The coefficient of viscosity `(eta)` of a fluid moving steadily between two surface is given by the formula `(f) = etaA dV//dx` where `f` is the frictional force on the fluid, `A` is the area in the fluid, and `dV//dx` is velocity gradient inside the fluid at that area. The SI unit of viscosity is given as :

Options

A. `kg m^(-1) s^(-1)`
B. `Nm^(-2) s`
C. Nil
D. Newtons

Explanation

To find the SI unit of viscosity (η), we start from the formula given:

\[ F = \eta A \frac{dV}{dx} \]

Where:
- \( F \) is the frictional force,
- \( \eta \) is the coefficient of viscosity,
- \( A \) is the area,
- \( \frac{dV}{dx} \) is the velocity gradient.

### Step 1: Rearranging the formula to solve for η
We can rearrange the formula to isolate η:

\[ \eta = \frac{F}{A \frac{dV}{dx}} \]

### Step 2: Identifying the units of each variable
Next, we need to identify the SI units of each variable:
- The unit of force \( F \) is Newton (N).
- The unit of area \( A \) is square meters (m²).
- The velocity gradient \( \frac{dV}{dx} \) has units of velocity per unit distance. The unit of velocity \( V \) is meters per second (m/s), and the unit of distance \( x \) is meters (m). Therefore, the unit of \( \frac{dV}{dx} \) is:

\[ \frac{m/s}{m} = s^{-1} \]

### Step 3: Substituting the units into the equation for η
Now we can substitute the units into the equation for η:

\[ \text{Unit of } \eta = \frac{\text{Unit of } F}{\text{Unit of } A \times \text{Unit of } \frac{dV}{dx}} \]

Substituting the units we identified:

\[ \text{Unit of } \eta = \frac{N}{m^2 \cdot s^{-1}} \]

### Step 4: Converting Newton into base SI units
Recall that 1 Newton (N) can be expressed in terms of base SI units:

\[ 1 \, N = 1 \, \text{kg} \cdot \text{m/s}^2 \]

Thus, we can rewrite the unit of η:

\[ \text{Unit of } \eta = \frac{kg \cdot m/s^2}{m^2 \cdot s^{-1}} \]

### Step 5: Simplifying the units
Now, we simplify the expression:

\[ \text{Unit of } \eta = \frac{kg \cdot m}{s^2} \cdot \frac{s}{m^2} \]

This simplifies to:

\[ \text{Unit of } \eta = \frac{kg}{m \cdot s} \]

### Conclusion
Thus, the SI unit of viscosity (η) is:

\[ \text{Unit of } \eta = \text{kg} \cdot \text{m}^{-1} \cdot \text{s}^{-1} \]