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

Explanation

To solve the problem of finding the force exerted by the wind on the roof of a house, we will use Bernoulli's principle. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand Bernoulli's Principle
Bernoulli's principle states that for an incompressible, frictionless fluid, the total mechanical energy along a streamline is constant. The equation can be expressed as:
\[ P + \frac{1}{2} \rho v^2 + \rho gh = \text{constant} \]
Where:
- \( P \) = pressure
- \( \rho \) = density of the fluid
- \( v \) = velocity of the fluid
- \( g \) = acceleration due to gravity
- \( h \) = height

In this case, we can neglect potential energy changes since the wind is blowing parallel to the roof.

### Step 2: Apply Bernoulli's Equation
We can apply Bernoulli's equation to the roof and the wind above it. Let:
- \( P_0 \) = atmospheric pressure (pressure inside the house)
- \( P_1 \) = pressure above the roof (where the wind is blowing)
- \( v_1 \) = speed of the wind = 40 m/s
- \( \rho \) = density of air = 1.2 kg/m³

According to Bernoulli's principle:
\[ P_0 + \frac{1}{2} \rho v_0^2 = P_1 + \frac{1}{2} \rho v_1^2 \]

Since the pressure inside the house is atmospheric pressure and the wind speed is given, we can rearrange this to find the change in pressure:
\[ \Delta P = P_0 - P_1 = \frac{1}{2} \rho v_1^2 \]

### Step 3: Calculate the Change in Pressure
Substituting the values into the equation:
\[ \Delta P = \frac{1}{2} \times 1.2 \, \text{kg/m}^3 \times (40 \, \text{m/s})^2 \]

Calculating:
\[ \Delta P = \frac{1}{2} \times 1.2 \times 1600 \]
\[ \Delta P = 0.6 \times 1600 \]
\[ \Delta P = 960 \, \text{Pa} \]

### Step 4: Calculate the Force on the Roof
The force exerted by the wind on the roof can be calculated using the change in pressure and the area of the roof:
\[ F = \Delta P \times A \]
Where:
- \( A \) = area of the roof = 250 m²

Substituting the values:
\[ F = 960 \, \text{Pa} \times 250 \, \text{m}^2 \]
\[ F = 240000 \, \text{N} \]
\[ F = 2.4 \times 10^5 \, \text{N} \]

### Step 5: Determine the Direction of the Force
Since the pressure inside the house is greater than the pressure above the roof (due to the wind), the force exerted by the wind on the roof will be directed upwards.

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

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