Grim Reaper Does a Back-of-the-Envelope Calculation

I have a secondary super-villain identity. People on Google+ called me:
Elke the Ripper or Master of the Scythe.

elkement-reaper[FAQ] No, I don’t lost a bet. We don’t have a lawn-mower by choice. Yes, we tried the alternatives including a reel lawn-mower. Yes, I really enjoy doing this.

It is utterly exhausting – there is no other outdoor activity in summer that leaves me with the feeling of really having achieved something!

So I was curious if Grim Reaper the Physicist can express this level of exhaustion in numbers.

Just holding a scythe with arms stretched out would not count as ‘work’. Yet I believe that in this case it is the acceleration required to bring the scythe to proper speed that matters; so I will focus on work in terms of physics.

In order to keep this simple, I assume that the weight of the scythe is a few kilos (say: 5kg) concentrated at the end of a weightless pole of 1,5m length. All the kinetic energy is concentrated in this ‘point mass’.

But how fast does a blade need to move in order to cut grass? Or from experience: How fast do I move the scythe?

One sweep with the scythe takes a fraction of second – probably 0,5s. The blade traverses an arc of about 2m.

Thus the average speed is: 2m / 0,5s = 4m/s

However, using this speed in further calculations does not make much sense: The scythe has two handles that allow for exerting a torque – the energy goes into acceleration of the scythe.

If an object with mass m is accelerated from a velocity of zero to a peak velocity vmax the kinetic energy acquired is calculated from the maximum velocity: m vmax2 / 2. How exactly the velocity has changed with time does not matter – this is just conservation of energy.

But what is the peak velocity?

For comparison: How fast do lawn-mower blades spin?

This page says: at 3600 revolutions per minute when not under load, dropping to about 3000 when under load. How fast would I have to move the scythe to achieve the same?

Velocity of a rotating body is angular velocity times radius. Angular velocity is 2Pi – a full circle – times the frequency, that is revolutions per time. The radius is the length of the pole that I use as a simplified model.

So the scythe on par with a lawn-mower would need to move at:
2Pi * (3000 rev./minute) / (60 seconds/minute) * 1,5m = 471m/s

This would result in the following energy per arc swept. I use only SI units, so the resulting energy is in Joule:

Energy needed for acceleration: 5kg * (471m/s)2 / 2 = 555.000J = 555kJ

I am assuming that this energy is just consumed (dissipated) to cut the grass; the grass brings the scythe to halt, and it is decelerated to 0m/s again.

Using your typical food-related units:
1 kilocalorie is 4,18kJ, so this amounts to about 133kcal (!!)

That sounds way too much already: Googling typical energy consumptions for various activities I learn that easy work in the garden needs about 100-150kcal kilocalories per half an hour!

If scything were that ‘efficient’ I would put into practice what we always joke about: Offer outdoor management trainings to stressed out IT managers who want to connect with their true selves again through hard work and/or work-out most efficiently. So they would pay us for the option to scythe our grass.

But before I crank down the hypothetical velocity again, I calculate the energy demand per half an hour:

I feel exhausted after half an hour of scything. I pause a few seconds before the next – say 10s – on average. In reality it is probably more like:

scythe…1s…scythe…1s…scythe…1s….scythe…1s….scythe…longer break, gasping for air, sharpen the scythe.

I assume a break of 9,5s on average to make the calculation simpler. So this is 1 arc swept per 10 seconds, 6 arcs per minute, and 180 per half an hour. After half on hour I need to take longer break.

So using that lawn-mower-style speed this would result in:

Energy per half an hour if I were a lawn-mower: 133kJcal * 180 = 23.940kcal

… about five times the daily energy demands of a human being!

Velocity enters the equation quadratically. Assuming now that my peak scything speed is really only a tenth of the speed of a lawn-mower, 47m/2, which is still about 10 times my average speed calculated the beginning, this would result in one hundredth the energy.

A bit more realistic energy per half an hour of scything is then: 239kcal

Just for comparison – to get a feeling for those numbers: Average acceleration is maximum velocity over time. Thus 47m/s would result in:

Average acceleration: (47m/s) / (0,5s)  =  94m/s2

A fast car accelerates to 100km/h within 3 seconds, at (100/3,6)m/s / 3s = 9m/s2

So my assumed scythe’s acceleration is about 10 times a Ferrari’s!

Now I would need a high-speed camera, determine speed exactly and find a way to calculate actual energy needed for cutting.

Is there some conclusion?

This was just playful guesswork but the general line of reasoning and cross-checking orders of magnitude outlined here is not much different from when I try to get my simulations of our heat pump system right – based on unknown parameters, such as the effect of radiation, the heat conduction of ground, and the impact of convection in the water tank. The art is not so much in gettting numbers exactly right but in determining which parameters matter at all and how sensitive the solution is to a variation of those. In this case it would be crucial to determine peak speed more exactly.

In physics you can say the same thing in different ways – choosing one way over the other can make the problem less complex. As in this case, using total energy is often easier than trying to figure out the evolution of forces or torques with time.

results-achieved-by-scythe-masterThe two images above were taken in early spring – when the ‘lawn’ / meadow was actually still growing significantly. Since we do not water it, the relentless Pannonian sun already started to turn it into a mixture of green and brown patches.

This is how the lawn looks now, one week after latest scything. This is not intended to be beautiful – I wanted to add a realistic picture as I had been asked about the ‘quality’ compared to a lawn-mower. Result: Good enough for me!

Scything: One week after

 

How to Introduce Special Relativity (Historical Detour)

I am just reading the volume titled Waves in my favorite series of ancient textbooks on Theoretical Physics by German physics professor Wilhelm Macke. I tried to resist the urge to write about seemingly random fields of physics, and probably weird ways of presenting them – but I can’t resist any longer.

