Floating balloon

An inflated toy balloon follows an erratic track when the air escapes freely. Would it be possible to let it float more or less stable in the air with a controlled outflow? I managed to do exactly that by putting a (glue stick) cap with a hole in the balloon outlet:

I started with a small hole. This did not elevate the balloon. By successively enlarging the hole I came to the point of floating. The mass of the cap makes sure that the balloon floats approximately upright.

An interesting question is whether we can calculate that the floating condition is met. For that, the thrust by the escaping air should equal the gravitation force on the balloon, the cap and the enclosed air.

I measured the following:

• Diameter of the inflated balloon: D~ 25 cm.
• Diameter of the hole in the cap: d~ 0.8 cm.
• Time the balloon floated: t~ 10 s. In this time all air escaped.
• Mass of balloon + cap: m~ 3 g.
• Over-pressure in the balloon (measured with a contra/Huygens barometer): 18 mbar= 1800 N/m².

Assuming the balloon is spherical, we compute the volume of the balloon as 4/3*pi*(25E-2/2)^3~ 8.2E-3 m^3. The gravitational force on the balloon, the cap and the mass of the air inside is therefore:

Fg~ (8.2E-3 * 1.3 + 3E-3)  * 9.8~ 0.13 N.

Here the density of air was assumed to be 1.3 kg/m^3. Note that the mass of the enclosed air is almost 4 times as much as the balloon+cap mass. One does not notice the air because its weight is balanced by the Archimedean upward force (that's also why a volume of air surrounded by air does not move spontaneously!).

The thrust for this case is given by this equation (from https://www.grc.nasa.gov/WWW/K-12/rocket/thrsteq.html):

Ft= me_dot * Ve + dp * Ae

The meaning of the symbols is:

Ft – thrust [N]

me_dot – mass flow rate through the hole [ kg/s ]

Ve – the air speed just outside the cap [ m/s ]

dp – the over pressure in the balloon [ N/m² ]

Ae – the area of the hole in the cap [ m² ]

The mass flow rate is me_dot~ (8.2E-3 * 1.3) / 10~ 1.1 E-3 kg/s.

The air speed just outside the cap can be computed by imagining the escaping air forming a cylinder of length L, with the ends having the area of the hole. The area of the hole is Ae~ pi * (0.8E-2 / 2 ) ^ 2~ 5.0 E-5 m^2. This cylinder goes through the hole in 10 s. So we have:

Ve~ (8.2E-3 / 5.0E-5) / 10~ 16.4 m/s.

We are now ready to compute the result of the thrust equation:

Ft= 1.1E-3 * 16.4 + 1800 * 5.0E-5~ 1.8E-2 + 9E-2~ 0.11 N.

This is quite close to the gravitational force Fg~ 0.13 N, notwithstanding assumptions as that the balloon is spherical, that we can see the escaping air as a cylindrical volume, etc. Of course, to validate the approach followed, this experiment should be repeated.

Note that the second term in the last equation is 5 times as large as the first term. This means that the thrust of the escaping air is negligible compared to the thrust due to the pressure difference between the balloon and the surroundings.

Civilised contempt

Recently I read a book by Carlo Strenger. The Dutch title can be translated as "Civilized Contempt - A Guide to Defending Our Freedom". The message of the book is essentially this: according to the author western people are nowadays too afraid of criticizing certain habits and beliefs of people from other cultures. They are too politically correct and too much into cultural relativism ("all cultures are equal"). Strenger's method of "civilized contempt" is to clearly state what views or habits of someone one does not like (the "contempting" part of the title), while at the same time not attacking the person itself (the "civilized"part). Apart from the question whether this separation will be felt as such by the person I wonder whether the book title is a good one. After all, "contempt" is quite a harsh word. I liked to know what the author's view is and therefore I sent the following e-mail to the author on March 17, 2018. Up till now I got no reply.

---
Dear Prof. Strenger,

I read your book "Zivilisierte Verachting. Eine Anleitng zur Verteidiging unserer Freiheit", That is, in the Dutch translation.

It is clearly written which makes it easy to agree or disagree with you. I agree with the latter part. However, I have one main objection. One that even puzzles me.

It is that the title "Ziviliesierte Verachting" ("Beschaafde minachting" in Dutch) is thus harsh in my opinion that it will hamper people to agree with the book and/or method or even read the book at all.

