A wind turbine converts some of the kinetic energy of the wind into electric energy. Suppose that the blades of a small wind turbine have length $L=4.0 \mathrm{m}$ (a) When a \(10 \mathrm{m} / \mathrm{s}(22 \mathrm{mi} / \mathrm{h})\) wind blows head-on, what volume of air (in \(\mathrm{m}^{3}\) ) passes through the circular area swept out by the blades in \(1.0 \mathrm{s} ?\) (b) What is the mass of this much air? Each cubic meter of air has a mass of 1.2 \(\mathrm{kg}\). (c) What is the translational kinetic energy of this mass of air? (d) If the turbine can convert \(40 \%\) of this kinetic energy into electric energy, what is its electric power output? (e) What happens to the power output if the wind speed decreases to \(\frac{1}{2}\) of its initial value? What can you conclude about electric power production by wind turbines?

Step by step solution

01

Calculate the area swept by the blades

The blades describe a circle with radius equal to the length of the blades, L, as they rotate. The area of this circle can be found using the formula for the area of a circle, A = πr^2, where r is the radius (L in this case). The area swept by the turbine blades is: A = πL^2

02

Calculate the volume of air passing through the area

To find the volume of air passing through the area during 1 second, we need to multiply the area by the wind speed. V = A * v where v is the wind speed.

03

Calculate the mass of air

To find the mass of the air, we need to multiply the volume of air by its mass per unit volume (in this case, 1.2 kg/m^3). m = V * ρ where ρ is the mass per unit volume (density).

04

Calculate the translational kinetic energy

To find the translational kinetic energy of the mass of air, we can use the formula: K.E. = 0.5 * m * v^2

05

Calculate the electric power output

To find the electric power output of the turbine, multiply the kinetic energy by the turbine's conversion efficiency (40% in this case). P = 0.4 * K.E.

06

Determine the power output when wind speed decreases

To find the power output when the wind speed decreases to half its initial value, we can repeat steps 2-5 with the new wind speed. Observe the new power output and compare it to the initial output to draw conclusions about electric power production by wind turbines.

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