Wind Turbine Blade Aerodynamic Work Solution

In renewable energy systems, wind turbine blades convert kinetic airflow energy into mechanical shaft power. Aerospace and energy engineers analyze the dot product between aerodynamic lift/drag force vectors and the local blade velocity/displacement vectors to evaluate instantaneous power generation.

A wind turbine rotor blade rotates in a vertical plane. Over an infinitesimal time interval $\Delta t$, a specific blade airfoil section undergoes a displacement vector of $\vec{d} = 2\hat{i} + 5\hat{j}\text{ meters}$. The combined aerodynamic lift and drag force vector exerted by the passing wind on this section is $\vec{F} = 120\hat{i} - 30\hat{j}\text{ Newtons}$.

1. Compute the mechanical work done by the wind on this blade section using the dot product.
2. Calculate the angle $\theta$ between the aerodynamic force vector $\vec{F}$ and the displacement vector $\vec{d}$.
3. Explain physically why the work is positive despite the negative $y$-component of force.

Step-by-Step Solution:

1. Calculate Mechanical Work ($W$): Using the algebraic dot product definition:

   $$W = \vec{F} \cdot \vec{d} = (120)(2) + (-30)(5) = 240 - 150 = 90\text{ Joules}$$

2. Calculate the Angle ($\theta$): Compute vector magnitudes:

   $$|\vec{F}| = \sqrt{120^2 + (-30)^2} = \sqrt{14400 + 900} = \sqrt{15300} \approx 123.69\text{ N}$$
   $$|\vec{d}| = \sqrt{2^2 + 5^2} = \sqrt{4 + 25} = \sqrt{29} \approx 5.385\text{ m}$$
   Apply geometric dot product formula:
   $$\cos\theta = \frac{\vec{F} \cdot \vec{d}}{|\vec{F}| |\vec{d}|} = \frac{90}{(123.69)(5.385)} = \frac{90}{666.07} \approx 0.1351$$
   $$\theta = \cos^{-1}(0.1351) \approx 82.24^\circ$$

3. Physical Interpretation: Because $\theta \approx 82.24^\circ < 90^\circ$, the net force vector forms an acute angle with the displacement vector ($\cos\theta > 0$). The positive forward thrust along $x$ ($240\text{ J}$) exceeds the negative resisting work along $y$ ($-150\text{ J}$), resulting in net positive energy transfer ($+90\text{ J}$) driving rotor rotation.

Meaning of the Answer & Real-Life Application:

Aerospace and wind energy engineers optimize blade pitch angles to ensure the aerodynamic resultant force maintains an acute angle ($\theta < 90^\circ$) relative to blade motion across varying wind speeds, maximizing positive work extraction and power output ($P = W / \Delta t$).