### Re: Why two Identical tapered cups placed one above other,is hard to seperate?

Date: Mon Jan 30 16:27:25 2006
Posted By: Joseph Weeks, Engineer
Area of science: Physics
ID: 1137691763.Ph
Message:
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What a great question.  To answer this question, we need to take a closer
look at nesting cups.  If you look at one of the cups from the side, you
can see that the cup is larger at one end, and then smaller at the other.
In terms of simple machines, the tapered cup is actually a cylindrical
wedge with the sharp end missing.  So two nested cups are really one wedge
within a second wedge.  This link: http://www.cosi.org/onlineExhibits/simpMach/sm1.swf
provides a good quick
review of simple machines including the inclined plane and the wedge.

The advantage of a simple machine, such as a wedge, is that it makes jobs
easier to perform by giving mechanical advantage.  In other words, an
inclined plane with a mechanical advantage of 5 would allow me to push a
heavy box up a slope by using a force five times less that would be needed
to lift the box, plus whatever additional force is needed to overcome
friction.  In this case, the plane increases 1 meter in height for every 5
meters in length.

OK, you noticed that I suddenly shifted from a wedge to an incline plane?
Well, a wedge is simply an inclined plane that I can move and shift around.
The tapered cups are inclined planes, or wedges, that generally have a

Let's say that the bottom of your cup has a diameter of 6 cm at the narrow
end, and a diameter of 8 cm at the top.  Let's further say that the cup is
12 cm tall.  The mechanical advantage of the cup as a wedge is 12 (the
height of the wedge), divided by 2 (the difference between the top and
bottom diameter) or 6.  That means that whatever force I apply to push the
cups together, that force is multiplied by a factor of 6.  So, if I apply
one newton of force in pushing the cups together, the resulting force
between the cups is six newtons.  So the first issue is how tapered the
cups are.

The key to your question is what happens when I try to pull the cups apart.
If vacuum were holding the cups together, unless the cups have an
air-tight seal, there is not much vacuum generated until the inner cup
moves with respect to the outer cup, generating lower pressure between the
cups.  So, if I moved the cups apart from each other very slowly to allow
air to fill the void between the cups, then the cups should separate more
easily than if I pulled them apart quickly.  But in my experience, that is
not the case; the problem is to get the cups to start moving in the first
place.  So the problem in separating the cups is one of static friction.

The formula for friction is F=uN where F=force, in this case the force to
separate the two cups, N is the Normal Force between two surfaces, and u is
the friction coefficient.  This link: http://www.carbidedepot.com/formulas-frictioncoefficient.htm
provides
friction coefficients between a number of solids.  You will notice from
this chart that the dry friction coefficient between two surfaces is much
higher than the wet coefficient.  So, let's say that you are washing
aluminum cups, and you put one on top of the other while there is a bit of
water to lubricate the two surfaces.  The lubricated coefficient between
two aluminum cups is around 0.3, and any force you apply to the cups will
wedge them together pretty well.  Then, if the lubricating layer drys out,
the friction coefficient is now around 1.35 or 4.5 times higher than when
it was wet.  So, let's say that you applied a one newton force in pushing
the cups together.  Because of mechanical advantage, the normal force is
around 6 newtons between the lubricated cups.  When you try and separate
the cups, you need to apply 6*1.35 or about 8 newtons to pull them apart,
just to overcome the static frictional forces.  So a lubricating film of
water or oil between the cups should make a world of difference in whether
it is easy or hard to separate the cups.

It is interesting to note that some materials, such as polystyrene and
teflon, have the same coefficient of friction whether they are dry or
lubricated.  Other materials, such as copper, iron, and aluminum, have a
large difference in the lubricated and dry coefficients of friction.

There may be other things that might cause your cups to stick together.
For example, if the outside cup is hotter than the inside cup when they are
nested together, the outside cup will contract as it cools, increasing the
normal force between the cups.  Also, if the outside cup has just been
removed from hot water and placed on the inner cup, there may be enough
water vapor inside to form a partial vacuum as the cups cool, pulling the
cups more tightly together.  Even though over time the partial vacuum might
go away (as air leaks in between the cups) the static frictional forces
would remain, making it more difficult to budge the cups apart.

So, in my opinion, the most important factors that determine how tightly
the cups will stick together include the taper of the cups, the materials
that are used in making the cups, and what materials are between the cups.

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