# Three wheels rotating at different speeds.

• Jan 23rd 2009, 12:10 PM
augmata
Three wheels rotating at different speeds.
Hello, everyone!

Suppose there were three wheels of equal size, rotating on the same axis, one point marked on the circumference of each wheel, and these three points lined up in one straight line. If the second wheel rotated twice as fast as the first, and if the speed of the third wheel was 1/π of the speed of the first, the initial line-up would never recur.

Is that true?
• Jan 23rd 2009, 12:56 PM
Mush
Quote:

Originally Posted by augmata
Hello, everyone!

Suppose there were three wheels of equal size, rotating on the same axis, one point marked on the circumference of each wheel, and these three points lined up in one straight line. If the second wheel rotated twice as fast as the first, and if the speed of the third wheel was 1/π of the speed of the first, the initial line-up would never recur.

Is that true?

Let $\displaystyle \dot{\theta_1}$ denote the rotational speed of wheel 1.

Let $\displaystyle \dot{\theta_2}$ denote the rotational speed of wheel 2.

Let $\displaystyle \dot{\theta_3}$ denote the rotational speed of wheel 3.

Hence!

$\displaystyle \dot{\theta_1} = \theta_1 t$

$\displaystyle \dot{\theta_2} = \theta_2 t$

$\displaystyle \dot{\theta_3} = \theta_3 t$

Now:

$\displaystyle \dot{\theta_2}=2\dot{\theta_1}$

Hence:

$\displaystyle \theta_2t=2\theta_1t$

$\displaystyle \theta_2=2\theta_1$

$\displaystyle \dot{\theta_3}=\frac{1}{\pi}\dot{\theta_1}$

Hence:

$\displaystyle \theta_3t=\frac{1}{\pi}\theta_1t$

$\displaystyle \theta_3=\frac{1}{\pi}\theta_1$

If they ever meet up again, then $\displaystyle \theta_1 =\theta_2=\theta_3$

This implies that

$\displaystyle \theta_1= \theta_2$

$\displaystyle \theta_1 = 2\theta_1$

It also implies that:

$\displaystyle \theta_1= \theta_3$

$\displaystyle \theta_1= \frac{1}{\pi}\theta_1$

How can a wheel pass through an angle, half that angle, and one '$\displaystyle \pi^{th}$' of that angle simultaneously?
• Jan 24th 2009, 09:20 PM
Quote:

Suppose there were three wheels of equal size, rotating on the same axis, one point marked on the circumference of each wheel, and these three points lined up in one straight line. If the second wheel rotated twice as fast as the first, and if the speed of the third wheel was 1/π of the speed of the first, the initial line-up would never recur.

Is that true?
It is true.

The first 2 wheels will line up whenever the first wheel reaches the starting point and not otherwise. So the first 2 wheels will line up only every time the first wheel has undergone a whole rotation. in this time, the third wheel will have undergone $\displaystyle 1/\pi$ rotations. Adding an integer number of $\displaystyle 1/\pi$ rotations will never produce a whole number because $\displaystyle 1/\pi$ is irrational. Therefore the 3 points will never line up again.