To solve the problem step by step, we will use the concepts of elasticity, specifically Young's modulus, and the relationship between stress, strain, and force.
### Step 1: Understand the problem and gather the given data
- Initial length of the rod, \( L_0 = 0.8000 \, \text{m} \)
- Initial radius of the rod, \( R_0 = 1000.0 \, \mu m = 1000.0 \times 10^{-6} \, \text{m} \)
- Final radius of the rod, \( R_f = 999.9 \, \mu m = 999.9 \times 10^{-6} \, \text{m} \)
- Young's modulus for aluminum, \( E = 70 \times 10^9 \, \text{N/m}^2 \)
### Step 2: Calculate the initial and final cross-sectional areas
The cross-sectional area \( A \) of a cylinder is given by the formula:
\[
A = \pi R^2
\]
- Initial area \( A_0 \):
\[
A_0 = \pi (R_0)^2 = \pi (1000 \times 10^{-6})^2 = \pi (10^{-3})^2 = \pi \times 10^{-6} \, \text{m}^2
\]
- Final area \( A_f \):
\[
A_f = \pi (R_f)^2 = \pi (999.9 \times 10^{-6})^2 = \pi (999.9^2 \times 10^{-12}) \, \text{m}^2
\]
### Step 3: Use the principle of conservation of volume
Since the density does not change, the volume before and after stretching must be equal:
\[
A_0 L_0 = A_f L_f
\]
Where \( L_f \) is the final length of the rod after stretching. Rearranging gives:
\[
L_f = \frac{A_0 L_0}{A_f}
\]
### Step 4: Calculate the change in length
The change in length \( \Delta L \) is given by:
\[
\Delta L = L_f - L_0
\]
### Step 5: Relate stress, strain, and Young's modulus
The stress \( \sigma \) is defined as:
\[
\sigma = \frac{F}{A_0}
\]
The strain \( \epsilon \) is defined as:
\[
\epsilon = \frac{\Delta L}{L_0}
\]
According to Young's modulus:
\[
E = \frac{\sigma}{\epsilon}
\]
Thus:
\[
F = \sigma A_0 = E \cdot \epsilon \cdot A_0
\]
### Step 6: Substitute the values
Substituting for stress and strain:
\[
F = E \cdot \left(\frac{\Delta L}{L_0}\right) \cdot A_0
\]
### Step 7: Calculate the force
Now we can calculate the force \( F \) using the values we have.
1. Calculate \( A_0 \) and \( A_f \):
- \( A_0 = \pi (1000 \times 10^{-6})^2 \)
- \( A_f = \pi (999.9 \times 10^{-6})^2 \)
2. Calculate \( L_f \) using the conservation of volume.
3. Calculate \( \Delta L \).
4. Substitute into the force equation to find \( F \).
### Final Calculation
After performing the calculations, we find:
\[
F \approx 44 \, \text{N}
\]
