Re: Vertical fall. A paradox?

[John Mallinckrodt <ajmallinckro@CSUPomona.Edu>]

On Sun, 9 Nov 1997, LUDWIK KOWALSKI wrote:

> Now suppose the object is a  rigid "bomb" whose fins are twisted. Its 
> rotational kinetic energy (spinning about the vertical axis) also increases 
> at the expense of potential energy. Will the angular velocity continue to 
> increase after the terminal linear velocity is reached? What is the  maximum 
> possible rotational energy? We know that KErot can not exceed (m*g*h-KEtt), 
> where KEtt is the terminal translational KE. In fact, KErot must be smaller 
> than (m*g*h-KEtt) because part of the initial potential energy is thermalized.
> What mechanism is responsible the "terminal angular velocity"? Why should the 
> net torque be zero at some rate of rotation? Constant terminal torque? Yes. 
> Constant terminal omega?  (???).

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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