{Stress} \times \text{Strain}$$ $$U = \frac{1}{2} \times (8.0 \times 10^{8}) \times (1.0 \times 10^{-5})$$

3. Simplify:

$$U = 0.5 \times 8.0 \times 10^{3}$$ $$U = 4.0 \times 10^{3} \, \mathrm{Jm^{-3}}$$

Answer

The strain energy per unit volume in the cable is $4.0 \times 10^{3} \, \mathrm{Jm^{-3}}$.