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 the electric energy. For wind speed V, the electrical power output will be proportional to?
A. $\text{V}$
B. ${{\text{V}}^{2}}$
C. ${{\text{V}}^{3}}$
D. ${{\text{V}}^{4}}$

Answer 1

Hint: The kinetic energy of the wind energy is tapped by a windmill in this problem, it is only able to convert a fraction of the wind energy into electrical energy. So we need to calculate the force acting on the windmill, which is making the windmill rotate. If we know the force acting on the windmill, we can calculate the power of the windmill by multiplying force with the velocity of the wind.

Step by step answer:

Suppose the wind strikes the windmill turbines as a cylindrical-shaped structure of area A and length V, which is the velocity of the wind. So the rate of change of volume of this hypothetical cylinder can be written as

 $\text{Volume}=\text{Area }\!\!\times\!\!\text{ Velocity}$ or
 $V'=A\times V$.

Here, $V$ is the velocity while $V'$ is the volume.
We know from Newton’s second law that the force acting on a body is the rate of change of momentum. We can write it mathematically as,
$\overrightarrow{F}=\dfrac{d\overrightarrow{P}}{dt}$
This can be rewritten as,
$\overrightarrow{F}=\dfrac{d\left( m\overrightarrow{V} \right)}{dt}=m\dfrac{dv}{dt}+V\dfrac{dm}{dt}$
Here, the velocity of the wind is constant, so the term $\dfrac{dV}{dt}$ is zero.
So we can write the force acting on the windmill as,
$\overrightarrow{F}=\overrightarrow{V}\dfrac{dm}{dt}$ …. Equation(1)

If the air or the wind has a density of $\rho $, then the rate of the mass of the wind that hits the turbine can be written as,
$\dfrac{dm}{dt}=\rho AV$…. Equation(2)
So the force acting on the body is, (substituting equation (2) in (1))
$\overrightarrow{F}=V\rho AV$
$F=\rho A{{V}^{2}}$
We now have an equation for the force acting on the windmill. So the power of the windmill can be found out by,
$\text{Power}=\text{Force}\times \text{Velocity}$
$\text{Power}=\left( \rho A{{V}^{2}} \right)\times \left( V \right)$
$\text{Power}=\rho A{{V}^{3}}$
Suppose the kinetic energy of the windmill is converted into electrical energy without any loss. The electrical power output of the windmill will be proportional to ${{V}^{3}}$.
So, the answer to the question is option (C).

Note: Power can be defined as the rate of doing work or work done in unit time. We have used a variation of that formula in our problem. So power can be written as,
$P=\dfrac{W}{t}$
The work done W can be written as the product of force and displacement. Also, displacement divided by time gives the velocity of the force. So we can write,
$P=\dfrac{W}{t}=F\times \dfrac{d}{t}=F\times v$
The SI unit of work is watt (W). Power is a scalar quantity.