Standing on Paper Cups
Why are you able to stand on three paper cups that
support a wooden board and not smash the paper cups and fall?
All materials have strength, derived from the bonds
formed by molecules etc. The shape can enhance the
Strength, as the cup is stronger than a flat plate, made of
The same Foam material.
But, strength has its limits, the 3 cups work for you , but could
they support your Dad?
Your question brought up another point the distribution of
The weight. The objective to evenly distribute the load to stay
Within the limits of the material.
You could try paper cups of the same size, they should be stronger
than the foam. Trial and error testing is what is used to
establish the strength of the material. From the testing,
rules of design are established.
Well, the cups stay normal-shaped and rigid until some collapsing
mechanism is triggered.
If someone standing on that board did a sideways wiggle or tried to
the sideways (shearing) force would probably trigger the paper cups to
fold and buckle sideways and collapse.
If the person holds properly still, at some larger weight maybe the glue
joint along one side of the cup would split,
then the cups could open wide and collapse.
Other than those, the cone or tapered-cylinder shape of the cups offers no
way to give in, at any weight,
until paper starts crushing on a small scale or tearing on a vertical line.
Most materials can be measured as to their
tensile strength, shear strength, bending strength, and compressive strength.
Compressive strength is the relevant number here.
The compressive strength of a block of wood or glued paper-fiber is more
than 5,000 psi (pounds per square inch).
So how many square inches of paper do you have holding up your weight there?
Suppose the paper cup's thickness thk= 0.010 inch, and the diameter D is
about 2 inches.
(Caveat- I lie a lot. You better measure the paper's thickness for yourself.
It might be only 0.005 inches.)
Circumference S = pi x Diam = 3.14 * 2 = 6.3 inch.
The paper cross-section area A = S x thk = 6 x 0.010 = 0.06 sq. inch
Ultimate compressive load L = area x strength = 0.06 sq inch x 5,000
= 300 pounds (per cup!)
So L x 3 cups = 900 pounds...
This says that if you put your whole class on the wooden board and nothing
else had gone wrong yet,
at 900 pounds one paper cup would tear left-from-right somewhere down one
side, to allow itself to collapse vertically.
Probably some other manner of buckling would occur before that.
But it shows that with the perfect structure and perfectly-distributed
many materials have surprisingly more strength than you usually imagine.
Can you think of ways to make it harder for the paper walls to fold in&out
and allow the cup to collapse?
You could probably hang your weight from a single paper cup glued between
two small wooden boards, too.
That would show the "tensile strength" of the paper.
Be sure it has got a limber rope from the center of each board, so there
is no shearing or bending force on the cup.
Do not suspend yourself higher than several inches, a distance from which
you can land on your feet safely
when the cups break totally unexpectedly.
I bet your 3 paper-cups are stronger, have one less mechanism of failure,
if they are between two wooden boards, and are thickly glued to each board
all the way around their rims.
That experiment is designed to show that paper, when made in the right
shape, actually has quite a bit of strength. You are normally unable to
stand on a paper cup because your foot will be placed just a bit unevenly
upon it, or will shift as you try to balance. By placing a board over three
cups, you have eliminated the sideways forces that would twist and bend the
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Update: June 2012