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A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?

Age

14 to 16

Challenge level

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving

Being curious Being resourceful Being resilient Being collaborative

Think about a coin with radius $1$ unit in the corner of a box. I roll the coin along the side of the box, making sure it doesn't slip.

How long would the box need to be for every point on the coin's circumference to touch the box as I roll it along?

Now imagine the coin rolling along two edges of a box like this:

If the two edges are the same length, how long do they need to be to ensure every point on the circumference touches? What if they're not the same length?

Now imagine rolling the coin around on a rectangular tray so that the edge of the coin is always in contact with the inside wall of the tray, and so that it always rolls and never slides. What proportion of the circumference touches the tray as it makes one complete circuit of a $4$ by $3$ tray?

Can you draw any diagrams to help to explain your thinking?

What is the smallest tray you can find where all of the circumference touches the tray at some point?

Well done Sam from Kings Wimbledon for your solution:

1) Coin radius 1: How long would the box need to be for every point on the coin's circumference to touch the box as I roll it along?

The left bottom quarter of the disc is not in contact, which is a radius' width. Then the whole circumference must roll along the base of the box, until another quarter of circumference is not in contact in the bottom right and corner.

So the disc must be two radii (from when the disc is at the sides) plus one circumference long. This is $l=1+1+(2\times\pi\times1)=2(1+\pi)=8.28$.

For any radius $r$, the length of the box is $l=r+r+(2\times\pi\times r)=2(r+\pi)$

2) If the two edges are the same length, how long do they need to be to ensure every point on the circumference touches? What if they're not the same length?

Call the length of the edges $l$. Then the amount of the circumference that touches each edge is $l-r$, as at the corner there will be a quarter of the circumference that will not be touched.

Obviously if $l-r> c$ then the whole circumference will touch both edges, so we want to find the minimum length an edge can have to touch the whole circumference. So you need

$2\times(l-r)+\frac{2\pi r}{4}=2\times(l-r)+\frac{\pi r}{2}=2\times(2\pi r)$

$2\times(l-r)=\frac{7}{2} \times \pi r$

$l=\frac{7}{4} \times \pi r +r$

$l=r\times(\frac{7}{4}\pi + 1)$

That is, $\frac{7}{8}$ of the circumference, and then plus the extra radius that is missed in the corner.

For edges of different lengths, obviously if both lengths are at least as long as $r\times(\frac{7}{4}\pi + 1)$ then it will work, and the whole circumference will be covered.

But the more interesting question is, if one of the edges is shorter than this figure, how long will the other need to be to make up for it, and ensure the whole circumference is still touched.

The answer is that if one length is less than $r\times(\frac{7}{4}\pi + 1)$, then the other length must be the entire circumference plus the extra radius lost in the corner. This can be seen from labeling four equidistant points on the disc and studying their movement as the disc rotates around the edges.

3) Rectangular tray: What proportion of the circumference touches the tray as it makes one complete circuit of a 4 by 3 tray?

Initially just looking at one side of the tray we can see how much of the circumference touches this. We can find the arc length that touches on any side by looking at the length of the side minus two radii from the corners. By noting the formula $c\times f=a$, the formula relating the cicumference, fraction of the circumference and the arc length of that fraction, we find $f=\frac{l-2r}{2\pi r}$.

So for a disc of radius $1$ in the $4 \times 3$ box, the proportion touched on the first side is $\frac{4-2r}{2\pi r}=\frac{1}{\pi}$, and then a quarter is missed, then $\frac{3-2r}{2\pi r}=\frac{1}{2\pi}$ and miss a quarter again. Then repeat for the last two sides. So after two sides $\frac{1}{\pi}+\frac{1}{2\pi}=\frac{3}{2\pi}$ of the circumference has been touched, and we have moved $\frac{3}{2\pi}+\frac{1}{2}$ of the circumference. Which is a distance of $3+\pi$, just less than the actual circumference of $2\pi$. So then the new bits that will be touched are $2\pi-(3+\pi)=\pi-3$ on each side from now on. So on the last two sides $2(\pi-3)$ of the circumfernce is touched, which is $\frac{2(\pi-3)}{2\pi}=\frac{\pi-3}{\pi}$ of the circumference. Adding this to the proportion touched on the first two sdies, the total proportion touched is $\frac{3}{2\pi}+\frac{\pi-3}{\pi}=\frac{2\pi-3}{2\pi}$ of the circumfernce. Which is a total arc length of $2\pi \times \frac{2\pi-3}{2\pi}=2\pi-3$.

Then as for every side the disc touches it touches anew $\pi-3$ of the circumference it will eventually touch the entire circumfernce.

Why do this problem?

This problem challenges learners to use their visualisation skills to gain an understanding of what happens when a coin rolls inside a rectangle, while providing a context for practising methods of calculation of circumferences and arcs.

Possible approach

Start by asking the learners to consider the question "How far forward would a bicycle travel if its wheels turned through one complete revolution?" Then show the first part of the problem, and ask what the difference is between the bicycle problem and the coin in the box problem - this should make it clear that the corners are key.

Now show the second diagram. Intuition may suggest that if the coin is travelling on two sides, each side would not need to be as long in order to get the whole circumference to touch, but having a corner where part of the circumference doesn't touch makes things interesting! Learners could draw corners on paper and roll a cardboard circle along them, highlighting on their circle the parts that touch and the parts that don't.

Now look at what happens when the coin rolls around the inside of a tray. Ask the learners to discuss in pairs whether all of the circumference of the coin will touch on one circuit of a $4$ by $3$ tray, and then share ideas.

One way of recording what happens is to draw a line $14$ units long (perimeter of tray) and mark all the key sections such as corners, and the points where the coin has made a complete revolution.

Key questions

How far does the centre of the coin travel as it makes one revolution?

What happens at the corners?

Possible extension

Try the problem Five Circuits, Seven Spins.

There are more ideas, explanations and problems to work on in the article A Rolling Disc.

How many times does the disc rotate about its own centre when it makes one revolution around the tray?

What happens if corners are not $90^\circ$?

Possible support

Physically manipulating a circular object inside a frame can make it clearer what's happening at the corners.

The problem Rolling Around may be a good starting point.

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Number and algebra

Geometry and measure

Probability and statistics

Working mathematically