Do objects have a tendency to fall in their most aerodynamic position? In one of my physics classes, we are using slow motion video to analyze the falling motion of an object and then find out the drag coefficient of that object etc.
I noticed while doing this that if I dropped our object (a styrofoam bowl with a cone of sorts attached to the top) from any position, it would always flip and fall tip down if dropped from high enough. There was of course a certain height when it would not flip and would fall slowly to the floor.
I asked my question to the professor and he said he wasn't completely sure and that the only relevant knowledge he had was that some high-atmosphere ice crystals which are basically square shaped fall with the square side down, rather than sideways.
No doubt there are many examples of objects not falling in their most aerodynamic position, so here's the way I'm picturing this. I imagine it is something like each object which falls in a particular orientation has reached a minimum (or maximum) on a potential energy curve. Except that I guess this curve would represent minimums in the drag force. And the reason then that many objects would not fall in their most aerodynamic position is because the energy barriers between that position and the next minimum are very small. I'm picturing a piece of paper falling for instance. Paper would fall fastest with its edge to the ground, yet it often teeters back and forth between that position and falling with its full area facing downwards.
So, any thoughts on this idea and just general answers to what determines which direction an object will face while falling?
 A: No (see falling leaves), and it's more likely to be quite the opposite since the the aerodynamic orientation is generaly unsteady and a small tilt is prone to make the orientation worser and worser.
A: The answer to this question is "hard" in the sense that a general object falling from general initial conditions may or may not reorient into a lowest drag orientation.
However, one can make some educated guesses by comparing the locations of the center of drag (more generally, the center of pressure) and the center of mass of the object.  Consider one commenter's example of a parachute -- nearly all the drag force is on the inner surface of the parachute -- meaning the center of drag is near the "middle" of the parachute.  However, the center of mass (since parachutes are generally much lighter than people) is below the parachute, near the person.  This has the effect of the person being suspended under the parachute.  If the person swings a little one way or the other, the center of drag is still above the center of mass and the system does not tumble.
For a styrofoam bowl with a conical top, the styrofoam is very likely to have little impact on the center of mass of the cone, but a large effect on the center of drag.  Once again, the falling preferred orientation is center of mass below center of drag.  However, this can be defeated -- make the cone out of styrofoam.  Another way to defeat this is to start with a very large angular velocity, so that the tendency of the mass to fall below the drag is overcome by the angular momentum.  The preferred orientation will eventually prevail, but there may be many revolutions before this happens.  (There is an optimization problem lurking here:  Make the cone too dense and its angular momentum will be hard to dissipate.  Make the cone too light and the centers of mass and drag will coincide so there will be no preferred orientation.)
An easy way to see that this should be hard is this.  The drag of the system depends on the orientation of the system, so the center of drag depends on the orientation of the system.  A good parachute will be designed so that a small swing by the person will lead to a small central force to bring the person back under the center of drag.  A bad parachute design would lack that small central force -- it could, even worse, lower net drag so that the parachute begins to fall faster, perhaps even faster than the person.  (This actually happens.  It's the worst scenario in what is called "stalling a parachute.")  Nevertheless, we can design for this.  Recall that SpaceShipOne was designed so that in its feather configuration it would fall in a particular orientation -- i.e., small displacements from that orientation would be passively corrected by the change in the center of drag.
A: It depends on the object's shape, mass distribution, and rigidity.
If the object's center of mass is directly leading its center of aerodynamic pressure while falling, it will be stable in that orientation.
If the object's centers of mass and pressure are coincident, or if the center of mass lies behind the center of pressure, it won't be stable in that orientation.
If the object is very flexible (such a piece of paper), its centers of mass and pressure can move all over the place, and it probably won't settle into a stable configuration.
A: An object will orientate itself with its centre of mass below its centre of pressure. Thus there is a right way and a wrong way to rig a parachute.
Rig the parachute with strings all around so that your mass will cause it to spread out horizontally, and you will live.
Attach yourself only to one edge of the parachute, and it will orientate itself vertically, so you will die.

A helicopter is another example of an object which will naturally fall slowly if its engine fails. it falls with the rotor at the top, and in this orientation the blades are horizontal, slowing its fall. 
On the other hand a rocket that has exhausted its fuel will fall with its fins at the top, and in this orientation the fins are vertical, so they do nothing to slow the fall.
A: The rigid bodies whose aerodynamics are most richly understood are airplanes.
Most airplanes are constructed to be aerodynamically stable: If you turn off the engines and trim the control surfaces appropriately once and for all, the plane will glide stably forward at a shallow angle of descent until it reaches the ground -- without any need for movement of the control surfaces along the way to keep its orientation right.
Technically, gliding is just one mode of falling -- that is, moving through the air under the influence of gravity.
An airplane will also have other stable modes of falling, though -- for example most can enter a stable spin, and T-tailed aircraft can enter a deep stall, also with the control surfaces in the same positions as for ordinary gliding.
These kinds of falls are stable at different vertical speeds, so you would probably say they are not equally aerodynamic.
