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Just Rolling Round

P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

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Coke Machine

The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design. Coins inserted into the machine slide down a chute into the machine and a drink is duly released. How many more revolutions does the foreign coin make over the 50 pence piece going down the chute? N.B. A 50 pence piece is a 7 sided polygon ABCDEFG with rounded edges, obtained by replacing AB with arc centred at E and radius EA; replacing BC with arc centred at F radius FB ...etc..

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Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

Contact

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

When the disc is in the corner there is one quarter of the circumference (that is a length $2\pi/4 = \pi/2$) which does not come in contact with the edge of the tray. As the disc rolls round one circuit:
an arc of length $2$ units comes into contact,
then an arc of length $\pi/2$ gets missed,
then an arc of length $1$ unit comes into contact,
then an arc of length $\pi/2$ gets missed,
then an arc of length $2$ units comes into contact,
then an arc of length $\pi/2$ gets missed,
then an arc of length $1$ unit comes into contact.

contact

Some points never touch, other points touch only once, other points touch twice.

As an extension to this problem, can you now work out how much of the circumference of the disc never touches the edge of the tray?

The article 'A Rolling Disc' discusses variations of this problem.