A wind with speed `40m//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 `(P_(air) = 1.2 kg//m^(3))`

Options

A. `4.8 xx 10^(5)N`, downwards
B. `4.8 xx 10^(5)N`, upwards
C. `2.4 xx 10^(5)N`, upwards
D. `2.4 xx 10^(5)N`, downwards

Explanation

To solve the problem, we need to calculate the force exerted by the wind on the roof of the house using Bernoulli's principle. Here’s a step-by-step breakdown of the solution:

### Step 1: Calculate the dynamic pressure exerted by the wind.
According to Bernoulli's principle, the dynamic pressure (P) exerted by the wind can be calculated using the formula:

\[ P = \frac{1}{2} \rho v^2 \]

Where:
- \(\rho\) is the density of air (given as \(1.2 \, \text{kg/m}^3\))
- \(v\) is the speed of the wind (given as \(40 \, \text{m/s}\))

Substituting the values:

\[ P = \frac{1}{2} \times 1.2 \, \text{kg/m}^3 \times (40 \, \text{m/s})^2 \]

Calculating this:

\[ P = \frac{1}{2} \times 1.2 \times 1600 = \frac{1.2 \times 1600}{2} = 960 \, \text{N/m}^2 \]

### Step 2: Calculate the force exerted on the roof.
The force (F) exerted by the wind on the roof can be calculated using the formula:

\[ F = P \times A \]

Where:
- \(A\) is the area of the roof (given as \(250 \, \text{m}^2\))

Substituting the values:

\[ F = 960 \, \text{N/m}^2 \times 250 \, \text{m}^2 \]

Calculating this:

\[ F = 240000 \, \text{N} = 2.4 \times 10^5 \, \text{N} \]

### Step 3: Determine the direction of the force.
The wind creates a pressure difference. The pressure inside the house is atmospheric pressure, while the pressure exerted by the wind on the roof is lower due to the wind speed. Therefore, the force exerted by the wind on the roof will act upwards, as the wind moves from an area of higher pressure (outside) to an area of lower pressure (inside the house).

### Final Answer:
The force exerted by the wind on the roof is \(2.4 \times 10^5 \, \text{N}\) upwards.

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