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I was driving the other day and couldn't help but look at all the wind mills in the wind farms. I know I am not doing the math correctly, but I want to know what is "really" happening when a windmill spins.

Suppose there is a wind mill with one blade that is 1 foot long. Now lets say it takes 1 second to complete a rotation. It travels in a circle. We can calculate the circumference of this circle by 2*R*PI => 2*1*PI which leads me to believe that the blade is traveling 2*PI feet per second.

But now let us take an arbitrary point on the blade, say .5. Now the circumference is 2*.5*PI which would then make the speed PI feet per second.

To me this means that different parts of the blade are traveling at different speeds, but intuitively this is not possible. Where is my logic wrong? I am more of a math guy, but I was very curious about this question and wanted to see the physics point of view.

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    $\begingroup$ "To me this means that different parts of the blade are traveling at different speeds, but intuitively this is not possible." Why not? Consider a rod spinning around a center. Clearly, its ends are moving, while its center is not. Thus the speed must change smoothly from the "high" speed at the ends to the "low"(no) speed at the center. $\endgroup$
    – ACuriousMind ♦
    Commented Oct 30, 2014 at 17:24
  • $\begingroup$ I would like this question if you explain why you believe your result is unintuitive. $\endgroup$
    – BMS
    Commented Oct 31, 2014 at 2:06
  • $\begingroup$ Because I did not understand the difference between angular and linear velocity. $\endgroup$
    – TheBlindSpring
    Commented Oct 31, 2014 at 2:24

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Different parts of the blades have different speeds, but all parts of the blades have the same instantaneous angular speed, that is all parts travel through the same angular displacement in a given amount of time. This is always true for a rigid body (when the angular speed is measured about the rotation axis).

As ACuriousMind pointed out in a comment, you shouldn't expect all parts of the blades to rotate with the same linear speed, since obviously the central axle does not move at all, while the edges of the blades move very fast. For a rigid body, their speeds are related as $v = r\frac{d\theta}{dt}$, that is the linear speed increases linearly with distance from the rotation axis, and is proportional to the angular speed.

For non-rigid bodies, differential rotation is also possible. One classic example is a whirlpool, which does not obey $v = r\frac{d\theta}{dt}$.

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