There are different ways to introduce special relativity. Typically, the Michelson-Morely experiment is presented first, as our last attempt in a futile quest to determine to absolute speed in relation to “ether”. In order to explain these results we have to accept the fact that the speed of light is the same in any inertial frame. This is weird and non-intuitive: We probably can’t help but compare a ray of light to a bunch of bullets or a fast train – whose velocity relative to us does change with our velocity. We can outrun a train but we can’t outrun light.

Michelson–Morley experiment

The Michelson–Morley experiment: If light travels in a system – think: space ship – that moves at velocity v with respect to absolute space the resulting velocity should depend on the angle between the system’s velocity and the absolute velocity. Just in the same way as the observed relative velocity of a train becomes zero if we manage to ride besides it in a car driving at the same speed as the train. But this experiments shows – via non-detected interference of beam of alleged varying velocities – that we must not calculate relative velocities of beams of light. (Wikimedia)

Yet, not accepting it would lead to even more weird consequences: After all, the theory of electromagnetism had always been relativistically invariant. The speed of light shows up as a constant in the related equations which explain perfectly how waves of light behaves.

I think the most straight-forward way to introduce special relativity is to start from its core ideas (only) – the constant speed of light and the equivalence of frames of reference. This is the simplicity and beauty of symmetry. No need to start with trains and lightning bolts, as Matthew Rave explained so well. For the more visually inclined there is an ingenious and nearly purely graphical way, called k-calculus (that is however seldom taught AFAIK – I had stumbled upon it once in a German book on relativity).

From the first principles all the weirdness of length contraction and time dilation follows naturally.

But is there a way to understand it a bit better though?

Macke also starts from the Michelson-Morely experiment  – and he adds the fact that it can be “explained” by the  Lorentz’ contraction hypothesis: Allowing for direction-dependent velocities – as in “ether theory” – but adding the odd fact that rulers contract in the direction of the unobservable absolution motion makes the differences the rays of light traverse go away. It also “explains” time dilatation if you consider your typical light clock and factor in the contraction of lengths:

Light clock

The classical light clock: Light travels between two mirrors. When it hits a mirror it “ticks”. If the clock moves relatively to an observer the path to be traversed between ticks appears to be longer. Thus measurement of time is tied to measurement of spatial distances.

However, length contraction could be sort of justified by tracing it back to the electromagnetic underpinnings of stuff we use in the lab. And it is the theory of electromagnetism where the weird constant speed of light sneaks in.

Contraction can be visualized by stating that like rulers and clocks are finally made from atoms, ions or molecules, whose positions are determined by electromagnetic forces. The perfect sphere of the electrostatic potential around a point charge would be turned into an ellipsoid if the charge starts moving – hence the contraction. You could hypothesize that only “electromagnetic stuff” might be subject to contraction and there might be “mechanical stuff” that would allow for measuring true time and spatial dimensions.

Thus the new weird equations about contracting rulers and slowing time are introduced as statements about electromagnetic stuff only. We use them to calculate back and forth between lengths and times displayed on clocks that suffer from the shortcomings of electromagnetic matter. The true values for x,y,z,t are still there, but finally inaccessible as any matter is electromagnetic.

Yes, this explanation is messy as you mix underlying – but not accessible – direction-dependent velocities with the contraction postulate added on top. This approach misses the underlying simplicity of the symmetry in nature. It is a historical approach, probably trying to do justice to the mechanical thought experiments involving trains and clocks that Einstein had also used (and that could be traced back to his childhood spent basically in the electrical engineering company run by his father and uncle, according to this biography).

What I found fascinating though is that you get consistent equations assuming the following:

  • There are true co-ordinates we can never measure; for those Galileian Transformations remain valid, that is: Time is the same in all inertial frames and distances just differ by time times the speed of the frame of reference.
  • There are “apparent” or “electromagnetic” co-ordinates that follow Lorentz Transformations – of which length contraction and time dilations are consequences.

To make these sets of transformations consistent you have to take into account that you cannot synchronize clocks in different locations if you don’t know the true velocity of the frame of reference. Synchronization is done by placing an emitter of light right in the middle of the two clocks to be synchronized, sending signals to both clocks. This is correct only if the emitter is at rest with respect to both clocks. But we cannot determine when it is at rest because we never know the true velocity.

What you can do is to assume that one frame of reference is absolutely at rest, thus implying that (true) time is independent of spatial dimensions, and the other frame of reference moving in relation to it suffers from the problem of clock synchronization – thus in this frame true time depends on the spatial co-ordinates used in that frame.

The final result is the same when you eliminate the so-called true co-ordinates from the equations.

I don’t claim its the best way to explain special relativity – I just found it interesting, as it tries to take the just hypothetical nature of 4D spacetime as far as possible while giving results in line with experiments.

And now explaining the really important stuff – and another historical detour in its own right

Yes, I changed the layout. My old theme, Garland, had been deprecated by wordpress.com. I am nostalgic – here is a screenshot –  courtesy to visitors who will read this in 200 years.

elkement.wordpress.com with theme Garland

elkement.wordpress.com using theme Garland – from March 2012 to February 2014 – with minor modifications made to colors and stylesheet in 2013.

I had checked it with an iPhone simulator – and it wasn’t simply too big or just “not responsive”, the top menu bar boundaries of divs looked scrambled. Thus I decided the days of Garland the three-column layout are over.

Now you can read my 2.000 words posts on your mobile devices – something I guess everybody has eagerly anticipated.

And I have just moved another nearly 1.000 words of meta-philosophizing on the value of learning such stuff (theory of relativity, not WordPress) from this post to another draft.

Non-Linear Art. (Should Actually Be: Random Thoughts on Fluid Dynamics)

In my favorite ancient classical mechanics textbook I found an unexpected statement. I think 1960s textbooks weren’t expected to be garnished with geek humor or philosophical references as much as seems to be the default today – therefore Feynman’s books were so refreshing.

Natural phenomena featured by visual artists are typically those described by non-linear differential equations . Those equations allow for the playful interactions of clouds and water waves of ever changing shapes.

