Unit of coefficient of viscosity is

Options

A. Ns/m
B. `Ns^(2)//m `
C. `Ns^(2)//m^(2)`
D. `Ns//m^(2)`

Answer

To determine the unit of the coefficient of viscosity (η), we can start from its definition and the formula that relates it to other physical quantities.

### Step-by-Step Solution:

1. **Understand the Definition**: The coefficient of viscosity (η) is defined as the ratio of shear stress to shear rate.

2. **Identify the Formula**: The formula for the coefficient of viscosity can be expressed as:
\[ \eta = \frac{\text{Force} \times \text{Radius}}{\text{Area} \times \text{Velocity}} \]
where:
- Force (F) is measured in Newtons (N)
- Radius (r) is measured in meters (m)
- Area (A) is measured in square meters (m²)
- Velocity (v) is measured in meters per second (m/s)

3. **Substitute SI Units**: Substitute the SI units into the formula:
\[ \eta = \frac{N \cdot m}{m^2 \cdot \frac{m}{s}} \]

4. **Simplify the Expression**:
- The unit of force (N) can be expressed as \(1 \, \text{N} = 1 \, \text{kg} \cdot \text{m/s}^2\).
- Substituting this into the equation gives:
\[ \eta = \frac{(1 \, \text{kg} \cdot \text{m/s}^2) \cdot m}{m^2 \cdot \frac{m}{s}} \]
- This simplifies to:
\[ \eta = \frac{1 \, \text{kg} \cdot \text{m}^2/s^2}{m^2 \cdot m/s} = \frac{1 \, \text{kg} \cdot \text{m}^2/s^2}{m^3/s} \]

5. **Final Simplification**:
- When you simplify this further, you get:
\[ \eta = \frac{1 \, \text{kg} \cdot s}{m} \]
- Therefore, the unit of the coefficient of viscosity is:
\[ \text{Pascal-second (Pa·s)} \quad \text{or} \quad \text{kg/(m·s)} \]

### Conclusion:
The unit of the coefficient of viscosity is \( \text{Pa·s} \) or \( \text{kg/(m·s)} \).

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