Re: Vertical fall. Aparadox?

[LUDWIK KOWALSKI <KOWALSKIL@alpha.montclair.edu>]

I think I now can understand why the torque is zero when the angular velocity 
and the terminal linear velocity match each other. But first let me repeat 
the description of the gadget, in case you missed it yesterday.

> 1) Take a cork from a champain bottle and turn its wider side down. Drop
>    it and it falls down along the vertical axis. There is no rotation.
> 2) With a sharp knife make a "meridional cut" in the cork and insert an
>    index cart into it. Drop the cork and see it falling without rotation.
>    The fins are parallel to the axis, the "fin angle" of John is zero.
> 3) Cut the card (only that part which is above the cork) along the axis.
>    Bend each half so that its plane is no longer parallel to the corks axis.
>    The fin angle is 30 to 45 degrees. Looking from the side you would see 
>    the card's profile as a letter Y. Something as below. Drop this "rigid"
>                           object and abserve its spinning. It is clear
>        \       /          that the terminal angular velocity is rapidly  
>         \     /           established when the "fins" are large. It can
>          \   /            be measured with a camcorder. If I had time
>           \ /             I would check John's formula against the real
>            I              data. The "mean radius" of the fins can
>          **I**            easilly be calculated for the flat rectangular
>          **I**            card sections. 
>          *****            What I would like to know now is the reason for
>          *****            saying that "the fins should slice down through
>         *******           the air without a net torque." The argument that
>        *********          "this is necessary to conserve energy" would not 
>        *********          satisfy me in this context.
>                                                          Ludwik Kowalski
>
>  P.S. The left blade above is closer to me than the right blade; do not 
>  forget that the blades were part of a single index card. The direction
>  of rotation is counter-clock-wise when looking from above.
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> > Ludwik,
> >
> > At terminal conditions the fins should slice through the air without any
> > net torque one way or the other.  I would expect a relationship something
> > like the following to hold:
> >
> >  (terminal angular speed) x ("mean radius" of the fins)
> >  ------------------------------------------------------ = tan(fin angle)
> >                (terminal linear speed)
> >
> > where the fin angle is the angle the fins make with the vertical. Slower
> > rotation than this would cause air resistance to both brake the fall and
> > increase the rotation; faster rotation would cause the air to both drive
> > the device downward and brake the rotation. 
> >                                                               John
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One way to understand terminal angular velocity is to think about a 
propeller on a roof. The frictional force (ball bearings) is usually very
small and it does not depend strongly on the rate of rotation. Suppose it
is negligible. The wind is blowing and the propeller is turning. The rate
of steady rotation depends on the wind velocity. Why does the net torque 
become smaller and smaller when the angular velocity is increasing in a 
wind whose speed is constant? 

Because the blades run away from the wind. When they run away as fast as the 
wind (I am referring to the torque-producing wind velocity component, the one
which is perpendicular to the axis of rotation) then the net torque is zero.
John's formula is a mathematical way of saying the same thing.
                                                                 Ludwik
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