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Chapter 11: Problem 73

The construction of a flat rectangular roof \((5.0 \mathrm{m} \times 6.3 \mathrm{m})\) allows it to withstand a maximum net outward force that is \(22000\) \(\mathrm{N}\). The density of the air is \(1.29\) \(\mathrm{kg} / \mathrm{m}^{3} .\) At what wind speed will this roof blow outward?

Short Answer

Expert verified
The wind speed required to blow the roof outward is approximately 32.91 m/s.

Step by step solution

01

Understand the Problem

The roof can withstand a maximum net outward force of 22000 N. We must find the wind speed at which the aerodynamic force (pressure difference due to wind) exceeds this limit and causes the roof to blow outward.
02

Calculate the Roof Area

Calculate the area of the roof using its dimensions:\[A = 5.0 \text{ m} \times 6.3 \text{ m} = 31.5 \text{ m}^2\]
03

Use Bernoulli's Equation to Find Pressure Difference

Bernoulli's principle explains that the pressure difference created by the wind can be described as: \[\Delta P = \frac{1}{2} \rho v^2\]where \(\rho\) is the air density and \(v\) is the wind speed.
04

Relate Pressure Difference to Net Outward Force

The net outward force due to wind can be expressed as:\[F = \Delta P \cdot A\]From Step 3, \(F = \frac{1}{2} \rho v^2 A\). We know \(F = 22000\, \text{N}\) from the problem.
05

Substitute and Solve for Wind Speed

Substitute the known values into the force equation:\[22000 = \frac{1}{2} \times 1.29 \times v^2 \times 31.5\]Now solve for \(v\) (wind speed):\[22000 = 20.3175 \times v^2\]\[v^2 = \frac{22000}{20.3175} \approx 1083.75\]\[v = \sqrt{1083.75} \approx 32.91\, \text{m/s}\]
06

Verify the Solution

Check the calculations for correctness. Recompute if necessary to ensure the wind speed is accurate.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Aerodynamic Force
Aerodynamic force is an essential concept in understanding how objects such as roofs, airplanes, or cars interact with air. This force results from the pressure difference generated by the movement of air over and around an object.

When wind blows over a flat roof, it creates a difference in air pressure: the pressure above the roof differs from the pressure below it. This variation in pressure causes a net force to act on the roof, which we refer to as aerodynamic force.

In practice, if the aerodynamic force becomes too great, it can cause structural damage, such as a roof being blown away. This underscores why engineering considerations and calculations involving aerodynamic force are vital in construction.
Pressure Difference
Pressure difference is an essential factor in aerodynamic situations. It is the driving force behind the aerodynamic force on an object. As per Bernoulli's Principle, a critical fluid dynamics principle, moving air has a lower pressure than still air.

Bernoulli's Equation, which is expressed as \( \Delta P = \frac{1}{2} \rho v^2 \), helps us recognize how the wind speed influences the pressure difference. Here, \( \rho \) stands for the density of air, and \( v \) is the wind speed.

Essentially, faster moving air on one side of a surface decreases the pressure, leading to a larger pressure difference across that surface. This concept is crucial in explaining how aerodynamic forces can push or pull surfaces like roofs, impacting their stability and structural integrity.
Wind Speed Calculation
Calculating the wind speed at which a structure, like a roof, will succumb to aerodynamic forces involves understanding and applying Bernoulli's Principle.

To determine this critical speed, we consider the maximum force the structure can withstand. For example, if a roof can handle a net outward force of 22000 N, we need to calculate the wind speed that produces this force, considering the roof's area and air density.
  • Calculate the area of the roof first (e.g., \( 31.5 \, \text{m}^2 \)).
  • Express the net force using the pressure difference formula \( F = \Delta P \cdot A \).
  • Substitute this into the pressure difference formula \( F = \frac{1}{2} \rho v^2 A \) and solve for wind speed \( v \).
From the calculation, substituting known values into the formula, we solve for \( v \), leading to a critical wind speed that's pivotal for understanding roof safety during storms.

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Most popular questions from this chapter

A room has a volume of \(120 \mathrm{m}^{3} .\) An air-conditioning system is to replace the air in this room every twenty minutes, using ducts that have a square cross section. Assuming that air can be treated as an incompressible fluid, find the length of a side of the square if the air speed within the ducts is (a) \(3.0 \mathrm{m} / \mathrm{s}\) and (b) \(5.0 \mathrm{m} / \mathrm{s}\).

To verify her suspicion that a rock specimen is hollow, a geologist weighs the specimen in air and in water. She finds that the specimen weighs twice as much in air as it does in water. The density of the solid part of the specimen is \(5.0 \times 10^{3}\) \(\mathrm{kg} / \mathrm{m}^{3} .\) What fraction of the specimen's apparent volume is solid?

A barber's chair with a person in it weighs \(2100\) \(\mathrm{N}\). The output plunger of a hydraulic system begins to lift the chair when the barber's foot applies a force of \(55\) \(\mathrm{N}\) to the input piston. Neglect any height difference between the plunger and the piston. What is the ratio of the radius of the plunger to the radius of the piston?

A blood transfusion is being set up in an emergency room for an accident victim. Blood has a density of \(1060 \mathrm{kg} / \mathrm{m}^{3}\) and a viscosity of \(4.0 \times 10^{-3} \mathrm{Pa} \cdot \mathrm{s}\). The needle being used has a length of \(3.0 \mathrm{cm}\) and an inner radius of \(0.25 \mathrm{mm}\). The doctor wishes to use a volume flow rate through the needle of \(4.5 \times 10^{-8} \mathrm{m}^{3} / \mathrm{s} .\) What is the distance \(h\) above the victim's arm where the level of the blood in the transfusion bottle should be located? As an approximation, assume that the level of the blood in the transfusion bottle and the point where the needle enters the vein in the arm have the same pressure of one atmosphere. (In reality, the pressure in the vein is slightly above atmospheric pressure.)

The vertical surface of a reservoir dam that is in contact with the water is \(120\) \(\mathrm{m}\) wide and \(12\) \(\mathrm{m}\) high. The air pressure is one atmosphere. Find the magnitude of the total force acting on this surface in a completely filled reservoir. (Hint: The pressure varies linearly with depth, so you must use an average pressure.)
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