Materials Included In Kit

Additional Materials Required

Safety Precautions

Clean up all spills immediately. The glue gun is hot during operation, take care not to make skin contact with nozzle or hot glue and always unplug when not in use. Please follow all laboratory safety guidelines.

Lab Hints

• Enough materials are provided in this kit for 15 student groups. The entire activity can reasonably be completed in one 50-minute class period.
• If you only have one triangular file, it may be most efficient to file the plastic test tubes prior to the lab.
• An extra cup and a large disposable weighing dish is provided to each group in order to prevent spills, however, the cups may be emptied over sinks or buckets in order for students to clearly observe the siphoning effect.
• It is important that the cups be sealed for leaks with the hot glue gun. Ensure that students are aware of its importance in order to prevent classroom spills.

Teacher Tips

• This activity is designed as a qualitative introduction to the siphoning effect and fluid pressure concepts. If the students have knowledge of Bernoulli’s equation, then a more quantitative analysis of pressure throughout the cup may be desirable.

Answers to Prelab Questions

1. Refer to Figure 1 in the Background section. What will happen to the liquid in the cup once it is filled to a point higher than the central column?

   All of the liquid contained in the bowl of the cup will drain out through the hole at the bottom of the stem.

2. What is Pascal’s law of communicating vessels?

   Pascal’s law of communicating vessels states that when liquid settles inside a set of connected containers, the fluid balances out to the same depth in all of the containers regardless of the shape and volume of the containers.

3. Read over the Procedure. Describe any safety hazards associated with this activity.

   The hot glue gun gets very hot and can be damaging to bare skin. Care must be taken so that the glue gun does not make contact with exposed skin, the glue gun must be propped using the stand and should be unplugged when not in use.

Answers to Questions

1. The static pressure of a fluid is calculated with the equation
   {14081_Answers_Equation_1}
   where ρ is fluid density, g is acceleration due to gravity, and h is the depth of the fluid in the container. With this relationship in mind, would a Pythagoras cup filled with mercury empty itself out in the same manner as water? How about with alcohol?

   The cup would empty out regardless of whether mercury, water or alcohol is used. As long as the fluid is of uniform density, the siphon would work in the cup.

2. If you had the same cup filled with water to a height below the height of the center column (see Figure 1 in the Background), what happens to the water? What would happen if a tiny vacuum were attached to the straw to pull a short column of water just into pipe B and then the vacuum was immediately detached? The water would be still and not drain from the cup if below the height of the column. If a column of water was pulled by vacuum just into the opening of pipe B, the same siphoning effect would take place and the entire liquid content in the bowl of the cup would drain.
3. In your own words, explain Bernoulli’s principle. An increase in the speed of a fluid also means a decrease in the pressure exerted by the fluid, likewise, a decrease in the speed of the fluid occurs with an increase of pressure exerted by the fluid.
4. Examine the following image:
   {14081_Answers_Figure_1}
   This toilet has no tank and therefore no flush handle. In your own words, describe how you could flush the toilet completely without the water tank.

   Quickly emptying an entire bucket of water into the bowl of the toilet would flush the contents because water would fill the siphon tube and the siphoning effect would cause the toilet to flush. Merely putting cups of water into the toilet bowl would not flush the toilet since water would spill over the edge of the siphon tube, never completely filling the tube.

Introduction

Toilets, gasoline thieves and a cup that punishes the greedy. What do these all have in common? Discover the answer and investigate the stunning effects of Bernoulli’s principle by building your very own Pythagoras cup!

Concepts

• Pressure
• Bernoulli’s principle
• Siphon

Background

The great Greek philosopher and mathematician Pythagoras of Samos (570 B.C.E–495 B.C.E) is credited to have invented the “greedy cup.” The story has it that Pythagoras was supervising workers at the water supply works on Samos and invented his peculiar cup to moderate the amount of wine the workers drank. As long as no one filled his cup past a critical depth, no issues would arise. However, should anyone let greed get the best of him and fill their cup past the allowed point, the entire contents would drain down through the bottom of the cup!