I once read about research into predicting the breaking up of marriages. From taped recording of conversations of partners it appeared that contempt being present was the best predictor for breaking up.

I myself automatically relate "minachting" to irrationality, spitting on the ground and not being civilized. Instead of expressing disdain I would rather say that I do not agree with someone's views, reject them, see them as dangerous, unfruitful, etc. But I would never use "minachting".

An amusing case is yesterday's reaction of our prime minister's Rutte's reaction to the most recent campaign spot of Wilderds' party PVV. The spot starts with the words "ISLAM IS" and then continues with a few minutes of adding words like "DISCRIMINATION", "VIOLENCE", "TERROR", etc. See https://www.pvv.nl/.

Rutte's reaction was: "I believe I represent the vast part of The Netherlands if I say that I think this is distasteful."

Well, maybe a good politician is by definition politically correct.
Nevertheless, my question is: don't you think your book's title is counter productive?

Yours sincerely,
Jos Groot

--- 

The value of a book is in my head, not in the text

The Myth Of The Myth Of Religious Neutrality

Now and then I read a book that goes against my own beliefs, in an attempt to not getting stuck in these beliefs. My experience is that this practice stimulates my thinking more than reading just another book that is in line with those beliefs and hence gives me pleasant feelings only. The most recent example is 'The Myth of Religious Neutrality' with the sub title 'An Essay on the Hidden Role of Religious Belief in Theories', by Roy A. Clouser (1991). I must admit I did not read it entirely – I speed red some of the later chapters - because I was mostly interested in the claim of the (sub) title: that theories are based on religious belief.

At first sight this claim is not unreasonable. Theories seem to me always based on some beliefs, albeit reasonable beliefs. For example belief in the validity of the logic used. But why should these beliefs be religious? Can Clouser proof the claim of the hidden role of religious belief?

Clouser was a professor of philosophy and religion. His book is therefore systematic, quite profound, focusing on fundamentals and therefore not for the faint of heart. Although he is clearly a Christian, he is not a 'fundamentalist' which is according to him someone who holds the 'encyclopedial assumption'. This assumption entails 'that sacred Scripture contains inspired and thus infallible statements about virtually every conceivable subject matter'. This was centuries ago a common view point. In stead, Clouser believes in a more subtle but very fundamental influence: Scripture does not dictate the contents of theories directly, but provides the basic assumption(s) they are based on. Hence the 'hidden role' of religious belief. Throughout the book it becomes apparent that this belief should better come from the Bible, according to Clouser.

How does Clouser proceed to prove the claim of the (sub) title? He starts out with some pleasantly clear, concise definitions. The one of 'religious belief' is a case in point: 'a religious believe is any belief in something or other as divine.' In this 'divine' means 'having the status of not depending on anything else', in other words, being self-existent. The Christian God is an (the only?) example. Three chapters are devoted to showing that theories in mathematics, physics and psychology are all based on a religious belief. That is, pagan beliefs, according to – again - his definition: a pagan belief is a belief that the divine is some part of the (God-)created universe, in other word, 'reality'. He gives as an example in physics Einstein's theoretical foundation. That is his (supposed) believe that the laws of logic and mathematics govern all reality (including human thought). This means that they are self-existent and thus divine. Hence Einstein's view is religious. In addition, because logic and mathematics are part of reality his religious belief is pagan.

In fact, this ends the part of the book 'proving' the claim of the (sub) title. Is the proof valid? I do not think so. A major weak point is that the claim is based on self devised definitions. One can proof anything from properly constructed definitions. Take the definition of the divine being self-existent: it is questionable whether there are non-hypothetical independent entities at all (of course the Christian God is an example according to Clouser). The 'religious' creeps in through this questionable definition.

After having shown that current theories make the wrong – pagan - assumptions Clouser devotes later chapters to biblical theories of reality and society. He skips mathematical, physical and psychological theories for two reasons: he does not have enough knowledge and his own 'theory of reality' comes into its own in social theories. The skipping further weakens his position.

In the afterword Clouser questions whether his religious, Biblical view does not divide people. He thinks this is not the case, again for two reasons. The first is that giving fundamental reasons for differences between theories will not lead to intolerance. Secondly, these reasons will bring forth a fruitful communication. What he does not mention is that he says that the Christian Bible provides the best ground for theories to be based on, which of course has the potential to divide people.

