|MadSci Network: Physics|
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 high mechanical advantage. 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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