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I'm writing up my physics coursework and I thought I'd try and find an equation described in the title. This is my attempt:

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Is it correct?

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2 Answers

I think that you are overestimating the importance of the blade mass. The blades rotate due to aerodynamic forces which are determined by blade geometry and size, and the fluid properties of the air impinging on them. The mass does not impact the aerodynamic properties of the blades. For a fixed wind speed, neglecting small effects like friction in bearings and the opposing torque of the generator, the blades will gather speed until their motion relative to the incoming wind reduces the angle of attack to approximately zero. At this point, the blades no longer generate lift thus the rotational speed of the blades will stay the same. This maximum "steady state" speed can be controlled somewhat by changing the pitch of the blades. In fact, that its one way to prevent the blades from rotating too fast in powerful winds. Ultimately, the mass should not affect the stady state blade speed. It will, however, affect how much time it takes for the blades to reach steady state speed since the torque is fixed by blade geometry and wind conditions. More mass means greater moment of inertia, which for fixed torque yields smaller acceleration. It is advantageous to make lighter blades because of sturctural considerations, not max attainable blade rotation rate. All of this is somewhat oversimplified because we ignored opposing torques and made no mention of aerodynamic drag, but the core idea doesn't require those considerations.

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That only gives you the linear velocity. Since the blade is rotating, the angular velocity $\omega$ = $v/r$. Also, since the blade is not a point object, you'd have to use the moment of inertia rather than the mass.

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Could you possible give me an example of how I would do that? Thank you. –  Theo Windebank Oct 31 '12 at 16:02
    
Well, the wikipedia article about moment of inertia that I linked to is quite clear... if you read that and you still have questions about this, I'll be glad to answer them. :) –  Kitchi Oct 31 '12 at 16:14
    
How do I algebraically manipulate the moment of inertia if its a summation with two variables? Can I assume m and r are constant so its just Nmr^2? –  Theo Windebank Oct 31 '12 at 17:01
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