The velocity of water in river is 180 km `h^(-1)` near the surface .If the river is 5 m deep,then the shearing stress between the surface layer and the bottom layer is ( coefficient of viscosity of water `eta =10^(-3)` Pa s)

Options

A. `10^(-2)N m^(-2)`
B. `10^(-3)N m^(-2)`
C. `10^(-4)N m^(-2)`
D. `10^(-5)N m^(-2)`

Explanation

To solve the problem of finding the shearing stress between the surface layer and the bottom layer of the river, we can follow these steps:

### Step 1: Understand the Given Information
- Velocity of water at the surface, \( v = 180 \, \text{km/h} \)
- Depth of the river, \( d = 5 \, \text{m} \)
- Coefficient of viscosity of water, \( \eta = 10^{-3} \, \text{Pa s} \)

### Step 2: Convert Velocity to SI Units
Convert the velocity from kilometers per hour to meters per second:
\[ v = 180 \, \text{km/h} = \frac{180 \times 1000 \, \text{m}}{3600 \, \text{s}} = 50 \, \text{m/s} \]

### Step 3: Determine the Change in Velocity
The change in velocity (\( dv \)) from the surface to the bottom layer can be taken as:
\[ dv = v - 0 = 50 \, \text{m/s} \]
(Here, the velocity at the bottom layer is assumed to be zero.)

### Step 4: Identify the Gradient of Velocity
The gradient of velocity (\( \frac{dv}{dx} \)) is calculated as:
\[ \frac{dv}{dx} = \frac{dv}{d} = \frac{50 \, \text{m/s}}{5 \, \text{m}} = 10 \, \text{s}^{-1} \]

### Step 5: Calculate Shearing Stress
The shearing stress (\( \tau \)) can be calculated using the formula:
\[ \tau = \eta \frac{dv}{dx} \]
Substituting the known values:
\[ \tau = 10^{-3} \, \text{Pa s} \times 10 \, \text{s}^{-1} = 10^{-2} \, \text{Pa} = 0.01 \, \text{N/m}^2 \]

### Final Answer
The shearing stress between the surface layer and the bottom layer is:
\[ \tau = 0.01 \, \text{N/m}^2 \]