A conundrum for you

Hi all,

First off I'd like to say that this is a rather mind-boggling matter and I've provided some badly drawn diagrams to try and help explain the situation, it's a bit of a read and it's certainly given me a great deal of troubled thought and I hope that some clever person can explain it for me, because I am utterly baffled.

Without further ado:

Consider a real, physical machine that seems to perform the impossible in a way that requires the involvement of infinity, imagine a simple sided shape like a Hexagon, imagine a symmetrical brass Hexagon rolling in the rather bumpy way of a non-circular wheel along a track.

Now let’s fix another Hexagonal wheel, around half the size, in the middle of the first wheel. Owing to a bit of cunning engineering this is also running along a track. The easiest way to imagine this is to think that the smaller Hexagon sticks out of the side of the big one so that the small Hexagon’s track is parallel to the other, but raised up and offset to one side. From the side it looks something like Figure 1 and from the end in profile view it looks like Figure 2.

Figure 1

Figure 2

Now we’re ready to begin. Rotate the big Hexagon one-sixth of a turn, so it goes from having one side flat on the track to having the next side flat down. Now check what’s happened to the smaller Hexagon. It too has rotated by one flat side- yet somehow the whole wheel has moved on by the size of the big wheel, much further than the length of the small Hexagon’s side. The leading edge of the small Hexagon’s side touching the track has moved forward the whole length of the big Hexagon’s side. How? It’s simple enough – as the big Hexagon’s side rotated it lifted the small Hexagon right off the track – it left a gap in the motion down the track of the small Hexagon equivalent to the extra distance as shown in Figure 3.

Figure 3

So far, so good and it doesn’t matter how many sides the wheel has, the same holds true. But now let’s look at the same picture with a pair of circular wheels. Presumably things are much the same.
Once again we have two wheels fixed together, one smaller than the other, each running on its own track, so that sideways on they look something like Figure 4 and on profile they look like Figure 5.

Figure 4

Figure 5

Now let’s rotate the big wheel a quarter turn. It’s no surprise that the smaller wheel makes a quarter turn too. Just as with the Hexagon, the point at which the small wheel touches the track has now moved forward along its own track the same distance as the point where the big wheel touches its track.

But there’s a real problem here. Unlike the Hexagonal wheel, the circles are smooth. They are continuous pieces of metal. If we made the machine well enough, there was no time when either wheel was out of contact with the track. So imagine following the point at which each wheel touches the track as we make the quarter turn. The big wheel has measured out a quarter of its circumference along the track. And the small wheel had a quarter of its, much smaller, circumference touch its track. Yet somehow, without ever seeming to leave the track, the smaller wheel has jumped on the full distance that the big wheel travelled. How did it manage to get so much further along? Where was the gap? (Figure 5)

Figure 5

If you aren’t quite sure what the fuss is about, imagine that we had the wheel in its starting position and undid a quarter of the rim of both the big and small circle. Flatten it out on the tracks and you will find that the big wheel’s undone rim covers the distance the whole wheel moved. But the small wheel’s rim is too short. How did it manage to cover the extra ground without ever coming away from the track the way the small Hexagon did?


Answers on a postcard.

If I am seeing this correctly, the small wheel does not leave the track, but (and assuming it fixed to the big wheel), moved along the track controlled by the larger.

Thus the small wheel will skid in it track as the larger revolves?
Apparently not (I'll reveal what my book says in time and if true it is quite astounding) tho I'm hoping some clever clogs who studies physics will know the answer and perhaps even come up with something else. As aforesaid, I only know the basic layman's terms so it doesn't take much to get me lost, physics is not my field but it is fascinating.

No, it's a rather messed up conclusion tbh, that's why I wanted to ask someone on here because I even doubt the book. I'll give it a day or so and see if this thread takes off and then give the answer.

I'm working on it! I'm currently experimenting with marking the circles. This would obviously then show that a full 90 degree turn would have the marks in the same place, and if not on tracks the small circle would just follow it, so is it simply that the tracks are there to mislead us into thinking some hocus pocus is at work when really it's just because the smaller circle is smaller it appears to travel further when really it doesn't...
Tricky, isn't it? It's still driving me a bit mad too. I've decided to draw another crap diagram in order to try and get my head round it cos I'm a bit thick when it comes to maths and I prefer to try and understand it like the ancient greeks, AKA I find it easier to look at pretty pictures.

So the problem is that the two circles move 90 degrees as shown in red, and they cover the same distance along the bars. But the smaller circle has a smaller surface area (shown in blue) than the larger circle's surface area (also shown in blue). How can the smaller circle travel the same distance along the bar as the larger circle when it's surface area is smaller?

That is the original question but phrased a bit differently... I think.



Book Reviewer
There seems to be a difference between the tracks which is proportional to the size of the wheels. All about circumferences cut in two and laid flat, me thinks.

I want to use the word tangent but I am less of a wiz than you, sir. So I could be number 1 dolt.
Best guess so far but I'm afraid it's not the answer. I'm not even sure I agree with the book tbh, that's why I posted the problem on here to see if someone can make any sense of it.

There's a major hint in the first paragraph.

It doesn't, or at least it might not. The combined wheels will only travel the length of the larger wheel circumference if there is greater friction between that wheel and the ground than there is between the raised platform and the smaller wheel.

If there is more friction with the smaller wheel than the bigger, the whole contraption will only move the distance of the smaller wheel circumference (for a full revolution).

As suggested, the other wheel slips, or is dragged. The idea that the hexagon is lifted is also a misnomer, it too is dragged as there will always be some point of it in contact with the rail.

That is why there is a differential on your car, to allow the wheels to turn at different rates when going round corners.
Is this not similar to the one about the top of a ships mast and it's relation to the deck of the ship on a circumnavigating trip around the world?

The point being that the top of the ship's mast must travel faster than the deck, ergo it's about time and distance.
First reply was right, the smaller wheel will skid as well as roll.
BrandySoured has it right. To achieve the same horizontal distance, the outer wheel has to turn through a greater circumference than the inner wheel because the track is further from the axle. The original situation with the hexagons is a red herring intended to make us believe that the small hexagon "keeps up" with the big one because it has lifted from the track when lifting from the track is more of a coincidence.
I've always wondered about this, it's exactly what happens to the cluster when the rear wheel of your ten speed goes round: Both the biggest cog, and the littlest cog (and all the rest inbetween) go one revolution...the same linear distance as the circumfrence of the wheel itself. Which is much bigger than any of the cogs. Argh.
I've always wondered about this, it's exactly what happens to the cluster when the rear wheel of your ten speed goes round: Both the biggest cog, and the littlest cog (and all the rest inbetween) go one revolution...the same linear distance as the circumfrence of the wheel itself. Which is much bigger than any of the cogs. Argh.
That I can understand!
Thread starter Similar threads Forum Replies Date
vvaannmmaann The NAAFI Bar 11
Proximo The NAAFI Bar 78
Ozduke Officers 26

Similar threads

Latest Threads