## square base cups

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spacefem
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### square base cups

I realize this forum is where we talk about quantum physics and bioengineering and all that, but for whatever reason the dumbest questions make me think of coming in here, hope nobody minds.

at the grocery store today buying plastic cups for a party, I noticed that red solo cups come in two varieties now: the traditional round base cups, and a newer version with a squared off base - rounded corners, but basically square. They're both round on top of course. This is google-able if you want pictures.

the square base cups claim that they are less likely to spill because square base.

I am skeptical. Why would a square base be less likely to spill? Given two cups, with the same area of bases, if one is square and one is round, I think round would better shape because all edges of the cup are the same distance from the center of gravity. the square cup would have some edge space closer to the center. the only way the square would be less likely to spill is if you happened to knock into the cup at one of the corners.

right?

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### Re: square base cups

For one thing, if we assume bumps come from random directions, being less likely to tip when hit near the corners means being less likely to tip in general.

For another, if you almost tip over one of the square-based ones, there's no chance it'll do that rolling-around wobble thing the round-based ones can do (like what happens to a coin after you spin it). During that wobble the cup is less stable and could be more easily knocked over by a second disturbance, whereas the square-based ones pretty immediately return to being flat.
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thoughtfully
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### Re: square base cups

Obviously, triangles are optimal then.

Step two (?)

Profit!

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Xanthir
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### Re: square base cups

Nah, triangles are really easy to tip, because the middle of each edge is very close (relatively) to the center, and hitting any of the corners (which stick out, and so are easier to hit than the rest of the cup) easily turns it over, with the opposite edge as the fulcrum of a lever. Squares suffer less from the first (center of the edge isn't nearly as close) and don't suffer at all from the second.
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Twistar
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### Re: square base cups

This sounds like a fun problem.

You can come up with some gaussian distribution of force vectors (directions and magnitudes) acting on different parts of the circular rim of the top of the cup. You can then probably calculate for any given vector whether the cup will tip or not based on the distance of the fulcrum edge to the center of the cup. Or some relevant centerline related to the force vector. With some simulation you could try it for different sided cups and determine the optimum cup shape! Or you could just make the cups and do experiments. I wonder how many cups of this sort someone would have to sell to pay a physicist to do this calculation. I guess none if they can just nerd-snipe the physicist.

SuicideJunkie
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### Re: square base cups

thoughtfully wrote:Obviously, triangles are optimal then.

Step two (?)

Profit!
Take that to the logical inductive conclusion, and you'll find that the paper cones from water dispensers are much better, but for the ultimate zero cornered beverage holder we should all be drinking out of bowls, since they are infinitely superior to standard cups.

SDK
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### Re: square base cups

SuicideJunkie wrote:Take that to the logical inductive conclusion, and you'll find that the paper cones from water dispensers are much better.

Either that or the standard circle.
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### Re: square base cups

A high-ball tumbler with four additional, solid cylinders around the base like the first stage of an Atlas rocket would be more stable right way up on the table but probably not so stable in the dishwasher. Vanes giving the same extra base size for less additional weight may be a better idea, as long as they're not prone to snapping off and becoming razor-sharp shards. Razor-sharp shards are usually considered a bad thing. If you're thinking of making them, maybe you could add five vanes and strengthen them with an outer pentagon for sale to US military types, make another design with five pairs of vanes to sell to Wiccans, one with six paired vanes for Jews, one with one of its four vanes extending further than the others for the Christians, one with everything for the Zen Buddhists, an approximation of a Mandelbrot set for that one guy, ...

... and no Weebles. Weebles would be a bad design for a drinking glass.
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lorb
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### Re: square base cups

Do the square base ones have same base area as the circle ones? same circumference? something else?
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Hypnosifl
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### Re: square base cups

spacefem wrote:Given two cups, with the same area of bases, if one is square and one is round, I think round would better shape because all edges of the cup are the same distance from the center of gravity. the square cup would have some edge space closer to the center. the only way the square would be less likely to spill is if you happened to knock into the cup at one of the corners.

Thinking about how torque works, it seems to me it should be the opposite--hitting the cup closer to its center of gravity should make it more likely to tip than hitting it farther away, assuming the force is the same strength in both cases and in a random direction. If you consider a vector from the center of gravity to the point you strike, then the torque is the product of the size of that vector and the size of the force vector, multiplied by the sine of the angle between them--so on average, a larger distance from center of gravity to the point that's hit should mean a larger torque (which is related to the fact that you can lift more weight with a lever if you're farther from the fulcrum, as Archimedes first noted). Of course this ignores the details of the forces between the base and the surface it's resting on (like the wobble effect gmalivuk mentions) so it isn't the whole story.