So fluid dynamics is more appealing to the artist than boring electromagnetic waves.

Grimshaw, John Atkinson - In Peril - 1879

Is there an easy way to explain this without too much math? Most likely not but I try anyway.

I try to zoom in on a small piece of material, an incredibly small cube of water in a flow at a certain point of time. I imagine this cube as decorated by color. This cube will change its shape quickly and turn into some irregular shape – there are forces pulling and pushing – e.g. gravity.

This transformation is governed by two principles:

  • First, mass cannot vanish. This is classical physics, no need to consider the generation of new particles from the energy of collisions. Mass is conserved locally, that is if some material suddenly shows up at some point in space, it had to have been travelling to that point from adjacent places.
  • Second, Newton’s law is at play: Forces are equal to a change momentum. If we know the force acting at time t and point (x,y,z), we know how much momentum will change in a short period of time.

Typically any course in classical mechanics starts from point particles such as cannon balls or planets – masses that happen to be concentrated in a single point in space. Knowing the force at a point of time at the position of the ball we know the acceleration and we can calculate the velocity in the next moment of time.

This also holds for our colored little cube of fluid – but we usually don’t follow decorated lumps of mass individually. The behavior of the fluid is described perfectly if we know the mass density and the velocity at any point of time and space. Think little arrows attached to each point in space, probably changing with time, too.

Aerodynamics of model car

Digesting that difference between a particle’s trajectory and an anonymous velocity field is a big conceptual leap in my point of view. Sometimes I wonder if it would be better not to learn about the point approach in the first place because it is so hard to unlearn later. Point particle mechanics is included as a special case in fluid mechanics – the flowing cannon ball is represented by a field that has a non-zero value only at positions equivalent to the trajectory. Using the field-style description we would say that part of the cannon ball vanishes behind it and re-appears “before” it, along the trajectory.

Pushing the cube also moves it to another place where the velocity field differs. Properties of that very decorated little cube can change at the spot where it is – this is called an explicit dependence on time. But it can also change indirectly because parts of it are moved with the flow. It changes with time due to moving in space over a certain distance. That distance is again governed by the velocity – distance is velocity times period of time.

Thus for one spatial dimension the change of velocity dv associated with dt elapsed is also related to a spatial shift dx = vdt. Starting from a mean velocity of our decorated cube v(x,t) we end up with v(x + vdt, t+dt) after dt has elapsed and the cube has been moved by vdt. For the cannon ball we could have described this simply as v(t + dt) as v was not a field.

And this is where non-linearity sneaks in: The indirect contribution via moving with the flow, also called convective acceleration, is quadratic in v – the spatial change of v is multiplied by v again. If you then allow for friction you get even more nasty non-linearities in the parts of the Navier-Stokes equations describing the forces.

My point here is that even if we neglect dissipation (describing what is called dry water tongue-in-cheek) there is already non-linearity. The canonical example for wavy motions – water waves – is actually rather difficult to describe due to that, and you need to resort to considering small fluctuations of the water surface even if you start from the simplest assumptions.

The tube

On the Relation of Jurassic Park and Alien Jelly Flowing through Hyperspace

Yes, this is a serious physics post – no. 3 in my series on Quantum Field Theory.

I promised to explain what Quantization is. I will also argue – again – that classical mechanics is unjustly associated with pictures like this:

Steampunk wall clock (Wikimedia)

… although it is more like this:

Timelines in Back to the Future | By TheHYPO [CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0) or GFDL (http://www.gnu.org/copyleft/fdl.html)], via Wikimedia Commons

 

 

 

 

 

 

 

 

This shows the timelines in Back to the Future – in case you haven’t recognized it immediately.

What I am trying to say here is – again – is so-called classical theory is as geeky, as weird, and as fascinating as quantum physics.

Experts: In case I get carried away by my metaphors – please see the bottom of this post for technical jargon and what I actually try to do here.

Get a New Perspective: Phase Space

I am using my favorite simple example: A point-shaped mass connected to an massless spring or a pendulum, oscillating forever – not subject to friction.

The speed of the mass is zero when the motion changes from ‘upward’ to ‘downward’. It is maximum when the pendulum reaches the point of minimum height. Anything oscillates: Kinetic energy is transferred to potential energy and back. Position, velocity and acceleration all follow wavy sine or cosine functions.

For purely aesthetic reasons I could also plot the velocity versus position:

Simple Harmonic Motion Orbit | By Mazemaster (Own work) [Public domain], via Wikimedia Commons

From a mathematical perspective this is similar to creating those beautiful Lissajous curves:  Connecting a signal representing position to the x input of an oscillosope and the velocity signal to the y input results in a circle or an ellipse:

Lissajous curves | User Fiducial, Wikimedia

This picture of the spring’s or pendulum’s motion is called a phase portrait in phase space. Actually we use momentum, that is: velocity times mass, but this is a technicality.

The phase portrait is a way of depicting what a physical system does or can do – in a picture that allows for quick assessment.

Non-Dull Phase Portraits

Real-life oscillating systems do not follow simple cycles. The so-called Van der Pol oscillator is a model system subject to damping. It is also non-linear because the force of friction depends on the position squared and the velocity. Non-linearity is not uncommon; also the friction an airplane or car ‘feels’ in the air is proportional to the velocity squared.

The stronger this non-linear interaction is (the parameter mu in the figure below) the more will the phase portrait deviate from the circular shape:

Van der pols equation phase portrait | By Krishnavedala (Own work) [CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0) or GFDL (http://www.gnu.org/copyleft/fdl.html)], via Wikimedia Commons

Searching for this image I have learned from Wikipedia that the Van der Pol oscillator is used as a model in biology – here the physical quantity considered is not a position but the action potential of a neuron (the electrical voltage across the cell’s membrane).

Thus plotting the rate of change of in a quantity we can measure plotted versus the quantity itself makes sense for diverse kinds of systems. This is not limited to natural sciences – you could also determine the phase portrait of an economic system!

