What is the S.I. unit of Poisson's ratio?

Options

A. kg `m^(-3)`
B. `Nm^(-2)`
C. `m^(-1)`
D. Being a dimensional number, it has no unit.

Answer

To determine the S.I. unit of Poisson's ratio, we can follow these steps:

### Step 1: Understand the Definition of Poisson's Ratio
Poisson's ratio (denoted as \( \nu \)) is defined as the ratio of lateral strain to longitudinal strain.

### Step 2: Define Strain
Strain is defined as the change in length divided by the original length. Mathematically, it can be expressed as:
\[ \text{Strain} = \frac{\Delta L}{L_0} \]
where \( \Delta L \) is the change in length and \( L_0 \) is the original length.

### Step 3: Identify the Types of Strain
In the context of Poisson's ratio:
- **Lateral strain** refers to the strain experienced in the direction perpendicular to the applied load.
- **Longitudinal strain** refers to the strain experienced in the direction of the applied load.

### Step 4: Express Poisson's Ratio
The formula for Poisson's ratio can be written as:
\[ \nu = \frac{\text{Lateral Strain}}{\text{Longitudinal Strain}} = \frac{\frac{\Delta L_{\text{lateral}}}{L_{0, \text{lateral}}}}{\frac{\Delta L_{\text{longitudinal}}}{L_{0, \text{longitudinal}}}} \]

### Step 5: Analyze the Units
Since both lateral strain and longitudinal strain are defined as a ratio of lengths (change in length to original length), they are both dimensionless quantities. Therefore:
\[ \nu = \frac{\text{Dimensionless}}{\text{Dimensionless}} = \text{Dimensionless} \]

### Step 6: Conclusion on Units
Since Poisson's ratio is a dimensionless quantity, it does not have any units. Thus, the S.I. unit of Poisson's ratio is:
\[ \text{No unit} \]

### Final Answer
The S.I. unit of Poisson's ratio is **dimensionless** (no unit).

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