To solve the problem, we need to calculate Young's modulus and the strain potential energy for the given metallic wire. Let's break down the solution step by step.
### Step 1: Calculate the Weight of the Hanging Mass
The weight \( W \) of the mass hanging from the wire can be calculated using the formula:
\[
W = mg
\]
where:
- \( m = 2 \, \text{kg} \) (mass)
- \( g = 10 \, \text{m/s}^2 \) (acceleration due to gravity)
Substituting the values:
\[
W = 2 \times 10 = 20 \, \text{N}
\]
### Step 2: Calculate the Cross-Sectional Area of the Wire
The cross-sectional area \( A \) of the wire can be calculated using the formula for the area of a circle:
\[
A = \pi r^2
\]
where \( r \) is the radius of the wire. The diameter of the wire is given as \( 0.3 \, \text{mm} \), so the radius \( r \) is:
\[
r = \frac{0.3}{2} = 0.15 \, \text{mm} = 0.15 \times 10^{-3} \, \text{m} = 0.15 \times 10^{-3} \, \text{m}
\]
Now, substituting the radius into the area formula:
\[
A = \pi (0.15 \times 10^{-3})^2 = \pi (0.0225 \times 10^{-6}) \approx 7.06858 \times 10^{-8} \, \text{m}^2
\]
### Step 3: Calculate the Stress in the Wire
Stress \( \sigma \) is defined as the force per unit area:
\[
\sigma = \frac{T}{A}
\]
where \( T \) is the tension (equal to the weight calculated in Step 1). Therefore:
\[
\sigma = \frac{20}{7.06858 \times 10^{-8}} \approx 2.83 \times 10^8 \, \text{Pa}
\]
### Step 4: Calculate the Strain in the Wire
Strain \( \epsilon \) is defined as the change in length divided by the original length:
\[
\epsilon = \frac{\Delta L}{L_0}
\]
where:
- \( \Delta L = 2 \, \text{mm} = 2 \times 10^{-3} \, \text{m} \)
- \( L_0 = 3 \, \text{m} \)
Substituting the values:
\[
\epsilon = \frac{2 \times 10^{-3}}{3} \approx 6.67 \times 10^{-4}
\]
### Step 5: Calculate Young's Modulus
Young's modulus \( Y \) is defined as the ratio of stress to strain:
\[
Y = \frac{\sigma}{\epsilon}
\]
Substituting the values calculated:
\[
Y = \frac{2.83 \times 10^8}{6.67 \times 10^{-4}} \approx 4.24 \times 10^{11} \, \text{Pa}
\]
### Step 6: Calculate the Volume of the Wire
The volume \( V \) of the wire can be calculated using the formula for the volume of a cylinder:
\[
V = A \times L_0
\]
Substituting the area and length:
\[
V = 7.06858 \times 10^{-8} \times 3 \approx 2.12 \times 10^{-7} \, \text{m}^3
\]
### Step 7: Calculate the Strain Potential Energy
The strain potential energy \( U \) per unit volume is given by:
\[
U = \frac{1}{2} \sigma \epsilon
\]
Total strain potential energy is:
\[
U_{total} = U \times V = \frac{1}{2} \sigma \epsilon \times V
\]
Substituting the values:
\[
U_{total} = \frac{1}{2} \times 2.83 \times 10^8 \times 6.67 \times 10^{-4} \times 2.12 \times 10^{-7}
\]
Calculating this gives:
\[
U_{total} \approx 19.97 \times 10^{-3} \, \text{J}
\]
### Final Answers
- Young's Modulus \( Y \approx 4.24 \times 10^{11} \, \text{Pa} \)
- Strain Potential Energy \( U_{total} \approx 19.97 \times 10^{-3} \, \text{J} \)
