Re: [Phys-L] kinematics objectives

[John Denker <jsd@av8n.com>]

On 05/09/2013 06:43 PM, Bill Nettles wrote:

> I have three squares 10 cm x 10cm: one of cardboard, one of aluminum
> and one of lead. I drop them in pairs, flat side facing the ground.
> Lead wins every time.

This is a nifty experiment.

> I've tried dropping the squares with the edges facing down, but the 
> cardboard flips too easily.

Flipping (i.e. aerodynamics stability) is a fixable problem.   It is
well worth fixing.  It allows you to demonstrate that a streamlined,
aerodynamically-stable object drops like a stone.

There are at least two ways of obtaining sufficient stability.

A) Let the object slide down a guide.

My preferred type of guide is a metal rod.  There are about ten 
reasons why you want to have a 10 or 12 foot rod in the classroom 
already.  It's good for demonstrating transverse *and* longitudinal 
wave propagation -- at dramatically different speeds -- among other
things.  The going rate for 1/8 diameter aluminum rod is less than 
a dollar per foot.

If you haven't got a suitable rod, you can use a wire.  Piano wire 
is best, but stranded picture-hanging wire will do.  Attach one end 
to a strong point in the ceiling, and then apply a *lot* of tension, 
to ensure that it stays straight.  Use a turnbuckle or a humongous
weight.

Feed the rod (or wire) through little round eyelets.

Consider attaching the eyelets to a carrier, and then using double-sticky
tape to attach the carrier to the object of interest.  This saves you the
trouble of getting the eyelets on/off the rod or wire during the course 
of the experiments.

If you want to have races, you need two (or more) separate guides.  OTOH
by putting more than one object on the same guide, you can observe
interference effects.  If you're clever you can use this to (mostly) 
nullify aerodynamic friction for one of the objects.

==========

B) You can fix the aerodynamic instability using aerodynamics, i.e. by
adding stabilizer fins to the back of the object.

             █████████        tail
                 █
                 █
                 █
                 █
                 █
                 █
                 █
             █████████
             █████████
             █████████         body
             █████████
             █████████

Properly engineering this requires a little bit of sophistication.  The
criterion can be stated (or misstated) in various ways.  In order of
increasing correctness:
  a) The center of area must be aft of the center of mass.
  b) The center of lift must be aft of the center of mass.
  c) The center of lift-prime must be aft of the center of mass,
   where lift-prime is the /derivative/ of lift with respect to
   angle of attack.
  d) Both (b) and (c).  Equilibrium and stability.

Building the tail out of lightweight materials obviously helps.
If you build the whole thing out of one material, you can give the
body more mass per unit area just by using a double thickness.  As
Mr. Spock would say, it's crude but effective.

Interestingly, though, the problem can be solved even if the entire 
structure has uniform mass per unit area.

For one thing, the tail as shown above has a larger aspect ratio
(more span per unit chord), and this gives it a steeper lift curve,
when you plot lift versus angle of attack.

Secondly, if you give the tail a proper airfoil section, rounded 
at the front and sharp at the back, this will give it more lift per
unit angle of attack ... and also more lift-prime (assuming you 
don't do the same for the body, i.e. assuming the body is just a 
flat plate).