When a liquid flows in a tube, there is relative motion between adjacent layers of the liquid. This force is called the viscous force which tends to oppose the relative motion between the layers of the liquid. Newton was the first person to study the factors that govern the viscous force in a liquid. According to Newton’s law of viscous flow, the magnitude of the viscous force on a certain layer of a liquid is given by
` F = - eta A (dv)/(dx)`
where A is the area of the layer ` (dv)/(dx) ` is the velocity gradient at the layer and ` eta ` is the coefficient of viscosity of the liquid.
A river is 5 m deep. The velocity of water on its surface is ` 2 ms^(-1) ` If the coefficient of viscosity of water is ` 10 ^(-3 ) Nsm ^(-2)` , the viscous force per unit area is :

Options

A. ` 10 ^(-4) Nm^(-2) `
B. ` 2 xx 10 ^(-4) Nm^ (-2) `
C. ` 4 xx 10 ^(-4) Nm^(-2)`
D. ` 5 xx 10 ^(-4) Nm^(-2) `.

Explanation

To solve the problem of finding the viscous force per unit area in the river, we will follow these steps:

### Step 1: Understand the given parameters
- Depth of the river (d) = 5 m
- Velocity of water at the surface (V_s) = 2 m/s
- Coefficient of viscosity (η) = \(10^{-3} \, \text{N s/m}^2\)

### Step 2: Determine the velocity gradient (dv/dx)
The velocity gradient is defined as the change in velocity (Δv) over the change in distance (Δx). In this case, the change in velocity is from the surface velocity to the bottom layer, which has a velocity of 0 m/s.

- Δv = V_s - 0 = 2 m/s
- Δx = d = 5 m

Thus, the velocity gradient can be calculated as:
\[ \frac{dv}{dx} = \frac{\Delta v}{\Delta x} = \frac{2 \, \text{m/s}}{5 \, \text{m}} = \frac{2}{5} \, \text{s}^{-1} = 0.4 \, \text{s}^{-1} \]

### Step 3: Use Newton's law of viscous flow to find the viscous force per unit area
According to Newton's law of viscous flow, the viscous force per unit area (F/A) is given by:
\[ \frac{F}{A} = \eta \frac{dv}{dx} \]

Substituting the values we have:
\[ \frac{F}{A} = (10^{-3} \, \text{N s/m}^2) \times (0.4 \, \text{s}^{-1}) \]

### Step 4: Calculate the viscous force per unit area
Now we perform the multiplication:
\[ \frac{F}{A} = 10^{-3} \times 0.4 = 0.4 \times 10^{-3} \, \text{N/m}^2 = 4 \times 10^{-4} \, \text{N/m}^2 \]

### Final Answer
The viscous force per unit area in the river is:
\[ \frac{F}{A} = 4 \times 10^{-4} \, \text{N/m}^2 \]

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