Aflat plate of area `0.05 m^(2)` is separated from another large plate at rost by a liquid layer of uniform thickness `1mu m`. The tangential force needed to move the smaller plate with a constant velocity of `10 cm s ^(-1) ` i s 20 N. Calculate the coefficent of viscosity of the liquid.

Explanation

To find the coefficient of viscosity of the liquid, we can use the relationship between the tangential force, the area of the plate, the velocity gradient, and the viscosity. The formula we will use is:

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

Where:
- \( F \) = tangential force (in Newtons)
- \( \eta \) = coefficient of viscosity (in Pascal-seconds or N·s/m²)
- \( A \) = area of the plate (in m²)
- \( \frac{dv}{dx} \) = velocity gradient (in s⁻¹)

### Step 1: Write down the given values
- Area \( A = 0.05 \, m^2 \)
- Thickness \( dx = 1 \, \mu m = 1 \times 10^{-6} \, m \)
- Velocity \( v = 10 \, cm/s = 0.1 \, m/s \)
- Tangential force \( F = 20 \, N \)

### Step 2: Calculate the velocity gradient \( \frac{dv}{dx} \)
The velocity gradient is defined as the change in velocity per unit distance. Since the plate is moving with a constant velocity, we can express the velocity gradient as:

\[ \frac{dv}{dx} = \frac{v}{dx} \]

Substituting the values:

\[ \frac{dv}{dx} = \frac{0.1 \, m/s}{1 \times 10^{-6} \, m} = 1 \times 10^{5} \, s^{-1} \]

### Step 3: Rearrange the formula to solve for viscosity \( \eta \)
We can rearrange the original formula to solve for the coefficient of viscosity:

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

### Step 4: Substitute the known values into the viscosity formula
Now, substitute the values into the equation:

\[ \eta = \frac{20 \, N}{0.05 \, m^2 \cdot 1 \times 10^{5} \, s^{-1}} \]

### Step 5: Calculate the viscosity
Calculating the above expression:

\[ \eta = \frac{20}{0.05 \cdot 1 \times 10^{5}} = \frac{20}{5 \times 10^{3}} = \frac{20}{5000} = 0.004 \, N \cdot s/m^2 \]

### Step 6: Convert to decapoise
Since \( 1 \, N \cdot s/m^2 = 10 \, decapoise \):

\[ \eta = 0.004 \, N \cdot s/m^2 = 0.004 \times 10 \, decapoise = 0.04 \, decapoise \]

### Final Answer
The coefficient of viscosity of the liquid is:

\[ \eta = 4 \times 10^{-3} \, decapoise \]