__Question 6.21__:*A*. (a) If the wind flows at a velocity

*v*perpendicular to the circle, what is the mass of the air passing through it in time

*t*?(b) What is the kinetic energy of the air? (c) Assume that the windmill converts 25% of the wind’s energy into electrical energy, and that

*A*= 30 m

^{2},

*v*= 36 km/h and the density of air is 1.2 kg m

^{–3}. What is the electrical power produced?

__Solution__:Area of the circle swept by the windmill =

*A*

*v*

Density of air =

*ρ*

**(a)**Volume of the wind flowing through the windmill per sec =

*Av*

Mass of the wind flowing through the windmill per sec =

*ρ*

*Av*

Mass

*m*, of the wind flowing through the windmill in time

*t*=

*ρ*

*Avt*

**(b)**Kinetic energy of air = (1/2) m v

^{2}

= (1/2) (ρAvt)v

^{2}= (1/2)ρAv

^{3}t

**(c)**Area of the circle swept by the windmill =

*A*= 30 m

^{2}

Velocity of the wind =

*v*= 36 km/h

Density of air,

*ρ*= 1.2 kg m

^{–3}

Electric energy produced = 25% of the wind energy

= (25/100) X Kinetic energy of air

= (1/8)

*ρ*A v

^{3}t

Electrical power = Electrical energy / Time

= (1/8)

*ρ*A v

^{3}t / t

= (1/8)

*ρ*A v

^{3}

= (1/8) X 1.2 X 30 X (10)

^{3}

= 4.5 kW

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