Question

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 proportional to :

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

Answer 1

The correct option is C $v^3$

Lets consider, $v=$ wind speed

Power, $P=Fv$. . . . . . . . . . . . . .(1)

Force, $F=\frac{dP}{dt}=\frac{d\left(mv\right)}{dt}$

$F=m\frac{dv}{dt}+v\frac{dm}{dt}$

And $\frac{dv}{dt}=0$

So, $F=v\frac{dm}{dt}$

$F=v\frac{d\left(\rho V\right)}{dt}$

where, $\rho =$ density of air which is constant

$V=$ volume

$F=v\rho \frac{dV}{dt}$. . . . . . . . . . . . (2)

From the equation of continuity,

$\frac{dV}{dt}=Area\times velocity$

$\frac{dV}{dt}=Av$

From equation (2),

$F=v\rho \times Av$

$F=\rho A{v}^2$

Substitute $F$ in equation (1), we get

$P=\rho A{v}^2\times v$

$P=\rho A{v}^3$

$P\propto v^3$

The correct option is C.