Question
Introduction
If you
think the answer is 2 cm (since the
wheel rotates two rounds), you are wrong!
This tricky question is taken from a previous year Asia Pacific Mathematical Olympiad for Primary Schools competition. In
the original question, there was no colouring. I added colours just to make the distinction
between the wheel and the track a bit clearer.
Visualisation
There is
some subtlety in this question. It is
something we often do not notice unless we really think hard about it. To get a handle on what is really happening,
let us imagine we made a mark on the
wheel in its original position, indicated by a blue dot at a twelve o’clock
position. Let us imagine rolling the wheel clockwise. Note that the centre of the wheel will move
in a bigger circle in an anti-clockwise (American:
counter-clockwise) direction within the orange circular track. Actually it does not matter which way you
roll, the centre of the wheel goes round the centre of the track in a direction
opposite to that of the turning wheel, which always remains in contact with the
track.
Since the blue dot on the wheel turns two rounds, by
the time the wheel reaches the bottom half, the blue dot must again be on top
or at a twelve o’clock position. Note
however, the point of contact between the wheel and the circular track
(indicated by a green dot whenever possible) is not the same as the blue dot! The green dot is actually a dynamic dot (it is not always the same
point on the wheel) whereas the blue dot is always the same dot on the wheel
but the wheel is being rotated.
Note that at the halfway point, the green dot is at
the bottom of the wheel while the blue dot is on top of the wheel. Notice also that, relative to the wheel, the
green dot goes in the opposite direction as the blue dot and they actually
crossed over somewhere along the way!
Actually the green dot has made one-and-a-half turns with respect to the
wheel. Remember that the green dot is
not a fixed point on the wheel, but it measures how much the wheel and the
track have been in contact.
As the wheel continues to roll back up, the green dot
makes another 1½ rounds.
So altogether the green dot moves through 3 rounds
while the blue dot rotates only 2 rounds around the centre of the wheel! The green dot is the one that matters.
Solution
circumference of track (measured by green dot) : circumference of wheel
= 3 : 1
Since radius is
proportional to circumference, the radius of the track is 3 cm.
Remarks
Yes, the
solution is that short. But the thinking
behind it is profound. But do we need to
draw all the diagrams as in the visualisation above? I did that to explain to you. Actually I imagined it in my mind. If imagination is difficult, you can act it
out by drawing a big circle and then using a small coin to simulate the
rotation around the track. I actually
drew a rough sketch by hand to convince
myself that my thinking was accurate.
Notice that
I used a proportionality argument. If
you know how to use the concept of proportion, you can make the working short
and sweet. There is nothing wrong in
using the formula circumference = 2p ´ radius.
This formula just says that the circumference of a
circle is proportional to its radius, and the constant of proportionality
is 2p. Your working
would look like this
circumference
of track = 3 ´ circumference of wheel
2p ´ radius of track = 3 ´ 2p ´ radius of wheel
After cancelling out the 2p, you would get
radius of
track = 3 ´ radius of wheel = 3 cm
You get the same conclusion, but using the
proportionality method, you do not need to bother about the 2p.
H01. Act it out
H02. Use a diagram / model
H04. Look for pattern(s)
H05. Work backwards
H06. Use before-after concept
H09. Restate the problem in another
way
Suitable Levels
* Primary School Mathematics Olympiad
* anybody game for a challenge relating to
imagination, circumference and lengths
