Spinning Egg Water Sprinkler

It is fun to spin a hard-boiled egg like a top, and because it is nice to spin I spin it on everywhere. It is quite intriguing when it met a shallow pool of water, several milimeters deep, the water unexpectedly climbs up the shell of the egg until it is about half way up it sprays out horizontally like a sprinkler.

At a glance this fountain effect seems very mysterious, but later I found out that the main idea behind it can be explained easily using basic physics. To avoid complexity, we can neglect any interaction with air and coriolis force, and also assume that the egg is spinning without precession. Thus the remaining forces responsible for this effect are adhesion force between water and the egg shell, centrifugal force, and gravity. When the egg is turning, the water tends to fly away due to centrifugal force but it also pulled by adhesion force toward the surface of the egg. The resulting motion will be the water climbing up the shell. The water rises until a certain point where the adhesion force no longer able to withstand the outward force, then water droplets sprinkle outward. The following is the quantitative explanation of this effect:

Simplified view of rotating egg

Where:

A_{n} is the normal component of adhesion force

A_{\tau} is the tangential component of adhesion force (acts like friction)

r_{f} is the maximum radius of projectile

r_{s} is the cross section radius where the water break up into water drops

We consider the forces acting on an infinitesimal portion of water with mass m in a rotating reference frame. By applying Newton’s second law, we can write the following equations:

\sum F_{ \tau }=m \omega^{2} r\cos \theta-m g\sin \theta - A_{ \tau }                                                                      (1)

\sum F_{ n }=(m \omega^{2} r\sin \theta+mg\cos \theta)-A_{n}                                                                 (2)

The water can climbs up if \sum F_{\tau} is positive. Assuming that the condition to stay on the surface is satisfied, all we need is a sufficiently large angular velocity.

The water can stay on the surface if \sum F_{n}=0. However if (mg\cos \theta+m \omega^{2} r\sin \theta)<0 the water will still remains in equilibrium because the  normal force begin to act. Therefore the water will leave the surface if \sum F_{n}>0, it will happen when  (mg\cos \theta+m \omega^{2} r\sin \theta) exceed the maximum value of A_{n}.

If the egg is spinning very fast, \sum F_{\tau} will reach the value of zero when \theta\rightarrow\frac {\pi}{2}. Above that point \sum F_{\tau} is negative, the water tends to flow back down. Therefore the water will accumulate around \theta=\frac {\pi}{2}, thus the mass in equation (2) will be replaced with a much larger mass, the value non-inertial force is increased. The water will continue to accumulate until the adhesion force no longer strong enough to hold. As a result, we can predict that for large \omega and adhesion force, the dripping will most likely to occurs at the point where the cross section radius of the egg is the largest. The dripping rate will balance the accumulation rate, so that the accumulated mass doesn’t change.
However this is not the case because egg’s adhesion with water is stronger than water’s cohesion. If the adhesion force is much stronger than cohesion, the mass will accumulate so much that the force responsible for maintaining the water is no longer adhesion, it is the cohesion between water. The water which is in contact with the egg shell remains there, while the outer part of the water drip away.

To prove these theories another experiment using wax coated egg(adhesion<cohesion) was conducted, unfortunately only small amount of water was attracted to spin together with the egg , that means there exist a coriolis force that was directed opposite to the centrifugal force, effectively it’s like the egg was spinning slower. So we need a higher angular velocity to observe the dripping effect, however it is quite difficult to reach high enough angular velocity using bare hand. I hope someday I can find a tool to provide high angular velocity….

The water drops ejected from the egg surface

If the dripping occurs when the cross section radius of egg is r_{s}, The drops will leave the shell with tangential velocity \omega r_{s}. The drops will follow parabolic trajectories, the time of flight is

t=\sqrt {\frac {2 y_{s}}{g}}

During this time, the drops will travel a distance  w r_{s}\sqrt {\frac {2 y_{s}}{g}} tangentially. Then by using a simple geometry we can get the radius of the water fountain

r_{f}=\sqrt{(r_{s})^{2}+(w r_{s}\sqrt {\frac {2 y_{s}}{g}})^{2}}=r_{s}\sqrt{1+\frac {2 y_{s}\omega^{2}}{g}}

 

Paper…

1. Optical Phenomena

It all started with a question “why wet spots look dark?”.

