Stupid question regarding rotation...

[Boerseun]

Right. Laugh at will. But will someone please enlighten me as to the following:

 

Say I take a wheel, and draw a line that intersects the axle. Now, let the wheel rotate. This wheel is standing vertically, as balanced on the nifty stand I built. At the very left side, where my line begins, points on the wheel move upwards, as measured against the line. Points to the right of the axle, move downwards.

 

As I move closer to the axle from the rim, the points are moving at right angles to my line, but slower and slower as I get nearer to the axle. The circles described by the points get smaller closer to the axle, but they still intersect my line at 90 degrees, regardless of the size of the circle.

 

Getting right close to the axle, the circles are getting very small, and the points seem to be moving very slow. Somewhere along the line, the points moving upwards, should come to a complete stop and then move downwards as I keep moving to the right. But try as I might, and under any magnification you can imagine, you won't find a spot thats standing completely still. They're either moving up or down. But somewhere along the line, they must flip direction. And if you keep the motion of the wheel in mind, and also keep in mind the fact that the points on the wheel slow down as you near the axle, there must be a point in the exact center of the wheel that's standing perfectly still? Even if you magnify the axle to atomic scales, you'll never find the 'magic' particle around which the wheel rotates. Every atom will either be a point on the line moving up or down on the line I drew that intersects the axle. The perfect center of the wheel should be a point thats smaller than the Planck scale. And even then, if we can reach string theory scales, the perfect center of the wheel around which everything rotates, will be even smaller than that. As a matter of fact, the point around which a simple wheel rotates, the point where the marked dots on the line stands perfectly still, must be infinitely small. But it must be there. There must be a point that's moving neither up nor down at right angles to the line, not moving at all, because as you near the center, the points continuously slow down - only to flip direction as you pass the center. But magnify as you like, you'll never find it.

 

Is there actually a center to a rotating wheel? Or am I wastin' my time pondering about silly crap?

[Jay-qu]

Think about what the middle would actually be - if you managed to skewer an atom straight down the middle then that atom would itself be spinning.

[Boerseun]

Think about what the middle would actually be - if you managed to skewer an atom straight down the middle then that atom would itself be spinning.

If you managed to skewer an atom, you'll end up with exactly the same issue. Just a very much smaller wheel, rotating around some infinitely small axle.

[Jay-qu]

lol thats exactly the answer i expected :confused:

 

So eventually maybe you will skewer a string or a quanta of space - then it would be meaningless to talk about rotation no?

[Boerseun]

I've got a sneaky suspicion that the exact point around which a body rotates can only be asymptotically described or approached.

 

Think about it:

 

The points on the wheel crosses the rim-to-rim line at exactly 90 degrees, at ever slower speeds. This can be imagined as follows:

 

Distance from center: Speed:

1m 100km/h

50cm 50km/h

1cm 1km/h

1mm 100m/h

0.001mm 10cm/h

 

...etc...

 

0.000000001 nanometers 0.00001 nanometers per hour

 

and so on.

 

All you'll be doing is to be adding digits. You know that if you cross the axle, the points will be running faster and faster, in the opposite direction. So, somewhere in your line, at the exact center, the points won't move at all, not? So how can anything rotate without winding itself up around its central stationary point?

 

I'm a dunce, I know.

[Jay-qu]

hehe, but your a good dunce - like the rest of us :confused:

 

You are right, its sort of like finding the smallest increment of space - you can always split it in half! Asymptopically approaching but never reaching your goal.

[Qfwfq]

Physically, no measurement is perfectly precise, no disk is a perfect circle, no mechanical axle or bearing is perfect. Apart from this, no disk is a perfectly homogeneous and rigid body, this is an idealization useful for large scale analysis. The meaning of large scale depends on the material and the stresses applied; in some cases even a few nanometres might not be a large enough scale.

 

That said, there will be some ideal geometric motion of rigid rotation that most closely describes the motion of the wheel. At the large scale of course. That certainly will have an axis of rotation but it don' gotta be a material part of the wheel. Like, the wheel could be mounted on a fixed axle but there's still a geometric axis of rotation. Suppose the axle is part of the wheel and even that a certain atom just happens to be right centred on it, and not even jiggling around in the heat, can we say that single atom has the same angular velocity of the wheel? Well that depends on how well the atom and its parts have been strutted to the neighboring ones, obviously.

 

I'm sure you've already objected that an imperfect axle and bearing screw this up enough, so have your wheel spinning in outer space, with no bearing, no gas around it and no more than a few measly photons to see it with. The axis certainly goes through the disk's centre of mass, even though its direction may be changing in time 'cause you didn't get the spin to be quite exactly along the principal axis of inertia.

[Farsight]

I think your problem might be to do with points Boerseun. We tend to think of a point as a point particle, a "something" that has tangible existence in its own right. But it doesn't. It's like the crease in my pants, it's a discontinuity. Perhaps you'd be better off thinking about a cone. Imagine a perfect cone, perfectly sharp.

 

 

Now try to describe the "point" of that cone. It isn't a thing, it's a place where something changes. And as Qfwfq was saying, you can't measure where it is with absolute precision. If you could measure to better and better resolution you'll narrow it down, but when you get to the subatomic level, the sharp point doesn't look so sharp any more, it looks blunt, or rounded, or blurred. You just can't nail it down with total certainty.

[Boerseun]

Thanks for the replies.

