The energy that a windmill can gather is proportional to the:

Correct option is B

The correct answer is (b) Square of its blade length. The energy a windmill can gather is directly related to the swept area of the blades, which is proportional to the square of the blade length. This relationship is key because the swept area determines how much wind the turbine can capture.
Key Concepts:
1. Swept Area (A):
· The area swept by the windmill blades can be calculated using: A=πr2 Where r is the blade length (radius).
· Therefore, the swept area is proportional to the square of the blade length.
2. Energy Capture:
· Since the energy captured by a windmill depends on the swept area, it increases as the square of the blade length increases.
Breakdown of Other Options:
1. (a) Second power of wind speed (v²):
· Incorrect. The power output of a wind turbine is proportional to the cube of the wind speed (v³), not the square.
2. (c) Third power of blade length:
· Incorrect. The swept area (and thus energy gathered) depends on the square, not the cube, of the blade length.
3. (d) Wind speed (v):
· Incorrect. Energy and power output depend on higher powers of wind speed, not linearly.
Information Booster 1. Wind Energy Formula:
· Power output from a wind turbine is given by: P=1/2ρAv3 Where:
· ρ = Air density
· A = Swept area = πr²
· v = Wind speed
2. Importance of Blade Length:
· Longer blades increase the swept area, allowing the windmill to capture more wind energy. This is why modern wind turbines have large blades.
3. Practical Considerations:
· Larger blade lengths are used in offshore wind farms to maximize energy capture due to consistent wind speeds.
Additional Knowledge · Wind Turbine Design:
· The design and length of windmill blades are critical for maximizing efficiency.
· Onshore turbines typically have blades up to 50 meters long, while offshore turbines can have blades exceeding 80 meters.
· Efficiency Limits:
· The Betz Limit states that no wind turbine can capture more than 59.3% of the kinetic energy in wind.