A wind with speed `40 m//s` blows parallel to the roof of a house. The area of the roof is `250 m^(2)` . Assuming that the pressure inside the house is atmospheric pressure, the force exerted by the wind on the roof and the direction of the force will be : `(rho_(air)=1.2 kg//m^(3))`

Options

A. `4.8xx10^(5)`N, upwards
B. `2.4xx10^(5)`N, upwards
C. `2.4xx10^(5)`N downwards
D. `4.8xx10^(5)`N, downwards

Explanation

To find the force exerted by the wind on the roof of the house, we can use Bernoulli's principle. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have a wind blowing at a speed of \( V = 40 \, \text{m/s} \) parallel to the roof of a house with an area \( A = 250 \, \text{m}^2 \). The density of air is given as \( \rho_{\text{air}} = 1.2 \, \text{kg/m}^3 \). We need to find the force exerted by the wind on the roof.

### Step 2: Apply Bernoulli's Equation
According to Bernoulli's equation, the pressure difference created by the wind can be expressed as:
\[ P_0 - P = \frac{1}{2} \rho V^2 \]
Where:
- \( P_0 \) is the atmospheric pressure,
- \( P \) is the pressure exerted by the wind,
- \( \rho \) is the density of air,
- \( V \) is the speed of the wind.

### Step 3: Calculate the Pressure Difference
Substituting the known values into the equation:
\[ P_0 - P = \frac{1}{2} \times 1.2 \, \text{kg/m}^3 \times (40 \, \text{m/s})^2 \]
Calculating \( (40 \, \text{m/s})^2 \):
\[ (40)^2 = 1600 \, \text{m}^2/\text{s}^2 \]
Now substituting this back into the equation:
\[ P_0 - P = \frac{1}{2} \times 1.2 \times 1600 \]
Calculating:
\[ P_0 - P = 0.6 \times 1600 = 960 \, \text{Pa} \]

### Step 4: Calculate the Force Exerted by the Wind
The force exerted by the wind on the roof can be calculated using the pressure difference and the area of the roof:
\[ F = (P_0 - P) \times A \]
Substituting the values:
\[ F = 960 \, \text{Pa} \times 250 \, \text{m}^2 \]
Calculating:
\[ F = 240000 \, \text{N} \]

### Step 5: Determine the Direction of the Force
The direction of the force exerted by the wind on the roof is upward, as the wind creates a lower pressure above the roof compared to the atmospheric pressure below.

### Final Answer
The force exerted by the wind on the roof is \( 240000 \, \text{N} \) upward.

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