A 120-V generator is rum by a windmill that has blades 2.0 m...

A $120-\mathrm{V}$ generator is rum by a windmill that has blades 2.0 m long. The wind, moving at $12 \mathrm{~m} / \mathrm{s}$, is slowed to $7.0 \mathrm{~m} / \mathrm{s}$ after passing the windmill. The density of air is $1.29 \mathrm{~kg} / \mathrm{m}^3$. If the system has no losses, what is the largest current the generator can produce?

Key Concepts

Electrical Power and Current Relationship

This principle involves the conversion of mechanical power extracted from the wind into electrical power. The basic relationship is given by P = IV, where P is the electrical power, I is the current, and V is the voltage. By knowing the power available from the wind and the generator's voltage, one can calculate the maximum current that can be produced by the generator.

Turbine Swept Area

The swept area of a windmill is the area covered by its blades as they rotate. It is typically calculated as the area of a circle with a radius equal to the blade length. The swept area is directly proportional to the volume of wind intercepted by the turbine and thus significantly influences the amount of energy that can be harnessed.

Wind Power Extraction

This concept connects the change in kinetic energy of the air to the power available from the wind. By calculating the difference in kinetic energy before and after the air interacts with the turbine and considering the mass flow rate, one can determine the total power that is theoretically extractable from the wind, assuming ideal (lossless) conditions.

Mass Flow Rate

Mass flow rate refers to the amount of mass passing through a given cross-sectional area in a unit of time. For a wind turbine, this is calculated using the density of air, the swept area of the blades, and the wind speed. It is a crucial factor because the energy that can be extracted from the wind depends not only on its speed but also on the quantity of air flowing through the turbine.

Kinetic Energy of Air

This concept involves understanding that moving air possesses kinetic energy determined by its mass and the square of its velocity. When wind passes through a device like a windmill, the change in its velocity corresponds to a change in kinetic energy, and this difference is available to be converted into mechanical and then electrical energy.

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