First slide

Power

Question

The blades of a windmill sweep out a circle of area A. If the wind flows at a velocity v perpendicular to the circle, then power developed by air is

Moderate

A

$\frac{1}{2} A ρ V$
B

$\frac{1}{2} A ρ v^{2}$
C

$\frac{1}{2} A ρ v^{3}$
D

$2 A ρ v^{3}$

Solution

m = Mass flow rate of air through the blades = ρ(Av). It is assumed that air comes to rest after passing through blades.

$\large \therefore$
The rate at which air is losing kinetic energy =
$\large \left ( \frac 12mv^2-0 \right )=\frac 12\rho Av^3$

$\large \therefore$
Power developed =
$\large \frac 12\rho Av^3$

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