Mathematical explanation for rim point moving backwards.

Question

The paradox is given in the chap. 1 of the book titled : Mathematical Fallacies and Paradoxes, by Bryan Bunch; as given here.

The book explanation has no mathematical formulation, say if states the path in terms of cycloid (also shown in Fig. 1.5, on page #7); then there is no formula for the same.

Also, using that the nearly flattened ends of the curve for inner circle's point B could have been given a mathematical explanation.

Further, there should be some mathematical explanation & quantification for the reverse loop attained by the fastest point on the rim, as shown in Fig. 1.7 on page #9, as also shown below.

Also, I am not clear if the reverse path is restricted to only the fastest point, or decreases quantitatively with each inner point till the center.

Answer 1

There are two motions involved.

One is the horizontal movement, another one is the clockwise rotation.

Let the initial position of the center of rotation be $(0, r)$, at time $t$, its location is $(vt, r)$. This point is not rotated.

Let $\gamma$ be a real number that will indicate the distance from the center for points that are initially at the same vertical position as the axis and let $\omega$ be the angular velocity. The locus can be written down as

$$(vt-\gamma r \sin (\omega t-\theta), r - \gamma r \cos (\omega t-\theta))$$

If we differentiate the first coordinate, we get

$$\frac{dx}{dt}=v-\gamma r \omega\cos (\omega t-\theta)=\omega \left(\frac{v}{\omega}-\gamma r \cos (\omega t-\theta)\right)=\omega y$$

Hence $\frac{dx}{dt}<0$ if and only if $y<0$.

That is any point that can go below the track will experience a negative velocity in its trajectory.

Answer 2

The figure for which you're seeking a mathematical formula is called a prolate cycloid. The Wolfram mathworld article just linked to gives the parametric equations of this curve for a rolling circle of radius $\ a\ $, with the point $\ A\ $ at a distance $\ b\ $ from the centre: $$ x = a\phi - b \sin\phi\\ y = a - b\cos\phi\ . $$ The parameter $\ \phi\ $, here, is the angle (in radians) through which the radius from the centre of the circle to the point $\ A\ $ has rotated, since that point was at its greatest negative extent in the $\ y\ $ direction. If you differentiate $\ x\ $ with respect to $\ \phi\ $: $$ \frac{dx}{d\phi} = a - b \cos\phi\ , $$ you can see that $\ \frac{dx}{d\phi} < 0\ $, whenever $\ cos\phi > \frac{a}{b}\ $. If $\ \phi_0\in \left(0,\frac{\pi}{2}\right)\ $ is the unique angle in the first quadrant satisfying $\ \cos\phi_0 = \frac{b}{a}\ $, the point $\ A\ $ will be moving backwards whenever $\ 2n\pi-\phi_0 < \phi< 2n\pi +\phi_0\ $ for some integer $\ n\ $.