Between a plate of area `100 cm^(2)` and another plate of area `100 m^(2)` there is a `1 mm`, thick layer of water, if the coefficient of viscosity of water is `0.01` poise, then the force required to move the smaller plate with a velocity `10 cms^(-1)` with reference to large plate is

Options

A. `100 dyn`
B. `10^(4)dyn`
C. `10^(6) dyn`
D. `10^(9)dyn`

Explanation

To solve the problem, we need to calculate the force required to move a smaller plate with a given velocity in a viscous fluid. We will use the formula for viscous force, which is given by:

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

Where:
- \( F \) is the viscous force,
- \( \eta \) is the coefficient of viscosity,
- \( A \) is the area of the plate,
- \( \frac{dv}{dz} \) is the velocity gradient.

### Step 1: Convert all units to CGS
- The area of the smaller plate is given as \( 100 \, cm^2 \).
- The area of the larger plate is \( 100 \, m^2 \) which we convert to \( cm^2 \):
\[ 100 \, m^2 = 100 \times 10^4 \, cm^2 = 10^6 \, cm^2 \]
- The thickness of the water layer is \( 1 \, mm \) which we convert to centimeters:
\[ 1 \, mm = 0.1 \, cm \]
- The coefficient of viscosity \( \eta \) is given as \( 0.01 \, poise \) (1 poise = 1 g/cm·s, so \( 0.01 \, poise = 0.01 \, g/cm·s \)).
- The velocity \( v \) is given as \( 10 \, cm/s \).

### Step 2: Calculate the velocity gradient \( \frac{dv}{dz} \)
- The velocity gradient \( \frac{dv}{dz} \) is calculated as:
\[ \frac{dv}{dz} = \frac{10 \, cm/s - 0 \, cm/s}{0.1 \, cm} = \frac{10 \, cm/s}{0.1 \, cm} = 100 \, s^{-1} \]

### Step 3: Substitute values into the viscous force formula
Now we substitute the values into the viscous force formula:
- \( \eta = 0.01 \, g/cm·s \)
- \( A = 100 \, cm^2 \)
- \( \frac{dv}{dz} = 100 \, s^{-1} \)

\[ F = 0.01 \, g/cm·s \cdot 100 \, cm^2 \cdot 100 \, s^{-1} \]

### Step 4: Calculate the force
Calculating the force:
\[ F = 0.01 \cdot 100 \cdot 100 = 100 \, g \cdot cm/s^2 \]

Since \( 1 \, g \cdot cm/s^2 = 1 \, dyne \):
\[ F = 100 \, dyne \]

### Final Answer
The force required to move the smaller plate with a velocity of \( 10 \, cm/s \) is \( 100 \, dyne \).

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