 

Perhaps I'm not expressing my problem with this clearly. I'm not battling with the points, what I'm battling with is that as the disk turns, every spot/part/point/particle on it will be in motion with it. And if you look closer and closer to the axis of rotation, the closer you get, the slower the constituent parts move. Until you get to the very center, where the direction flips around. So, looking again at my line that's being intersected at 90 degrees by the wheel, the speed at which the wheel crosses the line at 90 degrees decreases, until the direction it intersects flips over. So, between crossing the line at 90 degrees moving upwards, and crossing the line at 90 degrees moving downwards, we should have an infinitely small piece that stands dead still. But, like I described at an above post, being infinitely small, we can only asymptotically approach it.

 

It's like saying that if you have a line one centimeter in length, every mm will have the same properties, save for one infinitely small section that's properties will be completely foreign to the rest. You know it's there, on that 1cm line. And it must exist, because the flipping of the direction of the intersection's movement at 90 degrees demands it.

 

Rather strange, I reckon.

[Qfwfq]

Well, if you're just interested in the pure geometric matters, then it isn't all that wierd because you're talking about ideal entities but some things can strike the intuition as odd. A few amusing things to consider:

 

Consider your rotating disk, which can be described in terms of a rigid rotational motion. I the room's coordinates the rigid rotation is arond the axle, but imagine now that you travel across the room transversely to the axle. In your coordinates, what axis is the rotational motion around?

 

Imagine a bike wheel with a very rigid tube, or just a damn rigid wheel, rolling steadily along a perfectly smooth and rigid floor. Point-like contact, no deformation etc. What axis is it rotating around?

[Boerseun]

Well, a rigid wheel rolling on a rigid floor is rotating around its axle, of course - from the point of view of the floor.

 

But considering the rotation of the wheel as irrelevant based upon differing frames of reference doesn't hold up. For instance, if my point of view is on the wheel itself, I'm standing still? Not so. Centrifugal force comes into play, regardless of my point of view. The conservation of angular momentum implies that the only reasonable point of view that might be taken as such on a rotating wheel, will be exactly on that magical non-moving spot I'm talking about. Else, you'll sit on a 'non-moving platform' (your POV) but for some reason you'll always be pushed towards the rim.

 

Also, like airplane propellers having rpm limits due to breaking the sound barrier at the tips and cavitating, there are limits to the rotational speed of any object. No object can rotate at a speed that will cause the rim to break the speed of light. Thereby we can calculate the minimum sizes of pulsars, etc.

[Qfwfq]

Actually it is necessary to get the purely geometrical analysis straight, before taking care of the inertial or other mechanical aspects. Upholding these as objections to my quizzes is putting the cart before the horse. Besides I was not considering non inertial coordinate systems.

 

Well, since you don't appear to have it down pat, I'll give the game away and blurt it all. In both examples, the axis is parallel to the axle. In the first, it is vertically above or below, according to which way you're walking and which way the wheel is turning, by a distance such that tangent velocity in the room's coordinates cancels with your velocity. The second is a special case of the first, the axis is exactly at the point of contact with the floor.

 

Now, it's quite obvious that in neither case the axis is a fixed point of the wheel, it's called the instantaneous axis, it's at an ever changing position in the wheel. In the first case it is however fixed in the room; in the second it is also moving across the floor.

 

Wierd enough? And yet, it's a simple matter of the kinematics as described from different coordinates, without which you wouldn't be able to explain the "centrifugal force" that you raise as one of your objections and which isn't a real force anyway, only a matter of choice of coordinates. :eek_big:

[CraigD]

Is there actually a center to a rotating wheel? Or am I wastin' my time pondering about silly crap?

I think you’re failing to make a clear distinction between the ideal and the real.

 

As a few earlier posts note, in an ideal, geometric model, rotating things like wheels have precise center points that can be said to have zero velocity. You, however, seem to be specifically addressing a real wheel – something you might pull off you bicycle and attach to a nifty stand you’ve built, and examine with your own eyes and a real microscope (assuming you’ve clamped only one side of its axle and attached a bit of covering to the other side, so that there’s actually rotating wheel, rather than fixed axle, for you’re microscope to zoom in on.

 

If your stand and clamps and bearings are good enough, and your microscope low-resolution enough, you likely could find a point on the wheel that appeared stationary. Increasing your microscopic resolution, however, such as with a non-optical microscope like a scanning tunneling microscope, will eventually, I figure, exceed the precision of even the best possible bearings (stuff made out of diamond, etc), revealing the inherent mechanical “jerkiness” of real objects.

 

On top of this, we’re presumably talking about real-world objects here, which are much warmer than absolute zero, so on the scale of the best microscopes, the atoms in the solids of which your wheel is built will appear to be bouncing about pretty vigorously, drowning out the small motion of the wheel near its axis – making your exercise a bit like trying to pinpoint which bee in a swarm is moving at exactly the swarm’s average velocity at any given moment.

[Drum]

Hmmm .... for what its worth

 

I believe you are describing a radius

 

In the classical sense I don't think you can describe something as approaching 'infinitely small'. Any number or size greater than zero but less than one, can't really be described as infinite?? Otherwise we could say that we had found an infinite number, just above zero and just below one, which doesn't really make sense.

 

In the classical sense a radius turns around a 'stationary' point.

 

Once your radius falls below a certain size, say the 'orbit' of an electron it can no longer be described by classical maths. You are now in the land of uncertainity. The more accurate you wish to be with one property, the less accurate you must be with another.

 

If you state conclusively the location of your point, its velocity can only be described as a probability. If you state its velocity and direction of travel, you can only describe its location as a probability

 

or sumthin like dat .... :D

 

drum