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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