Modeling Plastic Cup Physical Behavior

[Bobbadillio]

I've noticed that plastic cups exhibit some interesting physical behavior.
Some paper cups(maybe all actually, I'm not sure) have concave bottoms, and when the liquid in the cup gets high enough, the forces from simply raising the beverage from the table to my mouth create enough force to pop the concave region into a convex one, and then it pops back to the concave position when it's at rest. This is a lot like the popping tops of baby-food cans, where once you've opened them, you can press down on the center and it pops in and out.
Now, when the water is high enough, the resting position of the bottom is the convex position, and lowering the cup causes the popping behavior.
What I'm wondering about, is how to accurately describe this behavior physically. I understand that there's roughly an even pressure at every point on the bottom of the cup due assuming that the bottom of the cup doesn't pop up/down very far compared to the height of liquid, but I think the stress on the bottom caused by liquid above each point should be very different, since each point is at a different distance from the edge, and I think the force should be considered as acting on some sort of springy lever arm attaching each point being pressed on to a point on the rim.
I look forward to responses :)

 

[JoAuSc]

It does make sense that the liquid would cause more stress on the middle of the cup's bottom than at a point near the rim, even if the force is the same in both places. The stresses should be radially symmetric while in either position. I'm guessing, though, that the bottom probably contorts to get from the up to down position or vice versa, breaking radial symmetry. The paper in the cup is not very compressible, especially not when it would rather bend, so the cup's bottom would seek to keep its surface area constant while it maneuvered from up to down. Both up and down positions are radially symmetric, and they seem to be the only positions that are radially symmetric while keeping the same surface area, so for the cup to pop from one position to the next, radial symmetry would have to be broken.

 

[sir-pinski]

Thank you for sharing your observations on the physical behavior of plastic cups. It is indeed fascinating how such a simple object can exhibit such complex behavior.

To accurately describe this behavior, we can use the principles of fluid mechanics and elasticity. The concave shape of the bottom of the cup is due to the weight of the liquid pressing down on it. As the liquid level rises, the pressure on the bottom of the cup increases, causing it to pop into a convex shape. This is because the bottom of the cup acts as a membrane and the increased pressure pushes it outwards.

When you lift the cup, the weight of the liquid is no longer pressing down on the bottom, causing it to return to its concave shape. This is similar to the popping tops of baby-food cans, where the pressure from the contents inside causes the top to pop out when opened.

The stress on the bottom of the cup is indeed different at each point, as you correctly pointed out. This is because the distance from the edge varies at each point, resulting in different levels of pressure. The force acting on the bottom of the cup can be considered as a distributed load, with each point experiencing a different amount of force.

To accurately model this behavior, we can use equations from fluid mechanics and elasticity to calculate the pressure and stress at each point on the bottom of the cup. This can help us better understand the forces at play and predict the behavior of the cup in different situations.

Overall, your observations and curiosity are a great example of how science can be found in everyday objects and how we can use principles and equations to accurately describe their behavior. Thank you for sharing your thoughts and I hope this response helps in understanding the physical behavior of plastic cups.