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Simon sent us this solution :
The obvious answer is that the outer circle (B say) rotating around
the stationary circle (A) revolves once but this is wrong. Looking
at the interactivity I could see that the mark on the circumference
started on the left of the outer circle and was on the left again
when the outer circle had only gone half way around. This means
that it must make two full turns as it rotates around the inner
circle. Now I need to explain why.
Let's start with each of the circles having a radius r. I thought
about the centre of the outer circle. When it makes a full circuit
of the inner circle the centre of the outer circle will have drawn
a circle of raduis 2r so it will have covered a distance of $2 \pi
\times 2r = 4 \pi r$.
So how many rotations has this outer circle made?
I next imagined the circle rolling along a line. How many
revolutions would be necessary for the centre to travel the
distance of $4\pi r$
In one revolution the centre will travel the same distance as the
circle (imagine a bicycle wheel) that is a distance of $2\pi r$. So
to travel a distance $4\pi r$ the circle would need to revolve
twice. This means the outer circle makes two full turns for every
single circuit of the inner circle.
Kevin of Langley Grammer School explained
what was happening for circles with different, as well as the same
radii.
The centre of the moving circle moves round the circumference
of a circle of radius $2r$, i.e. a distance $4\pi r$.
The centre of the moving circle moves a distance $2 \pi r$
when it makes one complete turn about its centre. Therefore when
the moving circle returns to $P$ it has made 2 complete
turns.
In this situation the ratio of the radii of the moving circle
to the non-moving circle was 1.
Let the non-moving circle have a radius of $r$, and the moving
circle have a radius of $nr$, so that the ratio of the radii is
$n$.
Therefore the centre of the moving circle moves along the
circumference of a circle with radius $(1+n)r$, i.e. a distance $2
\pi (1+n)r$. Therefore when the moving circle returns to $P$ it
will have made $$\frac{2\pi(1+n)r}{2 \pi nr} = 1+ \frac{1}{n}$$
turns. Therefore if the moving circle has a greater radius than the
non-moving circle then $n> 1$, and so $1/n$ would be less than
1, and so the number of turns would be less than 2. If the moving
circle has a smaller radius than the non-moving circle then $n<
1$, and so $1/n$ would be greater than 1, meaning that the number
of turns would be greater than 2.
It can also be seen that the moving coin makes at least 1
turn, regardless of the sizes of the circles, as 1/n is always
positive.
So, if the moving circle is half the radius of
the inner circle it will turn $1+2 = 3$ times
If the moving circle is one third the radius
of the inner circle - it will turn $1+3 = 4$ times.
Surprised?