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.


Poisson's ratio (\(\nu\)) is defined as the negative ratio of the lateral strain to the longitudinal strain in a material when it is subjected to a uniaxial load.

We can calculate it using the following formula:

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

Where:
- **Lateral Strain** is the strain in the direction perpendicular to the applied load (change in diameter).
- **Longitudinal Strain** is the strain in the direction of the applied load (change in length).

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

\[
\text{Longitudinal Strain} = \frac{\Delta L}{L_0} = \frac{2.01 \, \text{m} - 2.00 \, \text{m}}{2.00 \, \text{m}} = \frac{0.01 \, \text{m}}{2.00 \, \text{m}} = 0.005
\]

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

\[
\text{Lateral Strain} = \frac{\Delta d}{d_0} = \frac{0.01998 \, \text{m} - 0.02000 \, \text{m}}{0.02000 \, \text{m}} = \frac{-0.00002 \, \text{m}}{0.02000 \, \text{m}} = -0.001
\]

### Step 3: Calculate Poisson’s Ratio
Now, we can calculate Poisson's ratio using the formula:

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

### Conclusion:
The Poisson's ratio of the material is \( \nu = 0.2 \).

Would you like further details or have any questions?

Here are some related questions:
1. What is the significance of Poisson's ratio in material science?
2. How does the value of Poisson's ratio affect the behavior of materials?
3. Can Poisson's ratio be greater than 0.5 for any materials?
4. How does Poisson's ratio differ between brittle and ductile materials?
5. What are some typical Poisson’s ratios for common materials like steel or rubber?

**Tip:** Poisson’s ratio is typically between 0 and 0.5 for most materials, with rubber-like materials near 0.5 and metals around 0.3.

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.

Learn how to determine Poisson's ratio for a material subjected to a 100 kN tensile force. This problem involves calculating the longitudinal and lateral strains from changes in diameter and length. Ideal for college-level engineering and physics students.

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