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 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 m^(2)` v = 36 km/h and the density of air is `1.2` kg `m^(-3)` .What is the electrical power produced ?

Explanation

(a) Suppose area of the circle swept by the windmill is A, velocity of the wind is v and density of air = `rho `
Distance covered of air passing through circular area A ,
V = Ad
` :. V = Avt [ :. d = vt] `
` :. ` Mass of flowing air = volume `xx` density m =` Avtrho`
(b) Kinetic energy of air pasing through area of A in time t ,
`K = 1/2 mv^(2)`
`=1/2 xxAvt rho xx v^(2)`
` K = 1/2 A rho vt^(3)`
(c ) Input kinetic energy is energy of wind and output energy generated ,
` :. "Efficiency " eta = (" Output energy")/("Input energy")`
25 % = `("Electrical energy ")/(1/2Av^(3))`
` :. ` Electric energy produced
` =1/4 xx1/2 xx 0 xx 1.2 xx(10)^(3) xxt`
= 4500 t
where `v = (36xx1000)/(3600) = 10 ms^(-1)`
` :. "Electric power " = (" Electric energy")/("Time")`
= 4500 W
`=4.5 W`