# Pythagoras Cup Question

My TA professor has asked me to write a homework question about the Pythagoras cup. After posting this previously I realized I should have given some more background information and not leave the question so open-ended.

The Pythagoras Cup is a vessel with a hollow column in the center that is connected to a pipe that empties out of the bottom of the cup's stand. Its design is credited to Pythagoras but I found just as many sources saying someone else designed it as he did. Anyway here are some links that give more background information. Pythagoras Cup YouTube, Pythagoras Cup Wikipedia.

Here is the question I would like to ask the students, please let me know what I can improve or if you think it's fine.

An archeologist is on a dig in Greece when they come across a pottery vessel (I copied a picture of the cross section from Wikipedia). Upon further examination of the vessel, the archaeologist sees that the inner column is hollow, with small holes on both sides and pipe going through the center of the column and emptying out the stand. The archaeologist notices that upon filling the vessel with water to the top that it drains the entirety of its contents. The archaeologist repeats this but only fills the vessel half way and observes that the water stays completely inside the vessel.

Question 1: Explain why the vessel doesn’t empty itself when it is only filled half way but does when it filled to the top. What would happen if the archeologist filled the vessel with a small amount of a denser liquid before pouring the water in?Hint: Consider the forces acting on the fluid when the water is filled to the top and when it is only filled half way separately.

Solution to Q1: When the water is filled half way the hydrostatic forces from pressure and gravity are constant. See also Pascal’s Law of Communicating vessels. When the water is filled over the height of the central column the water inside the column has nowhere else to go but into the pipe. When the water enters the pipe gravity begins pulling it down which creates a siphon because of the difference in pressure. (I don't expect them to recognize that this is a siphon, but as long as they notice that the water in the inner column must be the same height as the water in the cup until the water goes over the top of the column then the weight of the water now in the central pipe creates a pressure difference "pulling" the water out I would give them full credit.) The weight of the water would not be enough to force the denser fluid into the central column.

Question 2: Calculate the mass flow rate of water leaving the vessel as a function of the time after it has been filled to the top. Assume that the vessel can be modeled as shown to the right (I've included a drawing I made in CAD of the cup) and that the vessel is cylindrical and the dimensions of the column are much smaller than the dimensions of the vessel. $D=4.5$ inches, $d_1=\frac{3}{4}$ inches, $d_2=\frac{1}{8}$ inches, $H=6$ inches, $h_1=3$ inches, $h_2=6$ inches, $h_3=2$ inches and $g=32.2 ft/s^2$. Hint: You may assume that the velocity of the water leaving the vessel is $V=\sqrt{2gy(t)}$ where $y(t)$ is the height of the water in the vessel measured from the bottom of the stand. The students have not been taught Bernoulli's equation yet so I gave them the velocity leaving the pipe. I also gave them more information then they need to solve the problem, the way I came up with an answer only needed $\rho$, $g$, $H$, $D$, and $d_2$.

Solution to Q2: Use the conservation of mass equation. $$\dot m_{in}-\dot m_{out}=\frac{d m_{CV}}{dt}$$

There is no mass entering the vessel so $\dot m_{in}=0$.

The amount of mass leaving the vessel is $\dot m_{out}=\rho V_{out}A_{out}=\rho\sqrt{2gy(t)}\frac{\pi}{4}d_2^2$.

The amount of mass in the vessel (ignoring the mass inside the column) is $m_{CV}=\rho\frac{\pi}{4}D^2y(t)$.

Plugging these into the conservation of mass equation gives $$-\rho\sqrt{2gy(t)}\frac{\pi}{4}d_2^2=\rho\frac{\pi}{4}D^2\frac{dy}{dt}$$ $$-\sqrt{2gy(t)}\frac{d_2^2}{D^2}=\frac{dy}{dt}$$ separating the ODE $$-\sqrt{2g}\frac{d_2^2}{D^2}dt=Kdt=\frac{dy}{\sqrt{y(t)}}$$ and integrating $$Kt+c_1=2\sqrt{y(t)}$$ the cup is filled to the top at the beginning so $y(t)$ measured from the bottom of the stand (the datum) at $t=0$ is $y(0)=H$. $$c_1=2\sqrt{H}$$ $$Kt=2\begin{pmatrix}\sqrt{y(t)}-\sqrt{H}\end{pmatrix}$$ the original question asked for the mass flow rate leaving the cup so plugging this answer into the $\dot m_{out}$ equation gives $$\dot m_{out}=-\rho\sqrt{2gy(t)}\frac{\pi}{4}d_2^2$$ $$\dot m_{out}=\rho\begin{pmatrix}\frac{g}{2}\frac{d_2^2}{D^2}-\sqrt{2gH}t\end{pmatrix}\frac{\pi}{4}d_2^2$$ the students can plug in the numbers from the problem but I'm mainly going to be looking for the above equation.

Please let me know what I can improve on.

