Which of the following is the unit of coefficient of viscosity

Options

A. `Ns/m^(-2)`
B. `Poise`
C. `kgm^(-1) s^(-1)`
D. All of these

Answer

To determine the unit of the coefficient of viscosity, we can follow these steps:

### Step 1: Understand the Definition
The coefficient of viscosity (η) is defined as the ratio of applied shear stress (τ) to the rate of shear strain (du/dy). Mathematically, it can be expressed as:
\[ \eta = \frac{\tau}{\frac{du}{dy}} \]

### Step 2: Identify the Units of Shear Stress
Shear stress (τ) is defined as force per unit area. The unit of force in the SI system is Newton (N), and the unit of area is square meters (m²). Therefore, the unit of shear stress is:
\[ \text{Unit of shear stress} = \frac{\text{Force}}{\text{Area}} = \frac{\text{N}}{\text{m}^2} \]

### Step 3: Identify the Units of Rate of Shear Strain
The rate of shear strain (du/dy) is the change in velocity (u) per unit distance (y). The unit of velocity is meters per second (m/s), and the unit of distance is meters (m). Therefore, the unit of rate of shear strain is:
\[ \text{Unit of rate of shear strain} = \frac{\text{m/s}}{\text{m}} = \text{s}^{-1} \]

### Step 4: Combine the Units
Now, we can combine the units of shear stress and the rate of shear strain to find the unit of viscosity:
\[ \text{Unit of viscosity} = \frac{\text{Unit of shear stress}}{\text{Unit of rate of shear strain}} = \frac{\text{N/m}^2}{\text{s}^{-1}} \]
This simplifies to:
\[ \text{Unit of viscosity} = \text{N} \cdot \text{s/m}^2 \]

### Step 5: Convert to Standard Units
In SI units, 1 Newton (N) can be expressed as kg·m/s². Therefore:
\[ \text{N} = \text{kg} \cdot \text{m/s}^2 \]
Substituting this into our equation gives:
\[ \text{Unit of viscosity} = \frac{\text{kg} \cdot \text{m/s}^2}{\text{s}^{-1} \cdot \text{m}^2} = \frac{\text{kg} \cdot \text{m}}{\text{s}^2 \cdot \text{m}^2} = \frac{\text{kg}}{\text{m} \cdot \text{s}} \]

### Step 6: Conclusion
Thus, the unit of the coefficient of viscosity is:
\[ \text{kg} \cdot \text{m}^{-1} \cdot \text{s}^{-1} \]

### Final Answer
The correct answer is option 4, as all three options provided in the question are correct.

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