Silent Acorns

Damn! you people keep coming up with better ideas while I'm typing up mine.

Cheers to Jesse for his excellent illustration. It's perfect down to the smallest movement.
:notworthy :notworthy

Silent Acorns

How about this question:

what happens if the ratio of the diameters of the two spheres is a whole number?

I think I know the answer but I'd like to hear your thoughts.

Jayjay

I don't know about whole numbers in specific, it seems that the reason why the orientation doesn't change in the original case is that there's a mapping from the path travelled in one ball to a path in the other ball that preserves distances and angles. The former is obvious from the way the two balls move relative to each other, but if you change the size of one of the balls, then the latter breaks, hence the orientation of the ball after moving it around changes.

Silent Acorns

First, the reason I said "a whole number" was to eliminate non-whole number ratios. In such cases a simple "once-around" trip will leave the ball in a new orientation. Also, it's obvious that if the moving ball is bigger than the fixed ball a "once around" trip will again leave the ball in a new orientation.

But what if:

D fixed
---------- = N, where "N" is a whole number ?
D moving

In this case a simple "once around" trip will bring yield the original orientation. The surface area of the larger fixed ball is N^2 times larger than the smaller fixed ball. It is quite possible for there to be N^2 points on the larger fixed ball for each point on the smaller moving ball.

That said, it's easy to come up with a counter example when the moving ball is three times smaller than the stationary ball. Simply move the ball down until it reaches the equator. The ball will be in its original orientation (i.e. it has rotated 360 degrees). Now move the ball a little along the equator (say 1 degree). It is now slightly off its original orientation. If you return the ball back to the north pole it will rotate another 360 drgrees and return slightly off its original orientation.

Jayjay

Hmm, I think the "once around" trips are a special case of a more general set of paths: regardless of the size of the balls, all full circular paths that are multiples of the moving ball's diameter will leave the ball in it's original orientation.

EDIT1: I don't think there's anything particularly interesting about the special case where d1:d2 is a whole number.

EDIT2: Actually I am not at all certain of the claim I made above...

Principia

I think we're going to analyze this problem to death.

There is probably some reason to think that other sizes of the fixed sphere can result in conserved orientation after rotating about a closed path.

In one extreme, say the fixed sphere had a radius of 0 (i.e. it is a point). I think in this case, the sphere is simply rotating rigidly about a point, and so this scenario holds.

At the other extreme, say the fixed sphere had an infinite radius (ik.e. it is a plane). Here, rotating the sphere about any closed path also returns it to its original orientation.

Now, about all those cases in between...

I think a good place to start is to see if a mapping can be generated such that each point of the fixed sphere maps uniquely to a point on the moving sphere. OK, I've had enough of this problem for one day.

Jayjay

Really?

Let's say the diameter of the ball is 1. Let's also imagine a stick in the ball that points up. Now, roll the ball 1/8 units forward, so that the stick is now in angle of 45 degrees up. Roll left 1/2 units, and the stick now points 45 degrees down. Roll back 1/8, and the stick is pointing directly forward, 0 degrees relative to the plane. Roll right 1/2 units, and the ball is in its original place but the stick is still pointing forward. Obviously, orientation is not conserved when rolling a ball on a plane.

Or did I make a mistake?

Principia

Nope, I am wrong on this too.

Friar Bellows

Well, for what it's worth this late in the thread, the answer is that the ball retains its original orientation. I can't think of a better explanation than Jesse's mirror (tronvillain's was also brilliant) so I'll leave it at that. You'll find other interesting answers on sci.physics.research.

fando

Hee hee.. yeah there are a ton of approaches. That's the fun thing about these kinds of problems, to learn how other people solve them. I'd like to contribute mine.

When I redid the problem, I did it for an arbitrary angle, delta theta. Basically, I drew this:



This is a bird's eye view of the rotation delta theta. Taking B to be the final place the circles touch after the rotation, we flip the right circle 180 degrees horizontally to simulate the effects of rotating the ball back up to the north pole. It will look like the ball on the left and it is the orientation it began with.