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'Five Circuits, Seven Spins' printed from https://nrich.maths.org/
Why do this problem?
This is a good problem for discussion and developing clear
visualisation and mathematical communication. It relates the angle
of rotation of a circle to a distance and is therefore of use in
exploring radians and the formula $s=r\theta$.
Possible approach
Students' abilities to visualise the meaning of this problem might
vary considerably. As such, this problem can appear to be difficult
until a clear approach to the solution is found. The behaviour of
the disc at the corners is likely to cause the most difficulty in
imagining the rotation. As a result, students might need to be
given a variety of visual devices to allow them to get started. For
example:
- Imagine looking down onto the tray and watching the disc rotate
about its centre.
- Imagine breaking the journey into a series of straight line
trips.
- Imagine that the disc is pinned down in the centre and the tray
is a track moved around the disc.
- Imagine that the edge of the disc is coated in ink. Which parts
of the tray would be coloured following a lap of the track?
- Roll a coin around a book and use the head on the coin as a
reference. Does the head rotate as it moves through a corner (i.e.
when moving from a horizontal to a vertical part of the the
journey).
This is the sort of problem which becomes much clearer once a
solution has been found. Once students have solved the problem they
should try to rewrite their answer and method as clearly as
possible, in a way which is both simple but complete.
It is possible to tackle this problem using degrees and the
formula for the circumference of a circle, but it is much simpler
to solve using radians and the formula $s=r\theta$.
Key questions
How far does the centre of a disc move in one revolution when
the plate is in contact with a straight edge?
What mathematics allows us to relate this distance to an
angle?
What units should we measure the angle of rotation in?
Why?
If the disc has rotated $7$ full times, how far must it have
rolled?
As the disc makes a single lap of the tray, what parts of the tray
will have made contact with the disc? How far is this?
Possible support
Consider the distance a bicycle travels when the wheels rotate
once.
Read the article
A Rolling Disc - Periodic Motion.
Possible extension
Try the problem
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