A tensile force of 100 kN is applied on a 0.02-m diameter and 2-m long rod. After applying the load, the diameter of the rod decreases to 0.01998 m and the length increases to 2.01 m. Determine Poisson’s ratio of the material.

To determine Poisson's ratio (\( \nu \)) of the material, we need to relate the longitudinal strain and the lateral (transverse) strain. The formula for Poisson's ratio is:

\[
\nu = - \frac{\text{Lateral strain}}{\text{Longitudinal strain}}
\]

### Step 1: Calculate the longitudinal strain
The longitudinal strain is the change in length divided by the original length:

\[
\text{Longitudinal strain} = \frac{\Delta L}{L_0}
\]

where:
- \( \Delta L = L_f - L_0 = 2.01 \, \text{m} - 2.00 \, \text{m} = 0.01 \, \text{m} \)
- \( L_0 = 2.00 \, \text{m} \)

So,

\[
\text{Longitudinal strain} = \frac{0.01}{2} = 0.005
\]

### Step 2: Calculate the lateral strain
The lateral strain is the change in diameter divided by the original diameter:

\[
\text{Lateral strain} = \frac{\Delta d}{d_0}
\]

where:
- \( \Delta d = d_f - d_0 = 0.01998 \, \text{m} - 0.020 \, \text{m} = -0.00002 \, \text{m} \)
- \( d_0 = 0.020 \, \text{m} \)

So,

\[
\text{Lateral strain} = \frac{-0.00002}{0.02} = -0.001
\]

### Step 3: Determine Poisson’s ratio
Now we can calculate Poisson’s ratio:

\[
\nu = - \frac{\text{Lateral strain}}{\text{Longitudinal strain}} = - \frac{-0.001}{0.005} = 0.2
\]

Thus, the Poisson’s ratio of the material is \( \nu = 0.2 \).

Would you like further details or have any questions?

### Related Questions:
1. What is the significance of Poisson’s ratio in materials science?
2. How does the value of Poisson’s ratio affect material behavior under tensile loading?
3. Can Poisson’s ratio be greater than 0.5? Why or why not?
4. How is Poisson’s ratio measured experimentally?
5. What is the relationship between Poisson's ratio and the modulus of elasticity?

### Tip:
Poisson’s ratio is typically between 0 and 0.5 for most materials, with values closer to 0 indicating brittle behavior and values closer to 0.5 indicating more ductile behavior.

Determine Poisson’s ratio for a rod subjected to a 100 kN tensile force. The problem involves calculating longitudinal and lateral strain based on the changes in length and diameter. A detailed step-by-step solution suitable for engineering students.

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