The cup appears to be a normal cup with the exception of a central column inside (see Figure 1).

The central column is positioned over the stem of the cup and over a hole at the bottom of the stem. A small open pipe (pipe B in Figure 1) runs from the hole to nearly the top of the central column, leaving a small gap. The central column has a hole at the bottom that connects the bowl of the cup to a short small pipe (pipe A) that also runs nearly to the top of the central column. When the cup is filled, liquid rises in the bowl of the cup and in pipe A at equal rates due to Pascal’s law of communicating vessels (see Figure 2). This law simply states that when liquid settles inside a set of connected containers, in this case the bowl of the cup and pipe A, the fluid balances out to the same depth in all of the containers regardless of the shape and volume of the containers. As the cup continues to be filled, nothing of interest occurs until the level of liquid rises to the top of the central column, therefore filling the small gap. As soon as liquid fills the gap, it begins to run down pipe B and out of the hole at the bottom of the stem due to gravity. The liquid continues to pour out through the hole until the entire contents have been emptied out (see Figure 2). The reason for this stunning effect can be attributed to Bernoulli’s principle. The principle states that the pressure exerted by a moving fluid is less than the pressure exerted by a stationary fluid. When the liquid begins to move through pipe B it exerts less pressure than the stationary fluid in the bowl of the cup. This creates an area of low pressure at the top of the central column (in the gap) and the higher pressure of the liquid in the cup plus the atmospheric pressure (allegedly) pushes the liquid up pipe A against the pull of gravity and then down pipe B. This process continues until the liquid can no longer be found in the cup but on the clothes of the greedy drinker! This is generally known as the siphoning effect and is the same principle that is used by many modern toilets to flush or by thieves using hoses to empty entire gas tanks.

Experiment Overview

Using simple materials, build your own Pythagoras cup and observe the siphoning effects caused by pressure differentials.

Materials

Prelab Questions

1. Refer to Figure 1 in the Background section. What will happen to the liquid in the cup once it is filled to a point higher than the central column?
2. What is Pascal’s law of communicating vessels?
3. Read over the Procedure. Describe any safety hazards associated with this activity.

Safety Precautions

Clean up all spills immediately. The glue gun is hot during operation; take care not to make skin contact with nozzle or hot glue and always unplug when not in use. Please follow all laboratory safety guidelines.

Procedure

1. Plug in the hot glue gun and set aside while it warms up.
2. Using a pair of scissors, perforate the bottom in the center of a clear plastic cup to create a hole with a 1-cm diameter. Note: Cut away edges around the hole to make as clean of a hole as possible.
3. Cut the disposable spatula to make a 14-cm long straw.
4. Turn the plastic test tube upside down and at the opening, file a hole about half a centimeter in height (see Figure 3).
   {14081_Procedure_Figure_3}
5. Measure the straw to a 9-cm length and lay a stream of glue along the long side of the straw. Quickly insert the straw into the test tube and press and hold against the side. Note: Do not cover the triangular opening with the straw (see Figure 4).
   {14081_Procedure_Figure_4}
6. Once the glue is dry, insert the test tube and straw combination into the perforated cup, straw end down, and guide the bottom of the straw through the hole you cut in step 2. Push down so that the bottom opening of the test tube is flush with the bottom of the cup.
7. Holding the center column firm, adhere the straw to the cup by adding glue to the underside of the cup around the straw. Cover any gaps with glue so that water will not seep through (see Figure 5).
   {14081_Procedure_Figure_5}
8. Allow the glue to dry completely.
9. Cut a 2" x 2" hole in the center of the weighing dish.
10. Place the weighing dish right side up on top of the second, unaltered plastic cup.
11. Place the Pythagoras cup on top of the weighing dish so the straw goes through the hole in the center of the dish and extends into the bottom of the cup. Note: Use this setup to prevent spills when testing.
12. Slowly add water to the Pythagorus cup. Note the level of the water when it begins to drain out through the straw.

Student Worksheet PDF

14081_Student1.pdf