A simple but effective anemometer to measure wind speed can...

A simple but effective anemometer to measure wind speed can be made from a thin plate hinged to deflect in the wind. Consider a thin plate made from brass that is $20 \mathrm{mm}$ high and $10 \mathrm{mm}$ wide. Derive a relationship for wind speed as a function of deflection angle, $\theta .$ What thickness of brass should be used to give $\theta=30^{\circ}$ at $10 \mathrm{m} / \mathrm{s} ?$

Key Concepts

Aerodynamic Drag

Aerodynamic drag refers to the force exerted by moving air (wind) on a surface. In the context of a deflecting plate, the wind exerts a pressure that can be quantified using dynamic pressure formulas based on the square of the wind speed, the density of the air, and the effective area. This drag force then creates a moment about the hinge, which is key in determining the plate’s deflection angle.

Moment Equilibrium

Moment equilibrium is a principle in mechanics stating that for a body to be in static equilibrium, the sum of all moments (torques) about any pivot point must be zero. In the anemometer, the moment produced by the aerodynamic drag is balanced by a restoring moment provided by the plate’s resistance to bending. Establishing this balance is crucial for deriving the relationship between wind speed and the deflection angle, and for determining the appropriate material thickness.

Beam Bending and Stiffness

Beam bending theory describes how structural elements deform under load, particularly through the concept of bending stiffness. The stiffness of a plate or beam is determined by its material properties, such as Young's modulus, and its geometric properties, like the moment of inertia, which for rectangular sections depends on the thickness raised to a high power. This theory helps in relating the applied moment due to wind drag to the deflection angle and in calculating the required thickness of the brass plate for a specific deflection.

Material Properties and Geometry

Material properties, specifically Young’s modulus, define how a material deforms under an applied load, while geometric dimensions such as thickness, height, and width determine the structural resistance to bending. In the design of the anemometer, selecting the appropriate brass thickness is critical because it directly influences the bending stiffness and therefore the deflection angle under a given wind load. This concept integrates both the mechanics of materials and geometric considerations to ensure the device's performance.

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