• Much better question! Although, still very open-ended. It would be helpful to know which aspects of the question you think may need improvement. Also, was there any guidance from your TA professor? And what level is the class? Won't your TA professor give you feedback? – sammy gerbil Aug 29 '16 at 14:07
• I haven't sent it to him yet. I wanted to post it here first. It's an undergraduate fluids class, in their second week of classes. – WnGatRC456 Aug 29 '16 at 14:16
• I think that the absence of viscosity in your question makes it all a bit hypothetical. When people have not been taught Bernoulli, perhaps the flow rate question is a bit "out of reach". Also - at the end (when the cup is empty and the column of water is being sucked up the syphon) won't the rate increase again? Your hint suggests to ignore that. Finally - feet, inches. Really. REALLY? At University level? Please. – Floris May 9 '17 at 14:58
• I'm voting to close this question as off-topic because while it is interesting, "is this a good homework question" is not a question about physics per se. This seems like a fit for an educators' site. – Floris May 9 '17 at 14:59
• This question was posted such a long time ago, and you're just now checking it out? Come on. – WnGatRC456 May 9 '17 at 15:00

Solution to Q1:

1. Your explanation is rather vague, especially for an undergraduate class. eg "When the water is filled half way the hydrostatic forces from pressure and gravity are constant." Which forces exactly? At what points? Don't you mean "balanced" or "in equilibrium" rather than "constant"? Later you say : "...the weight of the water now in the central pipe creates a pressure difference 'pulling' the water out." Pressure in liquids does not 'pull' - using this word is misleading, as it implies that cohesion and gravity ('the chain model') cause the siphon to work, which is not correct. See Siphon Misconceptions and Siphons Revisited.

2. Why don't you tell the students it is a siphon? Wouldn't that make the explanation a lot easier? Why don't you expect undergraduates to recognise a siphon? That is strange.

3. What about a diagram, as in the other answer here? "A picture paints a thousand words."

4. Where is your explanation of the sub-question about using 2 liquids of different densities? Why you are asking this is not clear. You need to make a point about this in your solution, and give a bigger hint towards this point in the question.

5. Another question you could ask is : Does the cup get drained completely, or does the column of water inside the cup (of height $h_2$) get left behind?

Question 2:

1. Why ask for mass flow rate? Isn't volume flow rate easier? I would give the volume of the cup and ask them how long it takes to drain from filled.

2. "Assume that the vessel can be modeled as shown to the right (I've included a drawing I made in CAD of the cup) and that the vessel is cylindrical..." Your drawing is not cylindrical! Do you mean it has cylindrical symmetry?

3. "..and the dimensions of the column are much smaller than the dimensions of the vessel." Is it necessary to tell the students to make such assumptions? Apart from giving them dimensions and Torricelli's Law for velocity, why not say :"State any assumptions you make"?

• These are the type of suggestions I was hoping for. Thanks. – WnGatRC456 Aug 29 '16 at 16:11
• For 7, I don't want them to consider the cups changing cross sectional area. – WnGatRC456 Aug 29 '16 at 16:14
• Your CAD cup looks great, but a simpler illustration of a cylindrical cup might be better... Sorry, I think maybe you are right after all about telling the students to neglect the cross-section of the central column - it is something you'd expect to take account of. – sammy gerbil Aug 29 '16 at 16:28
• I do have a simpler drawing I did in paint (similar to the Wikipedia figure) but I didn't think to use it. I might now since I didn't realize how confusing the cad figure was. – WnGatRC456 Aug 29 '16 at 16:42
• Another change I would make is to remove the fictitious story introduction, which I think is patronising at this level. – sammy gerbil Aug 29 '16 at 16:48

Based on only the information you supplied (not including the wikipedia link), I found it difficult to picture what exactly it was you were trying to describe. I would consider adding an image similar to the one below from wikipedia (or maybe you already did?):

In addition to clearly showing the different states of the system, the channel from the inside to outside of the cup is very clear. In contrast, the channel in the CAD drawing you supply looks... Discontinuous? There's a solid line separating the tube in the stand from the portion in the inner column, and the inner column looks sealed off from the portion from where the liquid sits. Perhaps this could be made clearer without the gradients, and instead opting for a solid black-white-grey color scheme.

Also, given the somewhat bizarre nature of this device, it might be worth calling it out by name in the problem. It would make question 1 easy to google, but it would be helpful those struggling with visualization to have something to look up. Plus, the bulk seems to be in question 2 anyways.

Other than the visualization issues, I think you are in good shape!

For clarification, I have included a crudely modified CAD drawing below.

• I considered putting the name of the device in the problem but then they could just look it up and find the answer without really thinking it through. Also, the CAD drawing is a cross section, that's why it looks discontinuous. But putting the 4 pictures from Wikipedia could work. – WnGatRC456 Aug 29 '16 at 2:51
• @NWernerC456 I will attempt to clarify. In the CAD drawing, I can't trace the path the liquid flows with a pen without drawing over what appear to be walls. In comparison, I can easily do this with the cross section I linked to. – user363165 Aug 29 '16 at 3:36
• Ok I see. What you did is you erased some solid lines. Those lines have to be there though because there is a surface. Otherwise it looks like it's just floating in midair. I think I'll just use the figure from Wikipedia. – WnGatRC456 Aug 29 '16 at 4:54