Addicts of popular culture memes might have guessed already which phase portrait needs to be depicted in this post:

Reconnecting to Popular Science

Chaos Theory has become popular via the elaborations of Dr. Ian Malcolm (Jeff Goldblum) in the movie Jurassic Park. Chaotic systems exhibit phase portraits that are called Strange Attractors. An attractor is the set of points in phase space a system ‘gravitates’ to if you leave it to itself.

There is no attractor for the simple spring: This system will trace our a specific circle in phase space forever – the larger the bigger the initial push on the spring is.

The most popular strange attractor is probably the Lorentz Attractor. It  was initially associated with physical properties characteristic of temperature and the flow of air in the earth’s atmosphere, but it can be re-interpreted as a system modeling chaotic phenomena in lasers.

It might be apocryphal but I have been told that it is not the infamous flap of the butterfly’s wing that gave the related effect its name, but rather the shape of the three-dimensional attractor:

Lorenz system r28 s10 b2-6666 | By Computed in Fractint by Wikimol [Public domain], via Wikimedia Commons

We had Jurassic Park – here comes the jelly!

A single point-particle on a spring can move only along a line – it has a single degree of freedom. You need just a two-dimensional plane to plot its velocity over position.

Allowing for motion in three-dimensional space means we need to add additional dimensions: The motion is fully characterized by the (x,y,z) positions in 3D space plus the 3 components of velocity. Actually, this three-dimensional vector is called velocity – its size is called speed.

Thus we need already 6 dimensions in phase space to describe the motion of an idealized point-shaped particle. Now throw in an additional point-particle: We need 12 numbers to track both particles – hence 12 dimensions in phase space.

Why can’t the two particles simply use the same space? Both particles still live in the same 3D space, they could also inhabit the same 6D phase space. The 12D representation has an advantage though: The whole system is represented by a single dot which make our lives easier if we contemplate different systems at once.

Now consider a system consisting of zillions of individual particles. Consider 1 cubic meter of air containing about 1025 molecules. Viewing these particles in a Newtonian, classical way means to track their individual positions and velocities. In a pre-quantum mechanical deterministic assessment of the world you know the past and the future by calculating these particles’ trajectories from their positions and velocities at a certain point of time.

Of course this is not doable and leads to practical non-determinism due to calculation errors piling up and amplifying. This is a 1025 body problem, much much much more difficult than the three-body problem.

Fortunately we don’t really need all those numbers in detail – useful properties of a gas such as the temperature constitute gross statistical averages of the individual particles’ properties. Thus we want to get a feeling how the phase portrait develops ‘on average’, not looking too meticulously at every dot.

The full-blown phase space of the system of all molecules in a cubic meter of air has about 1026 dimensions – 6 for each of the 1025 particles (Physicists don’t care about a factor of 6 versus a factor of 10). Each state of the system is sort of a snapshot what the system really does at a point of time. It is a vector in 1026 dimensional space – a looooong ordered collection of numbers, but nonetheless conceptually not different from the familiar 3D ‘arrow-vector’.

Since we are interesting in averages and probabilities we don’t watch a single point in phase space. We don’t follow a particular system.

We rather imagine an enormous number of different systems under different conditions.

Considering the gas in the cubic vessel this means: We imagine molecule 1 being at the center and very fast whereas molecule 10 is slow and in the upper right corner, and molecule 666 is in the lower left corner and has medium. Now extend this description to 1025 particles.

But we know something about all of these configurations: There is a maximum x, y and z particles can have – the phase portrait is limited by these maximum dimensions as the circle representing the spring was. The particles have all kinds of speeds in all kinds of directions, but there is a most probably speed related to temperature.

The collection of the states of all possible systems occupy a patch in 1026 dimensional phase space.

This patch gradually peters out at the edges in velocities’ directions.

Now let’s allow the vessel for growing: The patch will become bigger in spatial dimensions as particles can have any position in the larger cube. Since the temperature will decrease due to the expansion the mean velocity will decrease – assuming the cube is insulated.

The time evolution of the system (of these systems, each representing a possible system) is represented by a distribution of this hyper-dimensional patch transforming and morphing. Since we consider so many different states – otherwise probabilities don’t make sense – we don’t see the granular nature due to individual points – it’s like a piece of jelly moving and transforming.

Precisely defined initial configurations of systems configurations have a tendency to get mangled and smeared out. Note again that each point in the jelly is not equivalent to a molecule of gas but it is a point in an abstract configuration space with a huge number of dimensions. We can only make it accessible via projections into our 3D world or a 2D plane.

The analogy to jelly or honey or any fluid is more apt than it may seem

The temporal evolution in this hyperspace is indeed governed by equations that are amazingly similar to those governing an incompressible liquid – such as water. There is continuity and locality: Hyper-Jelly can’t get lost and be created. Any increase in hyper-jelly in a tiny volume of phase space can only be attributed to jelly flowing in to this volume from adjacent little volumes.

In summary: Classical mechanical systems comprising many degrees of freedom – that is: many components that have freedom to move in a different way than other parts of the system – can be best viewed in the multi-dimensional space whose dimensions are (something like) positions and (something like) the related momenta.

Can it get more geeky than that in quantum theory?

Finally: Quantization

I said in the previous post that quantization of fields or waves is like turning down intensity in order to bring out the particle-like rippled nature of that wave. In the same way you could say that you add blurry waviness to idealized point-shaped particles.

Another is to consider the loss in information via Heisenberg’s Uncertainly Principle: You cannot know both the position and the momentum of a particle or a classical wave exactly at the same time. By the way, this is why we picked momenta  and not velocities to generate phase space.

You calculate positions and momenta of small little volumes that constitute that flowing and crawling patches of jelly at a point of time from positions and momenta the point of time before. That’s the essence of Newtonian mechanics (and conservation of matter) applied to fluids.

Doing numerical calculation in hydrodynamics you think of jelly as divided into small little flexible cubes – you divide it mentally using a grid, and you apply a mathematical operation that creates the new state of this digitized jelly from the old one.

