Windmill Energy Gathering Proportionality

The power available from the wind passing through a windmill rotor is determined by the kinetic energy of the air and the rate at which it passes. The kinetic energy of a mass $m$ of air moving at speed $v$ is given by $\frac{1}{2}mv^2$.

The mass flow rate of air through the rotor depends on the air density ($\rho$), the area swept by the blades ($A$), and the wind speed ($v$). Specifically, the mass flow rate is $\rho Av$.

Therefore, the power ($P$) extracted is the kinetic energy per unit mass times the mass flow rate:

$P = \frac{1}{2}v^2 \times (\rho Av) = \frac{1}{2} \rho A v^3$

The area ($A$) swept by the blades is determined by the length of the blades ($L$). It is the area of a circle with radius $L$, so $A = \pi L^2$. Since $\pi$ and $\frac{1}{2}\rho$ are constants, the power is proportional to $L^2$ and $v^3$.

$P \propto L^2 v^3$

The question asks about the energy gathered. Energy is Power multiplied by time ($E = P \times t$). While energy also depends on time, the fundamental proportionality concerning the windmill's characteristics relates to its dimensions and wind speed.

Analyzing Proportionality Factors

Based on the physics formula $P \propto L^2 v^3$, the power (and thus the potential energy gathered over a given time) is proportional to:

• The square of the blade length ($L^2$).
• The cube of the wind speed ($v^3$).

Evaluating the options provided:

• Option 1: second power of wind speed ($v^2$) - Incorrect. Power is proportional to $v^3$.
• Option 2: square of its blade length - Correct. Power is proportional to $L^2$.
• Option 3: third power of blade length - Incorrect. Power is proportional to $L^2$.
• Option 4: Wind speed (v) - Incorrect. Power is proportional to $v^3$.

The energy gathered is directly related to the power the windmill can capture. Therefore, the energy gathered is proportional to the square of its blade length.