Copper Bezel
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### Re: square base cups

Wait, what? The base is acting as a lever arm in whatever direction the cup is struck, where the cup is struck defines another lever arm between that point and the point of the base on which the cup is tipping, and we'd ideally want to increase torque on the first and reduce it on the second, but really don't have control over the latter at all. Also, as lorb said, no one has defined whether diameter, area, or circumference is held constant, so there are no meaningful numbers to compare. And wobbling and its damping, weight distribution, elasticity, and probably friction in some form are all likely to be more dominant factors anyway.
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gmalivuk
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### Re: square base cups

Volume, height, and the size of the top rim of the cup all remain (approximately?) the same, which I suspect is sufficient to constrain the measurements of the base.
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Hypnosifl
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### Re: square base cups

Copper Bezel wrote:Wait, what? The base is acting as a lever arm in whatever direction the cup is struck, where the cup is struck defines another lever arm between that point and the point of the base on which the cup is tipping, and we'd ideally want to increase torque on the first and reduce it on the second, but really don't have control over the latter at all.

Without getting into the (important) details of how the table exerts torque on the base of the cup in response to its movements, the value of the torque just from the force you apply should be larger if you apply force farther from the center of mass, that's all I was talking about (in response to the OP's comment about the square cup being more likely to tip if you hit it on an edge closer to the center of mass). One way of thinking about this is that if the cup were just floating in zero G, not initially rotating, then if you apply a set amount of force for a set amount of time, you'll get a faster rotation rate if you apply the force to a point on the cup that's further from the center of mass, like a point near the corner of the base of a square cup.

Copper Bezel
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### Re: square base cups

Oh, okay! There it is!

Comment was that it's more likely to rock on an edge of the base closer to the center of mass. The rim shape stays the same in either version of the cup - the point of impact doesn't move.

Spoiler:
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jewish_scientist
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### Re: square base cups

The ratio* between the area of a square to the largest circle that fits inside that square is 4:pi. This mean that a square will have a side length less than a circle's diameter if their areas are equal.
_____________________________________________________________________________

If the cup's center of mass goes past the vertical plane that is tangent to the point or edge the cup is tipping on, then the cup will spill. Also, I am ignoring the force vector's Z-axis component because I do not know how that will effect the cup. If it makes you happy, just imagine every vector parallel to the cup's base.

Imagine drawing a ray that goes through the center of a circle. When it hits the far side of the circle, it will be perpendicular to the tangent line of that point. Therefor any force vector that passes through the cup's center of mass will be perpendicular to the vertical plane that is tangent of the point the cup is tipping on. This means that the X-axis component and the Y-axis component of the force is displacing the cup's center of mass toward that vertical plane. The result is that every force vector that passes through the cup's center of mass will achieve maximum possible displacement in the desired/undesired direction.

Imagine drawing a ray that goes through the center of a square. Only 4 out of infinite possible rays will be perpendicular to the far side of the square. Therefor a force vector that passes through the cup's center of mass will most likely not be perpendicular to the vertical plane that is tangent to the edge the cup is tipping on. This means only the one component of the vector displaces the cup's center of mass toward the vertical plane the cup is tipping on (the other component is trying to displace the center of mass by moving it parallel to the edge it is tipping on). The result is every force that passes through the cup's center of mass must lie in 2 out of infinite planes in order to achieve maximum displacement.

*An ancient rabbi made a a really cool proof of this based on the formula for the area of a triangle and the formula for the circumference of a circle.
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Copper Bezel
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### Re: square base cups

The ratio* between the area of a square to the largest circle that fits inside that square is 4:pi. This mean that a square will have a side length less than a circle's diameter if their areas are equal.

Right, but cups are an engineering problem, not a pure math game. The question is, what are the constraints on the size of the base of the cup, and how do those constraints apply in each case?

A cylindrical vessel is inherently more stable than any vessel that narrows at the bottom. It also maximizes volume per area footprint, which is why they are used to transport and store liquids. However, it's nonideal as a drinking vessel, and particularly a disposable one:
• It cannot be stacked inside another identical cup for storage and transportation.
• It has to be lifted to a higher angle to completely drain.
• If the material is flexible and uniform, the side walls have less inherent resistance to collapse than the inverted, truncated cone.

A meaningful answer to the question would have to take those factors into account and determine the actual footprints of a comparable round-based and square-based cup. On the second point, for instance, the only thing that matters is the shallowest angle available, so the square could effectively be circumscribed around the circle. That's not true of stackability at all - it's definitely going to constrain the square to have a smaller breadth than the diameter of the circle, but we don't know by how much without actually doing the geometry. And I wouldn't know where to start in considering the durability aspect.

Can anyone think of anything else that should be on that list?
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DanD
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### Re: square base cups

It comes down to:

If the round base circumscribes the square base, the round is more stable.
If the round base inscribes the square base (tangent), the square is more stable in all directions except perpendicular to the edges, where they are equal.
If the two overlap, then there is a trade off depending on the direction of load.

However, in order for mold-ability (and mold removal) and nesting to work properly, the draft angle of any given wall segment should be at least a minimum value. If you assume these cups are made with that minimum value wherever possible, and that the rim diameters are the same, and the height is the same, then the square base is less stable.

The logic being that the "corner" of the square cup is inside the lip by that minimum draft angle. Therefore, the straight edges of the square base have to be inside by more than that draft angle, and net result is a square that can be inscribed within the round base.

This same issue also means that the center of mass of the square cup is higher for the same volume of liquid, and that it hold a lower volume.