Since we are still discussing a classical world we do know positions and momenta with certainty. This translates to stating (in math) that it does not matter if you do calculations involving positions first or for momenta.

There are different ways of carrying out steps in these calculations because you could do them one way of the other – they are commutative.

Calculating something in this respect is similar to asking nature for a property or measuring that quantity.

Thus when we apply a quantum viewpoint and quantize a classical system calculating momentum first and position second or doing it the other way around will yield different results.

The quantum way of handling the system of those  1025 particles looks the same as the classical equations at first glance. The difference is in the rules for carrying out calculation involving positions and momenta – so-called conjugate variables.

Thus quantization means you take the classical equations of motion and give the mathematical symbols a new meaning and impose new, restricting rules.

I probably could just have stated that without going off those tangent.

However, any system of interest in the real world is not composed of isolated particles. We live in a world of those enormous phase spaces.

In addition, working with large abstract spaces like this is at the heart of quantum field theory: We start with something spread out in space – a field with infinite degrees in freedom. Considering different state vectors in these quantum systems is considering all possible configurations of this field at every point in space!

_______________________________________

Expert information:

I have taken a detour through statistical mechanics: Introducing Liouville equations as equation of continuity in a multi-dimensional phase space. The operations mentioned – related to positions of velocities – are the replacement of time derivatives via Hamilton’s equations. I resisted the temptation to mention the hyper-planes of constant energy. Replacing the Poisson bracket in classical mechanics with the commutator in quantum mechanics turns the Liouville equation into its quantum counterpart, also called Von Neumann equation.

I know that a discussion about the true nature of temperature is opening a can of worms. We should rather describe temperature as the width of a distribution rather than the average, as a beam of molecules all travelling in the same direction at the same speed have a temperature of zero Kelvin – not an option due to zero point energy.

The Lorenz equations have been applied to the electrical fields in lasers by Haken – here is a related paper. I did not go into the difference of the phase portrait of a system showing its time evolution and the attractor which is the system’s final state. I also didn’t stress that was is a three dimensional image of the Lorenz attractor and in this case the ‘velocities’ are not depicted. You could say it is the 3D projection of the 6D phase portrait. I basically wanted to demonstrate – using catchy images, admittedly – that representations in phase space allows for a quick assessment of a system.

I also tried to introduce the notion of a state vector in classical terms, not jumping to bras and kets in the quantum world as if a state vector does not have a classical counterpart.

I have picked an example of a system undergoing a change in temperature (non-stationary – not the example you would start with in statistical thermodynamics) and swept all considerations on ergodicity and related meaningful time evolutions of systems in phase space under the rug.

May the Force Field Be with You: Primer on Quantum Mechanics and Why We Need Quantum Field Theory

As Feynman explains so eloquently – and yet in a refreshingly down-to-earth way – understanding and learning physics works like this: There are no true axioms, you can start from anywhere. Your physics knowledge is like a messy landscape, built from different interconnected islands of insights. You will not memorize them all, but you need to recapture how to get from one island to another – how to connect the dots.

The beauty of theoretical physics is in jumping from dot to dot in different ways – and in pondering on the seemingly different ‘philosophical’ worldviews that different routes may provide.

This is the second post in my series about Quantum Field Theory, and I  try to give a brief overview on the concept of a field in general, and on why we need QFT to complement or replace Quantum Mechanics. I cannot avoid reiterating some that often quoted wave-particle paraphernalia in order to set the stage.

From sharp linguistic analysis we might conclude that is the notion of Field that distinguishes Quantum Field Theory from mere Quantum Theory.

I start with an example everybody uses: a so-called temperature field, which is simply: a temperature – a value, a number – attached to every point in space. An animation of monthly mean surface air temperature could be called the temporal evolution of the temperature field:

Monthly Mean Temperature

Solar energy is absorbed at the earth’s surface. In summer the net energy flow is directed from the air to the ground, in winter the energy stored in the soil is flowing to the surface again. Temperature waves are slowly propagating perpendicular to the surface of the earth.

The gradual evolution of temperature is dictated by the fact that heat flows from the hotter to the colder regions. When you deposit a lump of heat underground – Feynman once used an atomic bomb to illustrate this point – you start with a temperature field consisting of a sharp maximum, a peak, located in a region the size of the bomb. Wait for some minutes and this peak will peter out. Heat will flow outward, the temperature will rise in the outer regions and decrease in the center:

Diffluence of a bucket of heat, goverend by the Heat Transfer EquationModelling the temperature field (as I did – in relation to a specific source of heat placed underground) requires to solve the Heat Transfer Equation which is the mathy equivalent of the previous paragraph. The temperature is calculated step by step numerically: The temperature at a certain point in space determines the flow of heat nearby – the heat transferred changes the temperature – the temperature in the next minute determines the flow – and on and on.

This mundane example should tell us something about a fundamental principle – an idea that explains why fields of a more abstract variety are so important in physics: Locality.

It would not violate the principle of the conservation of energy if a bucket of heat suddenly disappeared in once place and appeared in another, separated from the first one by a light year. Intuitively we know that this is not going to happen: Any disturbance or ripple is transported by impacting something nearby.

All sorts of field equations do reflect locality, and ‘unfortunately’ this is the reason why all fundamental equations in physics require calculus. Those equations describe in a formal way how small changes in time and small variations in space do affect each other. Consider the way a sudden displacement traverses a rope:

Propagation of a waveSound waves travelling through air are governed by local field equations. So are light rays or X-rays – electromagnetic waves – travelling through empty space. The term wave is really a specific instance of the more generic field.

An electromagnetic wave can be generated by shaking an electrical charge. The disturbance is a local variation in the electrical field which gives rises to a changing magnetic field which in turn gives rise a disturbance in the electrical field …

Electromagneticwave3D

Electromagnetic fields are more interesting than temperature fields: Temperature, after all, is not fundamental – it can be traced back to wiggling of atoms. Sound waves are equivalent to periodic changes of pressure and velocity in a gas.

