Rotational Kinematics

Interesting on my FB feed today....



1. Setup
Vehicle (say motorcycle) is moving forward at 100 kph relative to the ground.
The wheel is rolling without slipping.
That means the point of the wheel in contact with the ground has zero velocity relative to the ground.
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2. Wheel motion
Think of a wheel as doing two motions at once:
1. Translation: the whole wheel (center) moves forward at 100 kph.
2. Rotation: the wheel spins such that the bottom point cancels out the forward motion and becomes 0 kph relative to the ground.
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3. Speeds at different points
Center of wheel: moves at 100 kph forward.
Bottom point: rotation gives -100 kph (backward relative to center) + 100 kph (forward from translation) = 0 kph relative to ground.
Top point: rotation gives +100 kph (forward relative to center) + 100 kph (forward from translation) = 200 kph relative to ground.
So the velocity around the rim relative to the ground looks like this:
Bottom = 0 kph
Center = 100 kph
Top = 200 kph
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4. Why it works
The math comes from rotational kinematics:
v_\text{point} = v_\text{center} + \omega \times r
Where:
(because no slip: tangential speed = center speed)
At the bottom, vectors oppose each other → cancel → 0.
At the top, vectors align → add → 200.
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So the picture is exactly right: when the bike is doing 100 kph, the top of the wheel is doing 200 kph relative to the ground, and the bottom is doing 0.

It needs to be pointed out that this view assumes use of the so-called intrinsic coordinate system, which is positioned in a particle of the bitumen at point of contact. This particle rides with a speed of 100 kph. An observer riding on this particle and looking at the adjacent tire would perceive the tire as fixed vs. own position (no slip), despite the apparent backward speed in an earth-bound coordinate system. Furthermore, looking from this particle, it would appear as if the wheel center revolves with Omega x r, and top of the tire revolves with Omega x 2r. These incremental speed vectors are parallel to the particle speed vector.

Different equations may be expressed depending on choice of the coordinate system, but if done correctly, the end result will be the same.

- Knut

The top of the tire is moving faster than 200 because the radius to the road is smaller due to tire compression.

hhh said:

the radius to the road is smaller due to tire compression.

Click to expand...

Tornado’s picture shows the opposite of that!

Aren't numbers fun?

Yes, the relative speed of the tire to the pavement is 0 at the point and instant of contact - a cool mathematical oddity.

Of course, an ant sitting on the tire will be moving at pi * diameter units per timeframe in more or less a circle and at some positive or negative portion of the forward motion of the tire regardless of where in the rotation the tire is at the time. This is like the fact that the Earth is turning on its axis, following an ellipse around the Sun, and the Sun is moving at more or less a straight line through space (actually rotating around the center of the galaxy while moving away from the center) all while the galaxy is moving through space. If you are walking, then you are moving in at least five directions at once relative to any fixed point in space. Since space is expanding, there's yet another direction.

If you would like your head to hurt, think about the speed of light through space. Then approaching the event horizon of a black hole. Then when crossing over the event horizon as viewed from the inside and outside of the event horizon. Then at the singularity.

OK, back to the search for a frame painter or powder coater...that makes my head hurt enough!

This is easier to understand when you look at what happens while running...your feet are not moving when in contact with the the ground while your whole body is moving with some forward velocity. Limbs and joints of various lengths and at various angles make calculations more complicated than a rotating wheel, but same principles apply.

Does this make it clear?

My grandfather started my brain hurt by asking if the center of a shaft turns … hmmm …

Richard Tool said:

My grandfather started my brain hurt by asking if the center of a shaft turns … hmmm …

Click to expand...

Grrrrrrrrrrrrr! Why did you put that in my head! I'm thinking that at the atomic level, no as the center is likely a void, but at the molecular level, yes. Of course, much like the premise of this thread, is it turning mathematically? There is no circumference at a diameter of zero and therefore no point of reference for "turning".

My major brain hurt is "How high is up". If you go in a perfectly straight line in any direction what happens when you go past the edge of the universe - more space? Put another way, no matter how hard physicists try, there is no end to space because what's on the outside of whatever they define?

marshg246 said:

there is no end to space because what's on the outside of whatever they define?

Click to expand...

Space may not be what you think it is.

Triton Thrasher said:

Space may not be what you think it is.

Click to expand...

Spoken like a physicist My point is what's outside whatever definition you use?

marshg246 said:

Spoken like a physicist My point is what's outside whatever definition you use?

Click to expand...

There's your problem...there can be no outside to the universe as we know it.it is beyond our everyday experience to conceive what can exist in a non dimensional space.

I have a headache, but the top illustration is not right.

Tornado said:

There's your problem...there can be no outside to the universe as we know it.it is beyond our everyday experience to conceive what can exist in a non dimensional space.

Click to expand...

There are physicists, and then there are THEORETICAL physicists!

The latter can be anything ranging from quite serious, to total clowns.

marshg246 said:

Spoken like a physicist My point is what's outside whatever definition you use?

Click to expand...

Things that don’t make sense to us.

Same sort of issues come up at very small sub-atomic scale as at very large bigger than visible “universe” scale.

The situation cannot be described in terms that we understand.

I’m in the wrong pub me !

Tornado said:

There's your problem...there can be no outside to the universe as we know it.it is beyond our everyday experience to conceive what can exist in a non dimensional space.

Click to expand...

One postulate is the multiverse. The trouble is, I can conceive what is non-existent outside the universe - more nothing. Even considering the multiverse or many universe theories, they all leave the fact that most consider negative space must be ignored. That which cannot be explained cannot exist is the basic thinking. But then not log ago, Black Holes were considered to be forever and now are not and were considered to be a singularity but clearly are not and it's not even proven that a singularity exists at their center (much like "does the center of a shaft turn").

Of course, what the hell do I know! Sixty years of thought and study have gotten me nowhere near understanding any of this better than anyone else - probably less than many!

I think Jerry's algorythm for related topics is getting confused also.

I recall in my undergrad physics classes, it was explained that a common misconception is that the universe is expanding into empty space from a central point...but in fact it is space itself that is expanding, and in all directions. There is no "center of the universe"...every point is moving away from all other points in all directions. Its like a muffin top, studded with raisins...all of them are moving apart as the muffin is baked and the batter rises.

So who would have a better understanding of all this - Albert Einstein or Stephen Hawking ?