Flipping beermats

Experiment and observation
Calculations by hand
Using a spreadsheet
Ian's mystery beermat
Design your own beermat

What makes a 'catchable' beermat?

At the second videoconference on 9 March, you will be asked to make a presentation on the work you have done since the first videoconference. Here are some things you could research:

• an analysis of the 'catchability' and 'flipping' properties of square and rectangular beermats
• what difference does it make if the mat is not uniformly thick
• what difference does it make if the mat is not rectangular
• work out how Ian's mystery beermat is made
• design a good beermat of your own for flipping and catching

Experiment and observation

During the first VC, you looked at flipping and catching beermats, and how they move through the air. In your follow-up experiments, you will need to decide what statistics you need to collect. For instance, how often do you need to repeat each experiment. Are your results consistent - that is, if you flip a mat 10 times, do you get the same result when you flip it another 10 times? If not, can you work out why not, and do something about it? So the first stage of your project work is to decide on your experimental method:

• how can you flip the beermats in a consistent way?
• how many times do you need to repeat each experiment to get a consistent average number of catches?
• how can you minimise error?

Use the beermats provided or cut out strong card to make sets of flipmats in square and rectangular shapes of various sizes - you need to cut them out accurately, so that they are genuinely square or rectangular and their dimensions are exact for future analysis. Start by experimenting on them to see if they are all equally catchable, or if some sizes and shapes are better than others. If this turns out to be the case, what shapes are the most catchable, what shapes are the least catchable? What happens if you glue two mats together, so that you have a single mat with double the thickness? What about if you glue a strip down one side, so that the thickness varies? Use your experiments to build up a picture about what properties in a mat make it easy to flip and catch.

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Calculating moments of inertia

Once you have a range of experimental observations, you can start analysing the properties of the flipmats to see how these tie up with what you have observed. When you flip a mat into the air it rotates. Sometimes it rotates in a stable way - the same rotation continues until it lands. Sometimes it rotates in an unstable way - it changes how it rotates until either it finds a stable way to rotate or it lands.

When an object moves in a line, its movement depends on its mass. Objects with more mass move more slowly, for a given push, and take longer to stop. For instance, it takes more power to move a large lorry than a small car, and it also takes the lorry a lot longer to stop from a given speed. When objects rotate, their movement depends on something called their moment of inertia - this is equivalent to the mass, but takes into account how the mass is distributed in an object.

On the right, you can see a diagram of a flipmat, which is 15cm wide by 9cm long and made of card 0.1cm thick. A centimetre grid has been drawn onto the flipmat, and the central vertical axis marked in red. Each square is then labelled with the distance of its centre from this axis.

Now imagine this flipmat is placed on the edge of a table, with the red axis along the table edge, and it is then flipped. To calculate the moment of inertia for this, we need to find the moment of inertia of each cell, then add them all together. Suppose we look first at the bottom left cell. To find its moment of inertia, we need to know its volume and how far it is from the axis.

Volume = 1 x 1 x 0.1 cm³ = 0.1 cm³

Distance from axis = 7 cm

Moment of inertia = volume x (distance)² = 0.1 x 7 x 7 = 4.9 units

The moment of inertia of all cells along the two extreme left and right edges is the same, as they are all the same volume, and all 7cm from the axis. Cells in the next strip in have moment of inertia = 0.1 x 6 x 6 = 3.6 units, and so on. Note that although each cell will have the same mass as every other cell, their moments of inertia vary, depending on how far from the axis they are.

Now working out the moment of inertia for 100 cells separately then adding them all together would be very slow. It is quicker to use this method (where I stands for moment of inertia):

Have a look at this and make sure you understand where all the figures in the first line come from, and that you follow how it has been worked out.

So if we flip our mat about its short axis we get a total moment of inertia of 252 units. What happens if we flip it about its long axis? Do a similar diagram, and see what result you get. You should find you get a quite different value for the moment of inertia.

The diagram on the right shows how to mark out the flipmat when one of the dimensions is an even number. Here a vertical axis is shown on a flipmat which is 10cm by 9cm. The centres of the nearest cells are 0.5cm from the axis, and so on out to the edge cells, which are 4.5cm from the axis. Doing a similar calculation gives a value for the moment of inertia for this flipmat about this axis of 74.25 units.

What is the corresponding value for the horizontal axis? Compare the size of the moment of inertia with your experimental observations - does the moment of inertia need to be low or high for stable rotation? Look at differently sized rectangles and work out the moments of inertia for their long and short axes. Can you see any connections between the values you obtain, and the observations you made on how they move when flipped? Which axis gives more stable flight? Which axis gives the lowest moment of inertia?

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Using a spreadsheet

Once you've got the hang of working out these moments of inertia by hand, you might like to have a go using this spreadsheet - flipmats20x20.xls. This will enable you to experiment with flipmats of varying size, and also those where you stick extra sheets of card together.

The image above shows the top part of the spreadsheet, set up for a flipmat which is 15cm wide and 9cm long. Hovering the mouse above each of the headings will give a comment which explains what it is about - above, you can see the comment for the row heights. The image below shows the next part of the spreadsheet. You can find out what each of the headings is (if it has a red triangle against it) by hovering the mouse over it. The blue figures show values you insert to alter the length, width, and thickness of your flipmat, and the weight of the card from which it is made. The orange figures show calculations the spreadsheet makes (very similar to those you did by hand, but in both directions) to find the values of the moments of inertia. The two figures which correspond to those you have worked out are I_xx and I_yy. The value of I_xx shown here is 252 units, the same as we found above. (You can ignore everything on the spreadsheet apart from the top yellow/blue cells and the blue and orange results cells below these. Everything in grey is where the spreadsheet performs calculations.)

To start with, see if you get the same results for the calculations you did by hand. Once you are satisfied that you know how to operate the spreadsheet, and that you get the same results either way, you can use the spreadsheet to test out all kinds of flipmats. To add extra strips of card to the flipmat, just double the number in the individual row heights or column widths (yellow cells).

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Ian's Mystery Beermat

You will be sent a 'mystery beermat'. Your task is to work out how it's made, using your research so far.

Design your own beermat

Now that you have all this experience on what makes a catchable beermat, you could design your own. You can choose any shape you like, any material you like - the only criterion is that it's good for flipping and catching!

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