Quantum Field Theory, however, should finally cover fundamental phenomena. QFT tries to explain tangible matter only in terms of ethereal fields, no less. It does not make sense to ask what these fields actually are.

I have picked light waves deliberately because those are fundamental. Due to historical reasons we are rather familiar with the wavy nature of light – such as the colorful patterns we see on or CDs whose grooves act as a diffraction grating:

Michael Faraday had introduced the concept of fields in electromagnetism, mathematically fleshed out by James C. Maxwell. Depending on the experiment (that is: on the way your prod nature to give an answer to a specifically framed question) light may behave more like a particle, a little bullet, the photon – as stipulated by Einstein.

In Compton Scattering a photon partially transfers energy when colliding with an electron: The change in the photon’s frequency corresponds with its loss in energy. Based on the angle between the trajectories of the electron and the photon energy and momentum transfer can be calculated – using the same reasoning that can be applied to colliding billiard balls.

Compton Effect

We tend to consider electrons fundamental particles. But they give proof of their wave-like properties when beams of accelerated electrons are utilized in analyzing the microstructure of materials. In transmission electron microscopy diffraction patterns are generated that allow for identification of the underlying crystal lattice:

A complete quantum description of an electron or a photon does contain both the wave and particle aspects. Diffraction patterns like this can be interpreted as highlighting the regions where the probabilities to encounter a particle are maximum.

Schrödinger has given the world that famous equation named after him that does allow for calculating those probabilities. It is his equation that let us imagine point-shaped particles as blurred wave packets:

Schrödinger’s equation explains all of chemistry: It allows for calculating the shape of electrons’ orbitals. It explains the size of the hydrogen atom and it explains why electrons can inhabit stable ‘orbits’ at all – in contrast to the older picture of the orbiting point charge that would lose energy all  the time and finally fall into the nucleus.

But this so-called quantum mechanical picture does not explain essential phenomena though:

  • Pauli’s exclusion principle explains why matter is extended in space – particles need to put into different orbitals, different little volumes in space. But It is s a rule you fill in by hand, phenomenologically!
  • Schrödinger’s equations discribes single particles as blurry probability waves, but it still makes sense to call these the equivalents of well-defined single particles. It does not make sense anymore if we take into account special relativity.

Heisenberg’s uncertainty principle – a consequence of Schrödinger’s equation – dictates that we cannot know both position and momentum or both energy and time of a particle. For a very short period of time conservation of energy can be violated which means the energy associated with ‘a particle’ is allowed to fluctuate.

As per the most famous formula in the world energy is equivalent to mass. When the energy of ‘a particle’ fluctuates wildly virtual particles – whose energy is roughly equal to the allowed fluctuations – can pop into existence intermittently.

However, in order to make quantum mechanics needed to me made compatible with special relativity it was not sufficient to tweak Schrödinger’s equation just a bit.

Relativistically correct Quantum Field Theory is rather based on the concept of an underlying field pervading space. Particles are just ripples in this ur-stuff – I owe to Frank Wilczek for that metaphor. A different field is attributed to each variety of fundamental particles.

You need to take a quantum leap… It takes some mathematical rules to move from the classical description of the world to the quantum one, sometimes called quantization. Using a very crude analogy quantization is like making a beam of light dimmer and dimmer until it reveals its granular nature – turning the wavy ray of light into a cascade of photonic bullets.

In QFT you start from a classical field that should represent particles and then apply the machinery quantization to that field (which is called second quantization although you do not quantize twice.). Amazingly, the electron’s spin and Pauli’s principle are natural consequences if you do that right. Paul Dirac‘s achievement in crafting the first relativistically correct equation for the electron cannot be overstated.

I found these fields the most difficult concepts to digest, but probably for technical reasons:

Historically  – and this includes some of those old text books I am so fond of – candidate versions of alleged quantum mechanical wave equations have been tested to no avail, such as the Klein-Gordon equation. However this equation turned out to make sense later – when re-interpreted as a classical field equation that still needs to be quantized.

It is hard to make sense of those fields intuitively. However, there is one field we are already familiar with: Photons are ripples arising from the electromagnetic field. Maxwell’s equations describing these fields had been compatible with special relativity – they predate the theory of relativity, and the speed of light shows up as a natural constant. No tweaks required!

I will work hard to turn the math of quantization into comprehensive explanations, risking epic failure. For now I hand over to MinutePhysics for an illustration of the correspondence of particles and fields.

Disclaimer – Bonus Track:

In this series I do not attempt to cover latest research on unified field theories, quantum gravity and the like. But since I started crafting this article, writing about locality when that article on an alleged simple way to replace field theoretical calculations went viral. The principle of locality may not hold anymore when things get really interesting – in the regime of tiny local dimensions and high energy.

Space Balls, Baywatch and the Geekiness of Classical Mechanics

This is the first post in my series about Quantum Field Theory. What a let-down: I will just discuss classical mechanics.

There is a quantum mechanics, and in contrast there is good old classical, Newtonian mechanics. The latter is a limiting case of the former. So there is some correspondence between the two, and there are rules that let you formulate the quantum laws from the classical laws.

But what are those classical laws?

Chances are high that classical mechanics reminds you of pulleys and levers, calculating torques of screws and Newton’s law F = ma: Force is equal to mass times acceleration.

I argue that classical dynamics is most underrated in terms of geek-factor and philosophical appeal.

[Space Balls]

The following picture might have been ingrained in your brain: A force is tugging at a physical object, such as earth’s gravity is attracting a little ball travelling in space. Now the ball moves – it falls. Actually the moon also falls in a sense when it is orbiting the earth.

Newton's cannon ball.

