In the above image you can see the large blue circle with the radius of $3$ and a smaller black circle wih the radius of $1$. The black circle does exactly two rotations around it's own axis and ends up at exactly the same place and angle it started with.
Now not all-sized circles end up in the same position after one rotation. It of course depends on their radius. So my question is:
If the outer (larger) circle has a radius $r_0$ and the inner
(smaller) circle has a radius of $r_1$, what is the smallest number
of rotations the inner circle has to do to around the the inside of
the outer circles to get the the same starting position and angle?
(sidenote: the red deltiod in the picture has nothing to do with the question)
Best Answer
The basic relation between the two circles is this: the point of contact travels the same distance along one circle as it does along the other.
So every time the inner circle returns to the same position, the point of contact has traveled a distance of $k \cdot 2\pi r_0$: $k$ times the circumference of the bigger circle. This is a $\frac{k \cdot 2\pi r_0}{2\pi r_1} = k \cdot \frac{r_0}{r_1}$ multiple of the circumference of the smaller circle.
So the inner circle will have the same orientation as at the start when $k \cdot \frac{r_0}{r_1}$ is an integer.
If $\frac{r_0}{r_1}$ is irrational, this will never happen, otherwise, it will happen after the inner circle goes around the outer circle $k = \frac{r_1}{\gcd(r_0,r_1)}$ times.