Wind Turbine Blade Aerodynamic Work

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.