Problem

(a) A force of $1.5 \times 10^{4} \, \mathrm{N}$ causes a strain of $1.4 \times 10^{-4}$ in a steel cable with a cross-sectional area of $4.8 \times 10^{-4} \, \mathrm{m}^2$. Calculate the Young’s modulus of the steel cable.
(b) The stress-strain graph is linear for this cable. Find the strain energy per unit volume stored in the cable when the strain is $1 \times 10^{-4}$.

Data

• Force: $F = 1.5 \times 10^{4} \, \mathrm{N}$
• Cross-sectional area: $A = 4.8 \times 10^{-4} \, \mathrm{m}^2$
• Strain (a): $\epsilon = 1.4 \times 10^{-4}$
• Strain (b): $\epsilon = 1.0 \times 10^{-4}$

Prerequisite Concepts

1. Young’s Modulus:

$$Y = \frac{\text{Tensile Stress}}{\text{Tensile Strain}}$$

Tensile Stress:

$$\sigma = \frac{F}{A}$$

2. Strain Energy Density:

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

Solution

(a) Calculate Young’s Modulus

1. Find Tensile Stress:

$$\sigma = \frac{F}{A} = \frac{1.5 \times 10^{4} \, \mathrm{N}}{4.8 \times 10^{-4} \, \mathrm{m}^2}$$ $$\sigma = 3.12 \times 10^{7} \, \mathrm{Nm}^{-2}$$

2. Calculate Young’s Modulus:

$$Y = \frac{\sigma}{\epsilon} = \frac{3.12 \times 10^{7} \, \mathrm{Nm}^{-2}}{1.4 \times 10^{-4}}$$ $$Y = 2.23 \times 10^{11} \, \mathrm{Nm}^{-2}$$

(b) Calculate Strain Energy Density

1. Find Stress for Strain $\epsilon = 1 \times 10^{-4}$: Using the relation:

$$Y = \frac{\sigma}{\epsilon}$$ $$\sigma = Y \times \epsilon = 2.23 \times 10^{11} \, \mathrm{Nm}^{-2} \times 1 \times 10^{-4}$$ $$\sigma = 2.23 \times 10^{7} \, \mathrm{Nm}^{-2}$$

2. Calculate Strain Energy Density:

$$\text{Strain Energy Density} = \frac{1}{2} \times \sigma \times \epsilon$$ $$\text{Strain Energy Density} = \frac{1}{2} \times 2.23 \times 10^{7} \, \mathrm{Nm}^{-2} \times 1 \times 10^{-4}$$ $$\text{Strain Energy Density} = 1.115 \times 10^{3} \, \mathrm{Jm}^{-3}$$

Answer

(a) The Young’s modulus of the steel cable is:

$$Y = 2.23 \times 10^{11} \, \mathrm{Nm}^{-2}$$

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

$$\text{Strain Energy Density} = 1.115 \times 10^{3} \, \mathrm{Jm}^{-3}$$