I saw this on a late night chat show in Australia while on sabbatical. It was done by Brian Greene from Columbia University though in his version the demonstration was about relativity. It is incredibly simple, potentially dramatic, and also gives you the opportunity to get soaked in front of your students which is often memorable and fun (for them anyway).
- One 2 liter soda bottle with cap (remove the label)
- Food coloring for visualization
- A knife or scissors for cutting a small hole in the base of the bottle
- Tape to temporarily cover the hole
- A stairwell or step ladder for additional height to drop from
- Cut a small hole in the side of the bottle near the base. The hole needs to be small enough that the bottle does not empty too rapidly but large enough that so that the water can be seen as it shoots out.
- Tape over the hole from the outside to seal it while setting up the demonstration
- Fill the bottle with water and food coloring and put the cap back on loosely (if it is tight then air cannot get in and the water will pulse out the base of the bottle rather than flow smoothly
- Hold the water bottle at a great height (stairwell, stepladder, etc.) Make sure there is no way for anyone to get underneath the bottle when you drop it.
- Remove the tape so that water is smoothly pouring out the base of the bottle
- Drop the bottle so that it falls freely. The water should stop flowing out as soon as you release it.
The water flows out of the base of the bottle because there is a pressure difference between the water inside the bottle (due to hydrostatic pressure) and atmosphere outside. Under solid body acceleration the hydrostatic pressure changes. Below is a very basic derivation of the vertical hydrostatic pressure equation under solid body acceleration.
Start by drawing a small element of fluid with vertical height dz and horizontal area dA (see figure 2a). As the fluid is accelerating as a solid body there are no shear stresses. Therefore, the forces acting on the element are its weight (mg=ρdzdA g) the pressure force up (PdA) and the pressure force down ((P+dP)dA) where we allow the pressure to vary vertically and take vertically upward as the positive z direction. The complementary kinetic diagram (figure 2b) simply has the mass times vertical acceleration (ma= ρdzdA)
Figure 2. (a) Force diagram showing the elements weight and pressure forces in the vertical direction. (b) Kinetic diagram showing the mass times acceleration. Up is positive.
Writing the equation of motion for the element in the z direction we get
ΣF=PdA – ρdzdA g – (P+dP)dA = ma = ρdzdA a
this simplifies to
dP/dz = – ρ (a + g)
again where a is positive upwards. For a static fluid (a=0) the standard hydrostatic pressure equation is recovered (dP/dz=-ρg). This hydrostatic pressure gradient produces the pressure difference between the water in the bottle and the atmosphere outside when you are holding the bottle.
When the bottle is released it initially accelerates at a rate a=-g (there is zero velocity and, therefore, no vertical drag force). In this case a+g=0 and the pressure in the water instantaneously drops to zero (atmospheric) everywhere. This means there is no longer a pressure difference across the hole in the base of the bottle and the water immediately stops flowing.
I did this in class down a 4 story stairwell. The students all stood around the stairwell when I dropped it. However, I did not use food coloring as I did not want to get students wet and stained. Also, when I did it in class the hole was too small so it was not easy to see on the video a student took. I repeated it later in the lab with a larger hole and food coloring. See photos below. The demonstration is essentially fool proof. As long as the cap is loose and the hole is big enough the only thing that can go wrong is that you get wet. One cautionary note., if you are dropping the bottle down a stairwell you need to have something for it to land in or it may disintegrate and cause a big mess. I dropped it into a large tub filled with sand.
figure 3. Images of the leaky bottle just before (left) and just after (right) dropping.
An index of all the demonstrations posted on this blog can be found here. Don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (email@example.com). I also welcome comments (through the comments section or via email) on improving the demonstrations.