anonymous
  • anonymous
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 blade into electric energy. For wind speed v, the electrical power output will be proportional to (Choose one among the four) 1. v 2. v^2 3. v^3 4. v^4
Mathematics
schrodinger
  • schrodinger
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anonymous
  • anonymous
In a unit of time, a unit of mass of wind will pass the blades. The wind has speed v, so the mass' kinetic energy is proportional to v^2. This energy is transfered in part to the blades, and it again proportional to v^2. Since energy is conserved, and we're told the generator converts a fixed fraction, that fixed fraction of energy is still proportional to v^2. The answer is v^2.
anonymous
  • anonymous
Thanks However, that is not exactly what I asked for. The question asked for the proportionality with the power, NOT Energy.
anonymous
  • anonymous
In my answer, I stated "In a unit of time"...this means an amount of energy proportional to v^2 has been transferred in this unit of time...that's power.

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anonymous
  • anonymous
But the book says its proportional to v^3
anonymous
  • anonymous
And has not provided any explanation
anonymous
  • anonymous
Ok...I know what's happened...
anonymous
  • anonymous
Kinetic energy is \[\frac{1}{2}mv^2\]For a mass of air flowing in a unit of time, we can write mass as a volume flow rate x time x density (=mass). Then\[E=\frac{1}{2}(Avt{\rho})v^2\]
anonymous
  • anonymous
The power is just rate of change of energy, which is means from the above, \[P=\frac{E}{t}=\frac{1}{2}(A{\rho})v^3 \]
anonymous
  • anonymous
So the power is proportional to the cube of velocity (but then again, it's also proportional to the square...technically speaking ;)).
anonymous
  • anonymous
Can you see how Av yields volume flow rate?
anonymous
  • anonymous
Yes I have got you

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