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  4. A wind-powered generator converts wind energy into electrical energy. Assume that the generator converts a fixed fraction of the wind energy intercepted by its blades into electrical energy. For wind speed v, the electrical power output will be most likely proportional to

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Q. A wind-powered generator converts wind energy into electrical energy. Assume that the generator converts a fixed fraction of the wind energy intercepted by its blades into electrical energy. For wind speed $v$, the electrical power output will be most likely proportional to

JEE MainJEE Main 2013Work, Energy and Power

Solution:

Power = F $\bullet$ v = Fv $F=v \big(\frac{dm}{dt}\big)=v \Bigg \{ \frac{d(\rho \times \, volume)}{dt}\Bigg \}$ $ \, =\rho v \Bigg \{\frac{d(volume)}{dt}\Bigg \}=\rho v(Av)=\rho Av^2$ $\therefore \, \, \, \, Power \, P=\rho Av^3 \, \, \, \, \, or \, \, \, P = v^3$