Apparently wet spots look darker if viewed from the same side as that of the light source, however if we observe light passing through same spot from the other side, it looks brighter instead.

So the word “dark” is not a very accurate word to describe how a wet spot looks like after all–Here is better description, a wet spot looks more transparent than a dry spot. Here are some possible explanations:

  1. Fibers sticking together
    When the paper/cloth gets wet, the water only gets into the fibers, it does not stay in the gaps between fibers. Then, because of the surface tension of water, the fibers tend to stick together, leaving bigger gaps between fibers and letting more light to pass.
  2. Light’s wavelength is shorter in water
    Assuming that the fibers are opaque, light beam passing through the paper will be diffracted by the fibers. Since light’s wavelength is smaller in water, the light is less diffracted in water, i.e. more focused. Thus, the intensity of light that passes through the paper will be greater. Another way to see this is, light with smaller wavelength will “see”  relatively bigger holes and accordingly easier to pass through them.
  3.  Refractive index of paper/cloth
    If the wavelengths of light within the visible spectrum are much smaller than the dimensions of the fibers, then we can say that certain wave train of light is in a fiber. The molecules composing the fibers will respond to the light by radiating electromagnetic wave. This “new light” will interfere with the “old light” in such a way which results in delayed light(advanced in phase)– This can be easily shown by using a phasor diagram. Thus, effectively, the light covers a smaller phase each second, in other words it has a lower phase velocity than the speed of light. As we know, the quantity that measures the effective phase velocity of light is called refractive index. Note again that the refractive index that we are referring to here is the refractive index of a single fiber, not the refractive index of a group of fibers. The refractive index of a fiber (cellulose) can be found by googling to be n=1.48. When we get the cloth wet, the air between the fibers is replaced by water–Water has index of refraction(n=1.33) much closer to that of fiber ( n=1.48) than air(n=1) has. Thus, each time a light beam encounters a fiber, it is bent much less than it would be with dry cloth. Therefore, in average, a light beam can pass through many more fibers before its deflected significantly. So more of the light ends up travelling forward into the fabric.

First, I need to kill the first theory…If the whole paper is immersed in water, It will still appears more transparent than it would be in air. Therefore the first theory failed to survive. If the whole gaps is filled with water, the surface tension force acting on a single fiber will be directed in all direction, thus there is  no effect at all, the fibers won’t fluff. The same thing happens with paintbrush’s bristles when you put it in water. The bristles will stick together after you pull it out from the water.

The two remaining theories predict that if we change the water with another liquid with higher refractive index, then the paper will be more transparent. To test this hypothesis, we did an experiment:

We take a piece of paper and we put several drop of water (n=1.33)  on a place and several drop of glycerin (n=1.47) on other place. At a glance, their transparency look almost the same, and the water spot is more wavy than the glycerin one and also glycerin is easier to permeate into paper than water do. Human’s eye capability to differ intensities is not reliable, so we measure the intensity of laser beam that pass through the paper with photodiode.

the photodiode is behind the paper

The measurement shows that the glycerin spot is more transparent than the water spot! Therefore it agrees with the two remaining theories!!

There is a tie breaker experiment…  If the “refractive index” theory’s effect is more dominant, a paper will be  invincible when immersed in liquid that has refractive index similar to that of cellulose (maybe it won’t work because some paper is colored), if the “diffraction” theory is right there will be no local maximum of transparency.

There is an argument that disadvantage the “diffraction” theory. Since diffraction depends on wavelengths it means that blue light, yellow light and red light is diffracted differently. Bluer light is less diffracted while redder light is strongly diffracted. Therefore it means that when white beam is diffracted, it will look slightly bluer. But in reality it doesn’t , so it is either the “diffraction” theory is wrong or our vision is not accurate enough. Yet another argument, it doesn’t matter whether it is more focused or not, the total  intensity of light that passes through the hole is still the same. More focused–>less interference, less focused–>more interference, it doesn’t make any difference.