Cannon ball and gravity. If the initial velocity is too small the ball traverses a parabola and eventually reaches the ground (A, B). If the ball is just given the right momentum, it will fall forever and orbit the earth (C). If the velocity is too high, the ball will escape the gravitational field (E). (Wikimedia). Now I said it – ‘field’! – although I tried hard to avoid it in this post.

When bodies move their positions change. The strength of the gravitational force depends on the distance from the mass causing it, thus the force felt by the moving ball changes. This is why the three-body problem is hard: You need a computer for calculating the forces three or more planets exert on each other at every point of time.

So this is the traditional mental picture associated associated with classical mechanics. It follows these incremental calculations:
Force acts – things move – configuration changes – force depends on configuration – force changes.

In order to get this going you need to know the configuration at the beginning – the positions and the velocities of all planets involved.

So in summary we need:

  • the dependence of the force on the position of the masses.
  • the initial conditions – positions and velocities.
  • Newton’s law.

But there is an alternative description of classical dynamics, offering an alternative philosophy of mechanics so to speak. The description is mathematically equivalent, yet it feels unfamiliar.

In this case we trade the knowledge of positions and velocities for fixing the positions at a start time and an end time. Consider it a sort of game: You know where the planets are at time t1 and at time t2. Now figure out how they have moved / will move between t1 and t2. Instead of the force we consider another, probably more mysterious property:

It is called the action. The action has a dimension of [energy time], and – as the force – it has all information about the system.

The action is calculated by integrating…. I am reluctant to describe how the action is calculated. Action (or its field-y counterparts) will be considered the basic description of a system – something that is given, in the way had been forces had been considered given in the traditional picture. The important thing is: You attach a number to each imaginable trajectory, to each possible history.

The trajectory a particle traverses in time slot t1-t2 are determined by the Principle of Least Action (which ‘replaces’ Newton’s law): The action of the system is minimal for the actual trajectories. Any deviation – such as a planet travelling in strange loops – would increase the action.

Principle of Least Action.

Principle of least action. Given: The positions of the particle at start time t1 and end t2. Calculated: The path the particle traverse – by testing all possible paths and calculating their associated actions. Near the optimum (red) path the variation does hardly vary (Wikimedia).

This sounds probably awkward – why would you describe nature like this?
(Of course one answer is: this description will turn out useful in the long run – considering fields in 4D space-time. But this answer is not very helpful right now).

That type of logic is useful in other fields of physics: A related principle lets you calculate the trajectory of a beam of light: Given the start point and the end point a beam, light will pick the path that is traversed in minimum time (This rule is called Fermat’s principle).

This is obvious for a straight laser beam in empty space. But Fermat’s principle allows for picking the correct path in less intuitive scenarios, such as: What happens at the interface between different materials, say air and glass? Light is faster in air than in glass, thus is makes sense to add a kink to the path and utilize air as much as possible.

[Baywatch]

Richard Feynman used the following example: Consider you walk on the beach and hear a swimmer crying for help. Since this is a 1960s text book the swimmer is a beautiful girl. In order to reach her you have to: 1) Run some meters on the sandy beach and 2) swim some meters in the sea. You do an intuitive calculation about the ideal point of where to enter the water: You can run faster than you can swim. By using a little more intelligence we would realize that it would be advantageous to travel a little greater distance on land in order to decrease the distance in the water, because we go so much slower in the water (Source: Feynman’s Lecture Vol. 1 – available online since a few days!)

Refraction at the interface between air and water.

Refraction at the interface between air and water (Wikimedia). The trajectory of the beam has a kink thus the pole appears kinked.

Those laws are called variational principles: You consider all possible paths, and the path taken is indicated by an extremum, in these cases: a minimum.

Near a minimum stuff does not vary much – the first order derivative is zero at a minimum. Thus on varying paths a bit you actually feel when are close to the minimum – in the way you, as a car driver, would feel the bottom of a valley (It can only go up from here).

Doesn’t this description add a touch of spooky multiverses to classical mechanics already? It seems as if nature has a plan or as if we view anything that has ever or will ever happen from a vantage point outside of space-time.

Things get interesting when masses or charges become smeared out in space – when there is some small ‘infinitesimal’ mass at every point in space. Or generally: When something happens at every point in space. Instead of a point particle that can move in three different directions – three degrees of freedom in physics lingo – we need to deal with an infinite number of degrees of freedom.

Then we are entering the world of fields that I will cover in the next post.

Related posts: Are We All Newtonians? | Sniffing the Path (On the Fascination of Classical Mechanics)

The Falling Slinky and Einstein’s Elevator

I have not known that this toy has a name at all. The ‘spring’ that can walk down the stairs is called Slinky:

2006-02-04 Metal spiral

We all know how the Slinky walks – but how does it fall?

This video might come as a surprise!

The authoritative article on The Falling Slinky is this one: Modelling a Falling Slinky – replacing it with a chain of masses connected by massless springs and calculating the trajectories of bottom, top and center of mass.

Edit: On replying to a comment, I searched for some written explanation by the professor featured in the video: And I found this great scientific paper an arxiv! The wave-like travelling of the “information that tension has collapsed” (as explained in the video) is put into equations.

I believe you could also explain it in the following way:

(1) If you are falling down you feel weightless – gravity is equivalent to acceleration.

You have for sure seen images of cartoon Einstein (the creator of General Relativity) in an elevator falling down in empty space (You find those images close to those cartoon spaceships emitting pulses of light from their cone ends)

(2) So the Slinky is not subject to gravity when it falls but the elastic force will contract it in the same way a Slinky stretched along a table would do.

Elevator gravity

Gravity is equivalent to acceleration.

Now we have to decide on the absolute position of the top or bottom of the Slinky: If it contracts will the bottom move up to the top or the other way round? Or will bottom and top move to the center? I think here we have to resort to considering the center of mass: From the observer’s frame of reference the COM needs to fall down as a point particle with the same mass. The top starting to move will make the COM ‘fall down’ and make the Slinky contract.

(The End)

Bonus Material: ‘Making of’ This Blog Post

Actually I came to this conclusion after playing with another thought experiment that did not work out well in the end.

