Re: flipping a coin.

Greg,

I glean three questions from your message.  I hope the discussion below 
answers them to your satisfaction.  I'm not really clear on your 
background, so please don't be offended if I'm writing below your level.

First: It will spin about its center of mass.  It is a well-established 
principle of physics that bodies spin about their center of mass.

Second: there is a fundamental property of motion called conservation of 
angular momentum.  Angular momentum is the rotational counterpart to linear 
momentum.  Linear momentum is often expressed mathematically as mass-times-
velocity.  Newton's second law of motion states that linear momentum is 
changed by the application of a force - mathematically, 

force = ((mass-times-velocity after force applied) minus
         (mass-times-velocity before force applied)) divided by elapsed-
time

On the other hand, linear momentum is conserved (doesn't change) if no 
force is applied.

Obviously, the direction of initial motion and the direction in which the 
force is applied are important in determining the resultant motion.  These 
days, we use vector notation to take such issues into account.

Angular momentum works much the same way, except it is more complicated :)  
Angular momentum depends upon how the mass of a body is distributed (it's 
rotational inertia) and how the body is rotating (about what axis and how 
quickly).  However, angular momentum does not depend on linear momentum and 
we can separate rotational motion from linear motion.

Just as you can see that linear momentum does not change if no force is 
applied, we may deduce (with the help of mathematics that I'm omitting) 
that angular momentum does not change if no torque is applied.  It is very 
important to understand that it is the angular momentum that does not 
change when no toque is applied; the angular velocity and the axis of 
rotation may change even if no torque is applied.

There's a nice animation at 
http://www.seas.gwu.edu/student/huaxc/cs206/cs206_3_2.html 
(note - link defunct as of 7/20/2006)

Now, will the angular velocity oscillate? no, not so long as the mass 
distribution doesn't change.  The axis of rotation will change, however.

Third: I don't think anyone has the capability to do the extension dynamic 
simulations to definitively answer this question, but I think we can make 
some pretty substantial headway by thinking it through.

Let's say that we're tossing a coin in such a way as to make it flip 
rapidly as it's center of mass traces a parabola through the air.  The 
ground on which it will land is very soft, like a grass-covered field.  (I 
make this assumption because the dynamics of impact and bouncing are very 
very complicated and, furthermore, I don't think they're pertinent to this 
issue)

If, each time we toss the coin, we do so with exactly the same motion and 
hold our hand in eactly the same place in space, the coin would land in the 
same amount of time and the same face would be up.  However, we are not 
able to be so precise in our actions (nor is nature, but let's avoid those 
perturbing forces for now)

Each toss of the coin imparts a certain angular velocity and requires a 
certain amount of time before it reaches the ground.  If we know the time 
and the angular velocity, we can compute how many times the coin will flip.  
If we also note which side was up before we toss it, we can clearly predict 
what side will be up when it hits the ground.

The time for the coin to reach the ground is determined by how hard we flip 
it (force applied), the mass (inertia) of the coin, and where our hand was 
when we flipped it.

The angular velocity of the coin depends on the torque applied and the mass 
distribution of the coin (rotational inertia).  The torque applied depends 
on where our thumb struck the coin relative to its center of mass.

Now our question is this: if the mass properties of the coin place the 
center of mass somewhere other than the geometric center of the coin, could 
this cause one face to turn up more often than the other?  Well, suppose 
you placed the coin in your hand such that the center of mass was furthest 
from your thumb.  This would increase the torque applied which would 
increase the angular velocity of the coin and, therefore, the number of 
times the coin flips over (as compared to a coin with the center of mass at 
the geometric center).  However, on the next toss the center of mass ends 
up closer to your thumb, having the oppositie effect.  I conclude that such 
a coin would see a greater variation in the number of times it flips over, 
but that this variation will not affect how often one side turns up over 
the other.

Troy
http://surf.to/tdg/