Re: [Phys-L] kinematics objectives

[John Denker <jsd@av8n.com>]

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

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

Previously, alas, I suggested letting the cardboard squares slide down
a guide.  That would work, but it's unnecessarily complicated.  There
is a simpler solution.  See below for details.

This is helpful, because it means you can demonstrate both the high-
friction case and the low-friction case without any complicated setup.

See also next message.

=============================
Details....

  Notation:  In the following x,y, and z are /body/ axes, attached
  to the object (not attached to the laboratory frame).

It turns out that /aspect ratio/ affects the stability.  Consider
a cardboard slat x=10.5 inches long by y=1.5 inches wide by z=1 mm 
thick.  Just now I did some experiments, dropping the slat from a
height of 2m, with the smallest face (yz) forward, i.e. with the long
axis (x) vertical.  We can call this the "javelin" orientation.  It
has an aspect ratio much less than unity, namely 1/7, meaning there 
is more chord than span.  It fell straight and true for 2m in 10 out 
of 10 trials.

The thing also fell straight and true in the "pancake" orientation,
i.e. the broad xy face forward and the tiny z axis vertical.

Contrast this with the results with the intermediate face (zx) forward, 
i.e. with the intermediate axis (y) vertical.  That gives an aspect
ratio greater than unity, namely 7.   We can call this the "wing"
orientation, because the thing looks like a normal wing.  It dropped 
only a few cm before the instability took over, resulting in rapid 
tumbling around the x axis.

Less extreme aspect ratios give less extreme results.  Consider 
a rectangle with x=10.5 inches y=7.5 inches and z=1 mm.  In the 
quasi-javelin orientation, i.e. with yz forward and x vertical, 
with good technique I observed significant instability in 7 out 
of 10 trials.  Good technique included minimizing
 -- curvature of the cardboard
 -- air currents, especially in the z direction
 -- initial velocity, especially in the z direction.

The pancake orientation was almost stable, although there was a 
slight tendency to flutter.  This was the classic falling-leaf 
flutter, not an outright tumble.

The quasi-wing orientation, with aspect ratio greater than unity,
i.e. with zx forward and y vertical, exhibited pronounced instability.
It fell a few tens of cm and then began tumbling around the long (x)
axis.