Question

A wind-powered generator converts wind energy into electric 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 proportional to

• A. $v$
• B. $v^2$
• C. $v^3$
• D. $v^4$

Answer 1

Answer: (C)

$\text{Power}=\overrightarrow{F}\cdot \overrightarrow{v}=\text{Fv}$
$\text{F}=\text{v}\left(\frac{\text{d}\text{m}}{\text{d}\text{t}}\right)=\text{v}\left\{\frac{\text{d}\left(\rho \times \text{volume}\right)}{\text{d}\text{t}}\right\}$
$=\rho \text{v}\left\{\frac{\text{d}\left(\text{volume}\right)}{\text{d}\text{t}}\right\}=\rho \text{v }\left(\text{Av}\right)=\rho {\left(\text{Av}\right)}^2$
$\therefore \text{Power }\text{P}=\rho {\text{Av}}^3\text{ or }\text{P}\propto {\text{v}}^3$