Wind turbines.

Step 1 of 5

The power in the wind

Each second a cylinder of air passes through the rotor, $v$⁠ long, its base the area $A$⁠ the blades sweep. Its mass is $\rho A v$⁠, and its kinetic energy gives the power in the wind:

$P_{\text{wind}} = \tfrac12 \rho A v^3$$P_{\text{wind}} =$$\tfrac12 \rho A v^3$⁠

The turbine takes a share $C_p$⁠ of it, the power coefficient. The Wind speed and the Rotor diameter are fixed for now.

Step 2 of 5

Twice the wind

The speed is cubed. Predict what will happen to the power when the wind doubles, then choose On under Show the power to see the curve.

Step 3 of 5

The Betz limit

Show the power shows how the air slows and spreads out as it passes the rotor. No turbine takes all the power in the wind.

Step 4 of 5

A smaller rotor

The area the blades sweep is a circle, $A = \pi \left(\dfrac{D}{2}\right)^2$$A =$$\pi \left(\dfrac{D}{2}\right)^2$⁠. Predict what will happen when the diameter halves.

Step 5 of 5

A gale

Every turbine has a rated power, the most its generator is built to give. The curve beside the Scene shows what happens above it.