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)

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