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Mysterious marbles

You will need:

  • Marbles of two different sizes.
  • 2 plastic glasses of identical size
  • Measuring beaker and water.
  • Pad, pen and paper.
  • Vernier callipers

Instructions


1. Fill one glass up to the brim with the small marbles, counting each one as you put it inside.

2. Repeat using the larger marbles in the other glass.

3. Fill both glasses full of water.

4. Gently tip the water of one container into the beaker and note the volume.

5. Repeat for the other container. Are the volumes of the glasses the same?
glass-with-marbles

There Is Fair Amount Of Geometry In This Problem.

 
Allowing for you filling the glass up to exactly the same height, i.e. just about to overflow, you should have found that both glasses hold the same amounts of water.

The reason? The spaces around the large marbles are larger, so you would expect that more water should be able to fit into the glass. However even though the glass with the smaller marbles had smaller spaces to fit the water, there were many more of them. It turns out that the total volume of spaces between the the two glasses works out to be the same.

Now, you've got to prove it! Use the following to help you out:
 
sphere-volume You need to work out the volume of the marbles within the glasses.
Marbles are spheres, so use the volume of a sphere can be calculated:

Volume = 4 / 3 x Pi x r3

where Pi = 3.14 and r is the radius of the sphere.
You can get the radius by using vernier callipers.

Dont forget your answer is a cubic measurement eg. cm3.
Now multiply the volume of each marble by the number of marbles used.

Now you have to workout the volume of the glass. Really, all this is the amount of water that filled the glass + the total volume of marbles within the glass. But, for the sake of 'previous' your experiment using mathematics, you can do the calculation to get a prediction of what the volume of water was going to be before you even do the experiment.

Each glass is going to be different, so the sake of making the calculation easier, let's assume that your glass is a cylinder
 
cylinder-volume The volume of a cylinder can be worked by calculating the area of the base and multiplying that number by it's height.

So, the base of the cylinder is a circle, therefore the area of the circle can be worked by the following:

Area = Pi x r2

where Pi = 3.14 and r is the radius of the circular base.
Dont forget your answer is a cubic measurement eg. cm3.
Now just multiply this number by the height of your glass

Now you know the total volume of the glass and the total volume taken up by the marbles.
So, the final calculation to find the volume of water surrounding the marbles is:

Volume of water = volume of glass - volume of marbles

Troubleshooting:

How accurately did you measure your marbles and glass?
Did you fill both glasses to the same equal height - of course, in practice it would be very hard to be exactly accurate.

This is a good demonstration of experimental design often showing differences between theoretical results and practical results, known as theoretical and practical yields.

Your job is to work out why your results differ to what you have on paper!
 
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