Clouser states that 'Having the right God is basic to all truth'. This looks to me as a basic assumption of himself. It seems to be the inspiration of his theory and the book in which it is described. A book which is unusual, faulty and nevertheless (or therefore?) thought provoking.

The tangent shaped track of a rain drop on a slanted tube

This picture made me think:

Fig. 1 - Constellation of metal tubes.

I took it on a railway station. The constellation of metal tubes keeps the first floor up in the air. A close up of the central tube makes clear what attracted my interest:

Look at the black tracks. Apparently rain drops fall down on the tube and create these dirt tracks while on their way down. All tracks converge to the gray strip.

I wondered whether it would be possible to compute the track of a drop. This is a typical physics problem. It turned out to be possible, as these pictures demonstrate:

Fig. 2 - Close up of Fig. 1.

The tube at the left is a different one than above and oriented vertically, to ease analysis. The blue track at the right resulting from theory resembles the track shape quite well. Note that it starts at the top at a position at the 'back' of the tube, visible in the plot right but invisible in the picture left.

Surprisingly, it turns out that the track resembles the plot of a tangent function. Straight lines, parabolas, circles, ellipses, sines, exponential functions and others - I know where one can literally see them in the real world. To my knowledge this drop track is the only example of a tangent function coming to life. Let's proceed to sketch the proof of this.

I used the Lagrangian formalism to solve the problem. This formalism is a generalization of Newton's laws. It allows one to use a suitable coordinate system, to handle constraints and to include frictional forces. For the latter I used a force which is linearly proportional to the speed of a drop. This is probably wrong, nevertheless it gives the reasonable result of above. Possibly the solution is not very sensitive to the type of friction one uses, as long as it leads to a constant velocity. Experiments show that drops move at a constant speed over an inclined surface.

For such a cylindrical problem it is only natural to use cylindrical coordinates (z, phi). The angle phi varies from 0 to 2xpi when traversing a circle at constant z. Surprisingly, when the friction is relatively large the problem can be solved analytically. These equations for the motion result in this case:

z= c1 * t

phi= 2 x atan(exp(-c2 * t + c3))

t is time in seconds. atan(x) is the inverse of tan(x). The  constants are:

• c1= -mgcos(theta)/k; m= drop mass [kg], g= gravitational acceleration (~9.8 m/s²), the frictional constant [kg/s], theta= the angle of the tube symmetry axis w.r.t. the direction of the gravitational force [deg].
• c2= mgsin(theta)/(kr); r= tube radius [m]
• c3= defines phi at t= 0 s.

Apparently the drop travels at constant speed c1 in the -z-direction, down the tube. No acceleration, hence no net force is excerted on the drop. This occurs because the frictional and gravitational force cancel each other.This is similar to a snow flake that falls at constant speed during quiet winter weather.

Fig. 3 - Tube pictured such that it appears to be vertical (left) and the theoretical result (right). Due to the rotation the gravitational force is directed parallel to the vector pointing from (0, 0, 0) to (0, 1, -1). I.e., the force vector points towards the reader, down right.

I guessed reasonable values for the parameters like the mass and tuned them subsequently to get reasonable resemblance between the analytic solution and the picture. The plots of z (left) and phi (right) I got are:

Fig. 4 - Plot of the cylindrical coordinates z (left) and phi (right) as a function of time t.

The blue curve of Fig. 2 (right) results from transforming these to (x, y, z) coordinates and plotting the result. phi asymptotically approaches 0 degrees. This corresponds to the dark band of Fig. 1 (left)where all drops converge to.

Note that one can recognize the tangent function in the right plot. This is due to the fact that atan(exp(t)) is approximately equal to a shifted and scaled version of atan(t) and hence the two have nearly the same shape. Because z is proportional to t, the track of drops on a tube thus resembles the tangent function. Check for yourself in the right plot of Fig. 3. As said before, this is the only case I know of where this function is visible in the real world.

A note on programs: I used Wolfram Alpha (https://www.wolframalpha.com/) to solve the differential equations and Octave (https://www.gnu.org/software/octave/) on Linux Mint (http://www.linuxmint.com/) to visualize the solution. The best things in life are free (https://www.youtube.com/watch?v=-GoYa1uNhYs)!

The road to reality - or beyond?

I held Roger Penrose's book “Road to reality” popularizing physics highly but I changed my my view a bit. This was due to discussing his short article “On the second law of thermodynamics” (1994) with some philosophy of physics students.

