The blades of a windmill sweep out a circle of area A. (a) If the wind flows at a velocity v perpendicular to the circle, what is the mass of the air passing through 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=30m^2`, `v=36kmh^-1` and the density of air is `1.2kgm^-3`, what is the electrical power produced?

Explanation

(a) Volume of wind flowing per second = Av
mass of wind flowing per second = Av`rho`
mass of air passing in t second = Av`rho`t
(b) Kinetic energy of air = `(1)/(2) mv^(2) = (1)/(2) ("Av"rho t ) v^(2)`
` = (1)/(2) Av^(3) rho t `
(c) Electrical energy produced = `(25)/(100) xx (1)/(2) Av^(3) rho` t
` = (Av^(3) rho t)/(8)`
Electrical power = `(Av^(3) rho t)/(8 t) = (Av^(3) rho)/(8)`
Now, A = 30 `m^(2) , v = 36 " Km "h^(-1) = 36 xx (5)/(18) m s^(-1)`
= = 10 `m s^(-1) , rho = 1.2` kg `ms^(-1)`
`therefore " Electrical power " = (30 xx 10 xx 10 xx 12 )/( 8) `W = 4500
W = 4.5 kW