Question

In a tornado or hurricane, a roof may tear away from the house because of a difference in pressure between the air inside and the air outside. Suppose that air is blowing across the top of a \(2000 \mathrm{ft}^{2}\) roof at \(150 \mathrm{mph}\). What is the magnitude of the force on the roof?

Short answer

The magnitude of the force on the roof is approximately 115,000 lb.

Step-by-step solution

Understand Bernoulli's Principle

Bernoulli's Principle can be used to solve this problem. It states that in a flowing fluid, an increase in velocity occurs simultaneously with a decrease in pressure.

Convert Units

The given velocity is in miles per hour. Convert this to feet per second. 1 mile = 5280 feet, 1 hour = 3600 seconds. So, \[ 150 \, \text{mph} \times \frac{5280 \, \text{ft}}{1 \, \text{mile}} \times \frac{1 \, \text{hour}}{3600 \, \text{s}} = 220 \, \text{ft/s} \]

Apply Bernoulli's Equation

The difference in pressure across the roof, due to air moving faster on the outside, is given by \[ \Delta P = \frac{1}{2} \rho v^2 \] where \( \rho \) is the air density (approximately 0.002378 \( \text{slugs/ft}^3 \) under standard conditions) and \( v \) is the converted velocity.

Calculate Pressure Difference

Substitute the given values into the Bernoulli equation to find the pressure difference:\[ \Delta P = \frac{1}{2} \times 0.002378 \, \text{slugs/ft}^3 \times (220 \, \text{ft/s})^2 \] \[ \Delta P = \frac{1}{2} \times 0.002378 \times 48400 \]\[ \Delta P \approx 57.5 \, \text{lb/ft}^2 \]

Calculate Force

The force on the roof can be determined using the pressure difference and the area of the roof. \[ F = \Delta P \times A \] Substitute the values:\[ F = 57.5 \, \text{lb/ft}^2 \times 2000 \, \text{ft}^2 \] \[ F \approx 115000 \, \text{lb} \]

Key concepts

Pressure Difference

The phenomenon of pressure difference plays a crucial role in understanding how forces act on surfaces in open environments, especially in events like tornadoes and hurricanes. When wind blows over a surface, such as a roof, the velocity of the wind above the surface can cause a reduction in pressure compared to the still air inside or below the structure.
According to Bernoulli's principle, this difference in velocity directly creates a difference in pressure. The faster the wind speed over a roof, the lower the pressure exerted by the air above it, compared to the air pressure inside. This pressure difference can lead to significant forces acting on structures and potentially cause damage, such as lifting the roof off of a house. It is important to understand this relationship to design buildings that can withstand such high wind events.

Air Density

Air density is a critical parameter in calculating the force exerted by wind on a structure. It represents the mass of air per unit volume, often given in units like slugs per cubic foot (slugs/ft³) in the imperial system.
In our problem, the standard air density of 0.002378 slugs/ft³ was used, which is typical of conditions at sea level and average temperature. The density of air can vary with altitude and temperature, affecting calculations. This is why accurate density values are essential in practical applications, as they directly influence the pressure calculation when applying Bernoulli's equation.
Therefore, always ensure that you have the correct value for air density when checking pressure differences, as this variable can significantly affect the magnitude of the resulting force.

Unit Conversion

Unit conversion is a fundamental skill in physics and engineering, ensuring that calculations are performed consistently and accurately. In this exercise, the speed of the wind was initially given in miles per hour (mph), which needed to be converted to feet per second (ft/s) to be compatible with other units used in the calculations.
To convert from mph to ft/s, we used the relationships: 1 mile = 5280 feet and 1 hour = 3600 seconds. Therefore:

• Multiply the speed in mph by 5280 to convert miles to feet.
• Divide by 3600 to convert hours to seconds.

In our example:

• \[150 \, \text{mph} \times \frac{5280 \, \text{ft}}{1 \, \text{mile}} \times \frac{1 \, \text{hour}}{3600 \, \text{s}} = 220 \, \text{ft/s}\]

Successful calculation requires proper unit management to ensure correctness and coherence of results.

Force Calculation

The calculation of force using pressure difference involves understanding basic fluid mechanics principles. The force acting on the roof due to wind is calculated by multiplying the pressure difference by the area over which the pressure acts.
Using the pressure difference from Bernoulli's equation:

• We found \[\Delta P \approx 57.5 \, \text{lb/ft}^2\]

This value represents the pressure exerted by the moving air on every square foot of the roof.
The area of the roof in the given problem was 2000 square feet:

• The total force is then given by \[F = \Delta P \times A\]
• Substituting the numbers: \[F = 57.5 \, \text{lb/ft}^2 \times 2000 \, \text{ft}^2\]
• This results in \[F \approx 115000 \, \text{lb}\]

By understanding these steps, we can see how a pressure difference translates into a sizeable force that can remove a roof from a building.