A metal plate of area `10^(3)" "cm^(2)` rests on a layer o oil 6 mm thick. A tangential force of `10^(-2)` N is appled on it to move it with a constant velocity of 6 cm `s^(-1)` . The coefficient of viscosity of the liquid is :-

Options

A. 0.1 poise
B. 0.5 posie
C. 0.7 poise
D. 0.9 poise

Answer

To find the coefficient of viscosity (η) of the liquid, we can use the formula for viscous force:

\[ F = \eta \cdot A \cdot \frac{dv}{dx} \]

Where:
- \( F \) is the tangential force applied (in Newtons),
- \( \eta \) is the coefficient of viscosity (in \( \text{N s/m}^2 \)),
- \( A \) is the area of the plate (in \( \text{m}^2 \)),
- \( \frac{dv}{dx} \) is the velocity gradient (in \( \text{s}^{-1} \)).

### Step 1: Convert the Area
The area of the plate is given as \( 10^3 \, \text{cm}^2 \). We need to convert this to square meters:

\[ A = 10^3 \, \text{cm}^2 = 10^3 \times 10^{-4} \, \text{m}^2 = 10^{-1} \, \text{m}^2 \]

### Step 2: Convert the Thickness
The thickness of the oil layer is given as 6 mm. We need to convert this to meters:

\[ \text{Thickness} = 6 \, \text{mm} = 6 \times 10^{-3} \, \text{m} = 0.006 \, \text{m} \]

### Step 3: Calculate the Velocity Gradient
The velocity of the plate is given as 6 cm/s. We need to convert this to meters per second:

\[ v = 6 \, \text{cm/s} = 6 \times 10^{-2} \, \text{m/s} \]

The velocity gradient \( \frac{dv}{dx} \) can be calculated as:

\[ \frac{dv}{dx} = \frac{v}{\text{Thickness}} = \frac{6 \times 10^{-2} \, \text{m/s}}{0.006 \, \text{m}} = 10 \, \text{s}^{-1} \]

### Step 4: Substitute Values into the Viscosity Formula
Now we can substitute the values into the formula:

\[ F = \eta \cdot A \cdot \frac{dv}{dx} \]

Rearranging for \( \eta \):

\[ \eta = \frac{F}{A \cdot \frac{dv}{dx}} \]

Substituting the known values:

\[ F = 10^{-2} \, \text{N}, \quad A = 10^{-1} \, \text{m}^2, \quad \frac{dv}{dx} = 10 \, \text{s}^{-1} \]

\[ \eta = \frac{10^{-2}}{10^{-1} \cdot 10} = \frac{10^{-2}}{10^{-1} \cdot 10} = \frac{10^{-2}}{10^{-1}} = 10^{-1} \, \text{N s/m}^2 \]

### Final Answer
\[ \eta = 0.1 \, \text{N s/m}^2 \]