Truncated Cone and Support
Date: Fall 2013
Below is a question that one of my students just ask. I feel like it should matter if the cup is upside down or right side up because if it is sitting on a table it will have forces from the object sitting on it and from the table below. If we do not consider the weight of the cup, the force down from the object on the cup should be same as the force up from the table so the cup is being crushed from both sides.
Here is the question:
Imagine a plastic cup that has a diameter that increases as the height increases. The mouth of the cup is wider than the bottom. Could this cup support more weight upside down or right side up?
In theory it should not matter. Whether the cup is upside down or right side up, there is a force balance. If you put a 20 N load on the cup, that 20 N load is carried all the way through the cup from the top to the bottom. That same 20 N is felt by both the top of the cup and the bottom no matter the orientation of the cup.
John C Strong
The cup (as you indicate) will have identical forces at each end. So,
considering this fact, it is obvious that whether the cup is upright or
upside down, each end will "see" no difference in the forces acting on
it, no matter which way it is oriented. Therefore, there can be no
difference in the cup's ability to support weight, no matter which way it
If we ignore the weight of the cup, then the orientation up or down should not matter. If the cup is not moving, then all the forces must balance. The weight creates a force F=mg where m=mass of weight and g=gravitation field intensity (about -10 N/kg). Consider the two sides of the cup (in either orientation):
1. Side touching weight. The weight is still exerting a force F down whether it is on the table or on the cup. The cup must push back up with the same force; otherwise the weight would fall.
2. Side touching the table. If you place the cup and the weight on a scale (as a substitute for the table), it will not matter which direction the cup is oriented. It will measure the same weight, or force, on it. Thus, the table must also push back with the same force as the weight pushes down; otherwise again, the cup would move.
Thus, the two sides will feel the same force, regardless of cup orientation. Now, this force is not distributed equally on one side or the other. The question then becomes, "what side will collapse first?"
Kyle J. Bunch, PhD, PE
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Update: November 2011