A metal plate of area 20 `cm^2`, is separated from a large plate by a layer of glycerine I mm thick. The coefficient of Viscosity of the glycerine is 20 poise. What is the honz?ntal force required to keep the plate moving with a velocity of 2 cm/s?
[poise=`10^(-1)N-s//m^2`]

Options

A. 0.04 N
B. 0.05 N
C. 0.06 N
D. 0.08 N

Explanation

To solve the problem, we need to find the horizontal force required to keep a metal plate moving with a certain velocity while separated from a large plate by a layer of glycerine. We will use the formula derived from Newton's law of viscosity.

### Step-by-Step Solution:

1. **Identify the given values:**
- Area of the plate, \( A = 20 \, \text{cm}^2 \)
- Thickness of glycerine layer, \( d = 1 \, \text{mm} \)
- Coefficient of viscosity, \( \mu = 20 \, \text{poise} \)
- Velocity of the plate, \( v = 2 \, \text{cm/s} \)

2. **Convert the units to SI:**
- Convert area from cm² to m²:
\[ A = 20 \, \text{cm}^2 = 20 \times 10^{-4} \, \text{m}^2 = 2 \times 10^{-3} \, \text{m}^2 \]
- Convert thickness from mm to m:
\[ d = 1 \, \text{mm} = 1 \times 10^{-3} \, \text{m} \]
- Convert velocity from cm/s to m/s:
\[ v = 2 \, \text{cm/s} = 2 \times 10^{-2} \, \text{m/s} \]
- Convert viscosity from poise to SI units (1 poise = \( 0.1 \, \text{N s/m}^2 \)):
\[ \mu = 20 \, \text{poise} = 20 \times 0.1 \, \text{N s/m}^2 = 2 \, \text{N s/m}^2 \]

3. **Calculate the velocity gradient:**
- The velocity gradient \( \frac{dU}{dY} \) is given by:
\[ \frac{dU}{dY} = \frac{v - 0}{d} = \frac{2 \times 10^{-2} \, \text{m/s}}{1 \times 10^{-3} \, \text{m}} = 20 \, \text{s}^{-1} \]

4. **Apply Newton's law of viscosity:**
- The formula for the tangential force \( F \) is:
\[ F = \mu A \frac{dU}{dY} \]
- Substituting the values:
\[ F = (2 \, \text{N s/m}^2) \times (2 \times 10^{-3} \, \text{m}^2) \times (20 \, \text{s}^{-1}) \]

5. **Calculate the force:**
\[ F = 2 \times 2 \times 20 \times 10^{-3} = 80 \times 10^{-3} \, \text{N} = 0.08 \, \text{N} \]

### Final Answer:
The horizontal force required to keep the plate moving with a velocity of 2 cm/s is \( 0.08 \, \text{N} \).

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