# Five Circuits, Seven Spins

##### Age 16 to 18 Challenge Level:

### 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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