2 circle with different radii covers same path on same number of turns... crack the fallacy

Question

I found this puzzle from an old issue of a science magazine জ্ঞান ও বিজ্ঞান ("Knowledge and Science")(August, 1977); submitted by Amal Dash, a teacher of Scottish Church school, Kolkata; as mentioned in the magazine.

Incidentally; the answer wasn't published there.

The original publication did not contained any sketch. I've added a few to make it easily readable.

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A wheel has an outer radius r₁ and an inner radius r₂. The wheel is rolling on a floor.

Normally, r₁ > r₂.

Before the wheel was being rolled, the topmost (from ground) point of outer margin (dot A) and inner margin (dot B) was marked with color dots.

After rolling; the wheel was stopped in a situation, when the 2 colored dots came upwards. So, in the mean-path; the wheel turned complete (integer) turns.

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Since the both circles on the wheel revolved n-complete turns; then point A (on big circle) should have been shifted 2nπr₁ and similarly the point B (on small circle) should have been shifted 2nπr₂.

But if so; since we can see after complete turns the point A and B came to similar orientation as initial, and so dots A and B shows same shift from their own initial positions in horizontal direction; so 2nπr₁ should be = 2nπr₂; ie. r₁ = r₂ . And we know very well that r₁ ≠ r₂.

So, where lies the fallacy? what was going inside the circle?

Answer 1

This may be a better question for the Math SE. Having said that ....

The fallacy is that the inner circle is not tracing out its own perimeter - it's along for the ride. You can see how quickly this breaks down if you consider the limit as $r_{\small_2}\to0$ that if both circles were independently moving $2n\pi r$ for their respective radii, as suggested in the fallacious argument, that this would leave the center-point of the wheel essentially motionless while the outer rim of the wheel moved as expected—a clearly nonsensical situation.

Since the inner surface of the wheel is not independent of the outer surface of the wheel, they both rotate together, but only point A on the outer circle is tracing out the perimeter of its $2n\pi r_{\small_1}$ circumference.

Answer 2

No different answer than User@Rubio; but a little bit graphical addition.

For me; the fallacy was a subconscious illusion that, as-if the wheel's movement does not depend upon which portion of it is touching the ground or track. Such as; as-if the wheel should have same movement whether it run as in OP or as in Fig.4 of following diagram. But that is NOT. _{And the best way to clear up this illusion is; to visualize and compare the Cycloids generated by the different spots on the wheel. In the following diagram (fig. 1 to 5); I did a try with simplest mensuration formulae and MS-Paint, considering r1= 40 pixels and r2= 20 pixels (as shown by horizontal and vertical ruler of ms-Paint on maximum zoom). In fig 2 and 3; I took 8 successive configurations (each differ 90 ° rotational phase). (In fig. 5 actually more samples were required, but not shown to keep the diagram readable.) . In each situation; at first the path moved by the wheel per 90 ° rotation has been calculated, and on that interval the wheel is repeatedly pasted.} The point B in Fig. 2 (Like OP) and fig. 3 (where a wheel of r2 radius, is running); are generating 2 different cycloid path. In Fig-1 the cycloid generated by B is horizontal-direction much more stretched, that is eating up that extra length. Fig. 6 Shows another condition, if the whole wheel (with point A and B); could be run on the inner circle (as in Fig. 5) we would get the same cycloid as Fig3 for inner circle; but for point A, a more special cycloid found, which is loopy and compressed (along horizontal direction) than that of fig.2. In Fig. 5 The colour variation (red to yellow) in larger circle is to cleanly trace the looping-back.

P.S. The use of ms-paint curve-tool done much on eye-gaze. so certainly there exists error. However all sorts of suggestions and criticism are welcome.

Answer 3

This is better understood if you first consider what is happening to the centre (I'm British, hence the original spelling) of the circle. Applying the $2 \pi r$ approach clearly doesn't work here as the circumference of a circle with no radius is zero. However, the axle of a wheel on a vehicle moves without describing any curved path.

Similarly, the point on the smaller circle is moving - partly owing to its placement on a rotating body and partly owing to the movement of the whole wheel - the contribution of which is seen by the axle's movement.