The article deals with the second law of thermodynamics. This law states that the entropy - the amount of disorder - of a closed system on average increases with time. It implies that the entropy of the universe goes up in the future. But what about the past? Penrose's solution is his "Weyl curvature hypothesis". This hypothesis states that the Weyl curvature is zero (or at least very small) at the Big Bang, and with it the entropy. More important than knowing what Weyl curvature is, is understanding what it explains:

1. The existence of the aforementioned second law.
2. The observation that the universe is homogeneous and isotropic at large scales.

The far more popular and rivaling inflation theory explains only 2. 

From the discussion it became clear that entropy has different definitions. The well established definition by Boltzmann is suited for the computation of the entropy of a gas. But what about that of the universe as a whole? Cosmology seems quite speculative.

Penrose is a good popularizer of mathematics and physics. His accompanying illustrations are unique. However, the danger is that the uninformed readers (as his readers will generally be) are seduced to see his speculative theories as being main stream. This occurs because he starts out in his books with accepted mathematical and physical material, slowly introducing his own speculations after that. Like the hypothesis of above and also his twistor theory. 

The question is whether Penrose leads one to reality or beyond.

Two ways of "counting infinity"

Do these sets have the same size? :

{1, 2, 3} and {1, 4, 9}

Is your answer the same if we extend the sets indefinitely, like this? :

{1, 2, 3, 4, ...} or {1, 4, 9, 16, ...}

Probably you determined the size by counting the number of elements in answering the first question. The sets {1, 2, 3} and {1, 4, 9} have each 3 elements so the answer to the first question is: yes, the two sets have the same size.

How to count the two infinite sets of the second question? This is impossible using conventional 1, 2, 3, --- counting. It would not end. Mathematician Cantor therefore proposed a method to find out whether two infinite sets have the same size, a way of “counting infinity”:

If we can pair all the elements of two sets than they have the same size. 

Because we can pair like this: {1, 1}, {2, 4}, {3, 9}, {4, 16}, ... Cantor's answer to the second question is the same as to the first: yes, the two sets have the same size. His method also works for finite sets, by the way. As such it is a generalization of the finite case.

But wait... something strange is going on. The set {1, 4, 9, 16, ...} is a proper sub set, a part of {1, 2, 3, 4, …} (not the same, hence “proper”). Nevertheless they have the same size. How can that be?

A mathematician's response might be that we should not speak of the size of a set but of its cardinality instead. That's the word Cantor introduced. This makes clear that we are dealing with the mathematician's world which differs from the ordinary world. In the ordinary world proper sub sets of a set are always smaller than the set - and infinite sets cannot be counted in this world. In mathematics proper sub sets of an infinite set can have the same cardinality as that set. Not the same size, which is just an everyday, non mathematical word.

So in the mathematical world it seems that we gain something - cardinality as a measure for the size of sets - and we loose something: that proper sub sets of a set are always 'smaller' than that set.

This is a kind of friction I was uneasy with for decades. I realized myself all too well the difference between 'size' and 'cardinality' of above and especially the more or less arbitrary choice of Cantor's pairing principle. Would another, maybe better choice be possible?

It felt as an impasse to me but probably not to William Byers, the author of the book 'How mathematicians think - using ambiguity, contradiction, and paradox to create mathematics'. His main point is that ambiguity etc. - things that somehow do not feel right - stimulate the formation of new mathematical ideas. Difficulties advance mathematics.

This is not an idea pertaining exclusively to mathematics. For example, koans are the Zen equivalent given by zen masters to their pupils. These are paradoxical puzzles like: “"What is the sound of one hand clapping?" To solve these one has to widen one's view of the world. A poetic, musical analog is this line from Leonard Cohen's song "Anthem": “There is a crack in everything. That's how the light gets in.” Imperfections are necessary for advancement, seems to be the suggestion.

Back to the original problem regarding infinity. How was it solved? About 100 years after Cantor the theory of “numerosities” was proposed by Vieri Benci. Like cardinality the numerosity is a measure for the size of a set. An important difference is that proper sub sets of a set have always a smaller numerosity than that set (while the cardinality can be the same).

Some more information can be found here: http://www.newappsblog.com/2014/03/counting-infinities.html. From the discussion there it appears to me that numerosity is a more complicated concept than cardinality – but I can be wrong.