A structural steel rod has a radius r =10mm and a length l.=1m When a force F=100kN is applied, it stretches it along its length. Young's modulus of elasticity of the structural steel is `2.0xx10^(11) Nm^(-2)`. What is the elastic energy density of the steel rod ?

Options

A. a. 1.59 mm
B. b. 2.32 mm
C. c. 0.159 mm
D. d. 3.18 mm

Explanation

To find the elastic energy density of the structural steel rod, we will follow these steps:

### Step 1: Understand the Formula for Elastic Energy Density
The elastic energy density (u) can be expressed as:
\[ u = \frac{U}{V} \]
where \( U \) is the elastic potential energy and \( V \) is the volume of the rod.

### Step 2: Express Elastic Potential Energy
The elastic potential energy \( U \) in a material can be given by:
\[ U = \frac{1}{2} \times \text{Stress} \times \text{Strain} \times V \]

### Step 3: Define Stress and Strain
- **Stress** (\( \sigma \)) is defined as:
\[ \sigma = \frac{F}{A} \]
where \( F \) is the applied force and \( A \) is the cross-sectional area of the rod.

- **Strain** (\( \epsilon \)) is defined as:
\[ \epsilon = \frac{\Delta L}{L} \]
where \( \Delta L \) is the change in length and \( L \) is the original length of the rod.

### Step 4: Relate Young's Modulus to Stress and Strain
Young's modulus (\( Y \)) relates stress and strain:
\[ Y = \frac{\sigma}{\epsilon} \]
From this, we can express strain in terms of stress and Young's modulus:
\[ \epsilon = \frac{\sigma}{Y} \]

### Step 5: Substitute Stress and Strain into the Energy Formula
Substituting the expressions for stress and strain into the energy density formula:
\[ u = \frac{1}{2} \times \sigma \times \epsilon \]
\[ u = \frac{1}{2} \times \sigma \times \left(\frac{\sigma}{Y}\right) \]
\[ u = \frac{\sigma^2}{2Y} \]

### Step 6: Calculate the Area of the Rod
The cross-sectional area \( A \) of the rod can be calculated using:
\[ A = \pi r^2 \]
Given \( r = 10 \text{ mm} = 10 \times 10^{-3} \text{ m} \):
\[ A = \pi (10 \times 10^{-3})^2 = \pi \times 10^{-4} \text{ m}^2 \]

### Step 7: Calculate Stress
Now, substituting the values into the stress formula:
\[ F = 100 \text{ kN} = 100 \times 10^3 \text{ N} \]
\[ \sigma = \frac{F}{A} = \frac{100 \times 10^3}{\pi \times 10^{-4}} = \frac{10^7}{\pi} \text{ N/m}^2 \]

### Step 8: Substitute Stress into the Energy Density Formula
Now substituting \( \sigma \) into the energy density formula:
\[ u = \frac{1}{2} \times \frac{(10^7/\pi)^2}{2 \times 10^{11}} \]

### Step 9: Simplify the Expression
Calculating:
\[ u = \frac{(10^{14}/\pi^2)}{4 \times 10^{11}} = \frac{10^{14}}{4 \times 10^{11} \pi^2} \]
\[ u = \frac{10^3}{4 \pi^2} \text{ J/m}^3 \]

### Step 10: Final Calculation
Using \( \pi \approx 3.14 \):
\[ u \approx \frac{10^3}{4 \times (3.14)^2} \]
\[ u \approx \frac{10^3}{39.56} \approx 25.3 \text{ J/m}^3 \]

### Conclusion
The elastic energy density of the steel rod is approximately:
\[ u \approx 2.5 \times 10^5 \text{ J/m}^3 \]