Question

In a hurricane, the air (density \(1.2 \mathrm{~kg} / \mathrm{m}^{3}\) ) is blowing over the roof of a house at a speed of \(110 \mathrm{~km} / \mathrm{h}\). (a) What is the pressure difference between inside and outside that tends to lift the roof? (b) What would be the lifting force on a roof of area \(93 \mathrm{~m}^{2} ?\)

Short answer

The pressure difference between inside and outside that tends to lift the roof is obtained in Pascal (Pa), and the corresponding lifting force on the roof of area 93 m² is calculated in Newton (N).

Step-by-step solution

Convert Speed

Convert the speed of the wind to m/s. The relation between km/h and m/s is 1 km/h = 0.277778 m/s. \[ v = 110 \, km/h = 110 \times 0.277778 \, m/s \]

Pressure Difference Calculation

From Bernoulli’s Equations, we can find the pressure difference due to the wind, using the equation \( ΔP = 0.5 \cdot ρ \cdot v^{2} \). Applying the values, we get \[ ΔP = 0.5 \cdot 1.2 \, kg/m^{3} \cdot (110 \times 0.277778)^{2} \, m^{2}/s^{2} \]

Force Calculation

Now, find the lifting force acting on the roof with the formula \(F = ΔP \cdot A\), where A is the roof area. Plugging in the given and calculated values, we get \[ F = (ΔP \cdot 93 \, m^{2}) \]. The final force value obtained must be converted into Newtons (N) to meet the conventional force measuring unit.

Key concepts

Understanding Pressure Difference

Pressure difference is a fundamental concept in physics, especially when analyzing the effects of wind on structures like roofs. In our exercise, we are using Bernoulli’s Equation to determine the pressure difference between the inside and outside of a house during a hurricane. This equation relates the speed of a fluid (in this case, air) and its pressure difference.
In the context of a hurricane, the wind speed increase causes a decrease in air pressure above the roof. Bernoulli’s Equation helps express this by the relationship:

• \[ ΔP = 0.5 \cdot ρ \cdot v^{2} \]

Where:

• \(ΔP\) is the pressure difference,
• \(ρ\) is the air density (given as 1.2 kg/m³), and
• \(v\) is the wind speed in meters per second.

This formula illustrates that faster wind speeds lead to a greater pressure difference, which is why hurricanes can potentially cause roofs to lift off houses.
Understanding the basics of pressure difference through Bernoulli's equation is crucial for aerodynamic analysis.

Exploring Aerodynamics and Its Effects

Aerodynamics is the study of how air interacts with solid objects like airplane wings, cars, and even buildings. It plays a critical role in understanding how hurricanes affect houses. The core idea is how wind speed and pressure exert forces on surfaces. In aerodynamics, focusing on wind patterns and vortices helps us predict structure responses under different wind conditions.
When high-speed winds blow over a surface like a roof, they reduce the air pressure above it, while the pressure below the roof remains higher, creating a net upward lift force. This lift can sometimes be enough to detach roofs. Aerodynamics also explains why streamlined designs reduce wind resistance, but for flat surfaces like roofs, the pressure differential is a major concern.
The principles of aerodynamics guide engineers in designing buildings to withstand strong winds by considering factors such as:

• Wind load calculations
• Structural integrity
• Material resilience

Therefore, understanding aerodynamics helps engineers create safer designs in hurricane-prone areas.

Force Calculation for Roof Safety

Calculating the force acting on a roof due to wind pressure difference is crucial for determining its structural safety. After understanding the pressure difference using Bernoulli’s Equation, we use it to compute the force that could potentially lift a roof off a house.
The formula used is:

• \[ F = ΔP \cdot A \]

where \(F\) is the lifting force, \(ΔP\) is the pressure difference, and \(A\) is the area of the roof.Given a roof area, the pressure difference calculated earlier is directly used to determine the lifting force on the roof.
For example, if the area of the roof is 93 m², the force calculation involves multiplying this area by the pressure difference (computed in the earlier steps).
Doing so helps predict whether roofs can withstand specific wind speeds or need reinforcement. Ensuring force calculations are precise is essential to prevent structural failures during hurricanes.