Now that last bit is important, because it most likely suggests that the square cup, if it has the same rated volume, is likely either wider at the rim or taller. The former would increase stability, the latter decrease it, relative to the nominal square cup, but in both cases you'd be better off with a larger round cup.

gmalivuk
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### Re: square base cups

That doesn't address the point I made at the beginning of the thread, about how a round base can wobble around before settling more than a square base.
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### Re: square base cups

And in particular, that wobble can cause the liquid contents to slosh, redistributing the weight in a way that destabilizes the base further and causes a full tip-over.
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### Re: square base cups

Xanthir wrote:And in particular, that wobble can cause the liquid contents to slosh, redistributing the weight in a way that destabilizes the base further and causes a full tip-over.
Indeed.

I have both types of cups left over from past parties. They have the same volume, same height, and same rim size, with the base adjusted accordingly. (So it fits neither within or around the circular base.) I wasn't interested in making a mess by tipping over full cups (or experimenting with the effects of sloshing contents), but empty, there seem to be two advantages of the square(ish) base:

1) As I mentioned before it won't wobble around in a circle like a coin will, though that's only relevant for off-center impacts, which wouldn't generally come from something bumping into a circular ring.

2) When pushed from any direction near the middle of a side of the base, the square-based cup will tend to rotate to keep one whole edge along the ground (and near the corner, the base extends farther from the centerline than the round cup, so is more stable even if it tips to balance on just one point. When it falls back from a position of having one whole edge along the ground, it tips past equilibrium in such a way that the opposite (whole) edge is along the ground before coming back to rest. This is a lot less likely to result in sliding, which often happens with the round cup because it only has one point in contact with the ground whenever it's not sitting completely flat.
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DanD
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### Re: square base cups

gmalivuk wrote:
Xanthir wrote:And in particular, that wobble can cause the liquid contents to slosh, redistributing the weight in a way that destabilizes the base further and causes a full tip-over.
Indeed.

I have both types of cups left over from past parties. They have the same volume, same height, and same rim size, with the base adjusted accordingly. (So it fits neither within or around the circular base.) I wasn't interested in making a mess by tipping over full cups (or experimenting with the effects of sloshing contents), but empty, there seem to be two advantages of the square(ish) base:

1) As I mentioned before it won't wobble around in a circle like a coin will, though that's only relevant for off-center impacts, which wouldn't generally come from something bumping into a circular ring.

2) When pushed from any direction near the middle of a side of the base, the square-based cup will tend to rotate to keep one whole edge along the ground (and near the corner, the base extends farther from the centerline than the round cup, so is more stable even if it tips to balance on just one point. When it falls back from a position of having one whole edge along the ground, it tips past equilibrium in such a way that the opposite (whole) edge is along the ground before coming back to rest. This is a lot less likely to result in sliding, which often happens with the round cup because it only has one point in contact with the ground whenever it's not sitting completely flat.

So it sounds like assumption

If you assume these cups are made with that minimum [draft angle] wherever possible

is the issue. For the square base not to fit inside the round base with the same rim height and diameter, the draft angle must be closer to vertical on the square base. There are reasons why a square cup could be manufactured with a shallower draft angle (related to moving mold components), but I have a hard time believing they'd be applied to something that inexpensive. As a result, it seems likely that a more efficient round cup could be produced that would be more stable.

I do not believe the rotation around the corner is a significant factor in stability (a cup hit hard enough to tip will tip on the corner or the edge). Likewise, while reduced sliding decreases the chance that the cup will be knocked off a surface, it increases the chance that it will tip (sliding tends to bring the base back under the CG.

The rolling wobble is a factor, but a rare one, as you suggested. In a standard impact free motion of the fluid will tend to reduce the tendency to tip. (Inertia of the liquid fights the result of the initial impact)

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### Re: square base cups

DanD wrote:is the issue. For the square base not to fit inside the round base with the same rim height and diameter, the draft angle must be closer to vertical on the square base. There are reasons why a square cup could be manufactured with a shallower draft angle (related to moving mold components), but I have a hard time believing they'd be applied to something that inexpensive. As a result, it seems likely that a more efficient round cup could be produced that would be more stable.
I suspect that while such a cup could be produced, it wouldn't be as user-friendly, because the shallower angle all the way around would make circular cups quite a bit harder to pull apart, while a shallower angle just near the corners of the square-based cup doesn't affect their separability as much.
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Copper Bezel
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### Re: square base cups

And a shallower angle is still easier to drink out of. The extreme case of a "difficult" vessel would be a soda can, although that's not helped by the design of the top. The soda can has a narrower diameter, but you still can't very well hold it at 90° to your face.
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thoughtfully
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### Re: square base cups

Even worse are cans that bulge out! They are cool in their transparency, though. The sloshing liquid was visible. Some sort of stiff plastic with an aluminum top.

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Copper Bezel
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### Re: square base cups

Oh God, yes. Adorable little balls of happiness, but definitely a bit of a pain. Stating the obvious, when you and your pretty friend have gone through a few of them, the mechanics start getting really awkward.
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