I think the “refractive index” theory makes a lot of sense, but I am not sure.

Another interesting phenomenon will show up if you put a wet paper on a surface and stick it; the transparency will increases significantly. The explanation comes like this, if we stick a paper into a surface, the light beam can get out through the same hole it goes in. Almost all the light that comes in can comes out, effectively it is like passing through the layer once, just like when you can see through a cloth that’s right in front of your eyes. We can see that the change in the intensity is quite large, because the effects is not added up, but it is multiplied. So when your shirt gets wet, lifting the shirt away from your skin will pretty much reduce the transparency.

2. Mechanical Phenomena

While playing with papers to find out some ideas to answer previous question, I found five more questions to think of:

  1. Why is it easier to tear wet paper than dry paper?
    Paper is made of small cellulose fibers tangled together like spaghetti, they can stick together by relying on friction force only. If you put water on it, it will lower the friction force between those fibers, thus they can slide over more easily, hence much easier to be tore.
  2. How does paper make sound when it is teared?
    I have two explanation for this:
    a) The sound is produced by snapping the fibers. When we are tearing the paper, we stretch some part of the paper until it reach the breaking tension of the fiber and suddenly the fibers snapped. The fibers are elastic, when it is stretched and suddenly cut into two, it is like stretching a spring and release it. Thus the fibers will vibrate as a spring, the greater it is stretched, the bigger the sound will be.
    b) The sound is produced by the stick-slip movement between fibers. It is the same effect that happen to your screeching door, it is a rapid transition between moving and stopping. Thus the frequency of sound produced is related to the time characteristic of the moving and stopping process.
  3. Why dry paper make more sound than wet paper?
    a) When the paper is wet, the water gets into the fiber , increasing its mass. Thus the frequency of vibration is much more lower than before, and maybe get out from human’s frequency range. And also because the friction is lower, the fibers can slide over easily, therefore not many fibers will get snapped.
    b)  When the paper is wet, the coefficient of friction between fibers is greatly reduced. Thus not much kinetic energy of the fibers is converted into sound energy.(screeching explanation)
  4. Why dry paper produces greater sound if we tear it faster?
    There will be more fibers snapped per second or there will be more screeching fibers at the same time. The sound from different parts of the paper interfere constructively, the wavelength of sound wave in the human’s hearing range is maybe long enough to account this effect.
    – The tearing is fast enough that the elastic wave in the paper has not traveled quite far in the tearing process. Thus the tearing force is focussed on a small region ,therefore those fibers within the region have very large amplitudes. And also the smaller the number of fibers participating in the vibration, the higher the frequency will be.
  5. Why wet paper is more stretched than dry paper?
    When the paper is dry, the fibers are pulled one side or the other which then creates tension. The entangled fibers are bent to some radius of curvature, because they are elastic they want to go back to their original shape(larger radius of curvature).They tend to get apart but friction force holds them together. When the paper gets wet, it lowers its static friction, therefore some strongly pulled fibers that previously unable to defeat the friction force are now able to slip off. Thus the paper will appears stretched and more wavy, and retain its shape even thought it is dried.
  6. What is the difference if we fold the paper before tearing it?
    Usually the voice will be smaller. It is because we cannot tear it in a short time. A folded paper can produce larger voice than a non folded paper of the same size if we can tear it fast enough.
Edit:I found a better explanation that seems more convincing to me. I tried sliding two edges of paper perpendicularly, surprisingly the sound produced by doing that is pretty similar with the one produced by tearing paper. It seems that the sound produced is due to the transversal vibration of the whole paper, the friction or the fibers snapping merely acts as driving force. and notice if you hold the paper near the sliding point, the frequency of sound produced becomes higher because the shorter the vibrating paper’s length the higher the frequency is(only short wavelengths of standing wave are allowed).Now which one gives more dominant driving force, friction or snapping fibers?
I think friction is more dominant, because sliding the edges gives a similar sound even without involving any snapping process. Also if we slide it faster, the frequency will be higher.