# Thread: Circle again!!-- the nature's trace??

1. For this discussion to begin, you all must know what is a "frustrum of a cone".. It almost looks like a cone whose apex has been cut out.. and now, it looks like a 'cylinder'whose one base end is small and the other is big..

It all happened when i unknowingly hit a steel glass (the one used for drinking, which is in the shape of cone's frustrum), which then went on rolling on the floor.... But wait.. i noticed something keen.. the rolling glass didnt go straight, but it rolls in an arc like fashion... And i realized, that this motion, if made to complete, it forms a circle..!! the arc motion is extrapolated to form a circle... CIRCLE !! It again makes me go mad and think deep..

This happened some 4 years before, after noticing which, i formulated a physical theory and worked out some mathematical equations and values, which i tested with a limited number of observations, and proved correct myself.. but unfortunately, my papers have missed... and i didnt care it for long time because i've been forced to join medicine by my parents, despite my aim to become a theoretical physicist... Now i feel that however small a discovery, it should be brought out, to be tested, and be published.. And i feel bad for missing those papers.. Anyway, i could reconstruct it, but takes a while.. and hope i wil do it.. as for now, i would explain how that phenomenon happens.... the mathematical part, which i had to reconstruct, and hence wont deal with it..

In a cone's frustrum, it has two bases, both in a circular shape, but one circular base is smaller than the other.. (better you imagine its picture)
And here in this case, these two circular bases are attached as a part of cone's frustrum and hence these two circular bases should roll(on the floor) simultaneously, rather than independently...

Consider two SEPARATE circular wheels, one larger and the other smaller. The larger wheel, in one complete rotation, covers a larger distance (on the road). but the smaller wheel, in one complete rotation, covers only a smaller distance. Inorder to reach the distance covered by one rotation of larger wheel, the smaller wheel has to perform multiple rotations (the number of rotations required depend upon the size variation between the two wheels).

But in an object with the shape of cone's frustrum, these two circular bases(one big, the other small), are continuous and attached with each other, and hence cannot perform different number of rotations, as in the previous wheel experiment.. Hence, when this object is rolled on a flat surface, its two circular bases have to face a 'MUST' obey condition:
1) They both should perform same number of rotations and should move simultaneously.

By this condition, it results that 'larger base' of the frustrum would cover a larger distance and the 'smaller base' would cover a smaller distance..............., but how an 'object (frustrum)' could cover two different distances simulataneously.

The answer lies in CIRCLE.. This object rolls on a circular path. But Why should this object choose a circular path??

In this circular path, there are two concentric circles (inner & outer)...!! The 'smaller base' would cast a smaller circle(inner circle- smaller distance), and the 'larger base' at the other end, would cast a larger circle(outer circle- larger distance). these two happen simultaneoulsy, and the whole object also moves along a circular path.. By doing so, the conditions are fulfilled.
* both the 'bases' perform equal number of rotations
* larger base covers a larger distance, and smaller base covers a smaller distance
You can't fulfill these conditions in any other motion, but only in a circular path..!!

Based on the difference in size of its two bases, the object may cast a smaller or larger circular path..
If the difference is too high, the object would cast a smaller circular path.. if the difference is small, it would cast a larger circle.. and if the difference is too small and insignificant, then you would get a very very large circle, which on short course, appears to be a straight line..
As the difference approaches zero, the trace approaches to become a straight line... And if there is no difference, then it would cast a straight line..!!

You could see a cylinder roll along a straight path, it is because the two bases are equal sized(difference=0)... But if there is difference in size between the bases, it should go along a circular path, however small the difference may be...

Mathematical formulations of the theory needs to be reconstructed.... hope would post them in near future.....

2.

3. But wait.. i noticed something keen.. the rolling glass didnt go straight, but it rolls in an arc like fashion... And i realized, that this motion, if made to complete, it forms a circle..!! the arc motion is extrapolated to form a circle... CIRCLE !! It again makes me go mad and think deep
Congratulations ! You noticed something that most people would have figured out in elementary school...

4. Come on friend.. but i did not stop with just observation.. no matter when an observation was made.. what really matters is that in which way you observe it, and whether it ended simply as an observation, or it formed an hypothesis..!! And you could not argue that newton realized apple fell at a late age, but we all know it at a very young age..!

5. Mathematical formulations of the theory needs to be reconstructed....
Looks like a trivial application of simple geometry (hint: similar triangles).

I suppose your enthusiasm is commendable. But maybe try and keep your posts a bit shorter and stick to the point (as far as there is one).

6. Thanks friend.. I might have cut short a large part, if i could use pictures... and that is why i find difficulty in transferring the information..

7. Originally Posted by arun karthik
Thanks friend.. I might have cut short a large part, if i could use pictures... and that is why i find difficulty in transferring the information..
We do not need any pictures for this; it's elementary Euclidean geometry - there is nothing new or unknown here.
I really don't get what your point is.

8. @markus: i think you are more of an argument rather than a discussion.. And i agree that u are present day einstein.. :P

9. I agree with Markus. This is a trivial and rather obvious fact.

Simplify it to a single wheel on an axle. If you lay the end of the axle on the ground, the wheel will go round in a circle with that point as the centre. If you put a smaller wheel on that end, then it raises the axle up and both wheels will go round circles centred on the point where the axle would end if you extended it. <yawn>

10. And i agree that u are present day einstein.. :P
If that was the case I wouldn't be on this forum right now, but out there doing research and publishing papers. Alas, as things stand, I am just an average guy with an interest in the sciences.
I am sorry if I did offend, however, what you are presenting here is elementary geometry. You do not need to be Einstein to understand that a cone segment will trace out a circle if rolled on the ground - that's completely trivial. You tried to make it sound as if it's a major new discovery, and that a "theory" needs to be spun around it; that's quite simply not the case.

If you want to make this non-trivial, then try to scale up this experiment, and conduct it in a non-stationary curved space-time background. You'd be surprised at just how non-trivial things become then...

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