I imagined the connection between the individual segments of the slinky becoming weaker and weaker until the Slinky ends up as a pile of separate rings. The rationale for this was that a Slinky is quite a weak spring – but in the explanation given above some restoring force is crucial.

The rings would be connected by strings that just keep the whole thing from falling apart. The strings would not be elastic. Thus there is hardly any elastic force, they wouldn’t be any oscillations when the bottom of the ‘string Slinky’ has been released.

Now all the rings except the top ring are suspended – each ring connected to its superior.  When the top ring is released it starts to fall – there would be no restoring force. I expect this thing to fall without any change in shape – in contrast to the Slinky – if the experiment is done in vacuum and everything is balanced carefully. In real live it would rather twist and tumble.

Probably I should test that at Christmas time –  connecting ring-shaped cookies with silvery or golden yearn would be both decorative and very close to my mental model.

Christmas Cookie Cutters

You could also use Christmas cookie cutters and connect them by silvery yarn – or by angel hair.

The Twisted Garden Hose and the Myth of the Toilet Flush

If you have wrapped your head around why and how the U-shaped tube in the flow meter (described in my previous post) is twisted by the Coriolis force – here is a video of a simple experiment brought to my attention by the author of the quoted article on gyroscope physics:

You could also test it yourself using a garden hose!

Accidentally you can observe this phenomenon so clearly because the typical angular frequencies of manual rotation result in a rather strong Coriolis force – in contrast to other every day phenomena that are falsely attributed to the Coriolis force associated with the rotation of the earth.

It is often stated – and I even found this in lecture notes and text books – that the Coriolis force is accountable for the unambiguously different sense of rotation of vortices in water flowing down the sink of your bathtub or toilet: In the Northern hemisphere water should spin anti-clockwise, in the Southern hemisphere clockwise. Numerous articles debunk this as an urban legend – I pick a random one.

On principle the statement on the sense of rotation is correct as the rotation of hurricanes is impacted by the Coriolis force. But for toilet flushes and the like the effect is negligible compared to other random factors impacting the flow of water. As pointed out in this article the momentum of leaves thrown into a bowl of water at a location near the equator of the earth (often used in demonstrations of entertain tourists) do have more impact than the Coriolis force.

Near the equator the Coriolis force is nearly zero, or more precisely: Since it is both perpendicular to the velocity and the axis of rotation the Coriolis force would be directed perpendicular to the surface of the earth – but there is no component of the Coriolis force that would allow water to flow North-South or East-West. Thus very near the equator the force is infinitesimally small – much smaller than the forces acting on, say, middle European or Austrian bathtubs. And even with those the Coriolis force does not determine the spin of rotation unambiguously.

How to estimate this impact – and why can we observe the twist in the garden hose experiment?

The size of the acceleration due to the Coriolis force is

2 times (angular frequency [rad/s]) times (component of the velocity [m/s] perpendicular to the axis of rotation)

The angular frequency in radians per second is 2 Pi times the number of rotations per second. Thus the angular frequency of the rotation of the earth is about 0,0000727 radians per second. The frequency of the motion of the garden hose was rather several turns per second, thus about 1 radians per second.

Imagine a slice or volume element of water flowing in a sink or a garden hose: Assuming a similar speed with an order of magnitude of 1 meter per second. The Coriolis force differs by several orders of magnitude

  • Bathtub vortex: 0,00015 m/s2
  • Garden hose: 2 m/s2

On the other hand, the acceleration due to gravity is equal to 9,81 m/s2.

The garden hose in the video moves under the influence of gravity – like a swing or pendulum – and the Coriolis force (The additional force due to motion of the hands of the experimenter is required to overcome friction). Since the Coriolis force is of the same order of magnitude as gravity you would expect some significant impact as the resulting force on every slice or volume element of water is the vector sum of the two.

It is also important to keep track of the origins of the components of the velocity:

The radial flow velocity (assumed to be about 1 m/s) in the hose is constant and simply dictated the by the pressure in the water line. There is no tangential flow velocity unless caused by the Coriolis force.

In case of the bath tub the assumed 1 m/s do not refer to the velocity of the tangential motion in the vortex, but to the radial velocity of the water flowing “down” the sink. The tangential velocity is what would be caused by the Coriolis force – ideally.

Any initial velocity is is subject to the initial conditions applied to the experiment.

Any random tangential component of the flow velocity in the vortex increases when the water flows down:

If there is a small initial rotation – a small velocity directed perpendicular to the symmetry axis of the flush – pronounced vortices will develop due to the conservation of angular momentum: As the radius of rotation decreases – dictated by the shape of the bathtub or toilet – angular frequency needs to increase to keep the angular momentum constant. Thus in your typical flush you see how a random and small disturbance is “amplified”.

However, if you would conduct such experiment very carefully and wait long enough for any disturbance to die out, you would actually see the vortices due to Coriolis force only.[*] I have now learned from Wikipedia that it was an Austrian physicist who published the first paper on this in 1908 – Ottokar Tumlirz. Here is the full-text of his German paper: Ein neuer physikalischer Beweis für die Achsendrehung der Erde.  (Link edited in 2017. In 2013 I haven’t found the full text but only the abstract).

Tumlirz calculated the vortices’ velocity of rotation and used the following setup to confirm his findings:

Tumlirz-1908-Coriolis-Experimental-Setup

My sketch of the experimental setup Ottokar Tumlirz used in 1908 as it is described in the abstract of his German paper “New physical evidence on the axis of rotation of the earth”. The space between the two plates is filled with water, and the glass tube is open at the bottom. Water (red) flows radially to the tube. The red lines are bent very slightly due to the Coriolis force.

Holes in a cylindrical tube – which is open at the bottom – allow water to enter the tube radially. This is not your standard flush, but a setup built for preventing the amplification of tangential components in the flow. Due to the Coriolis force the flow lines are not straight lines, but slightly bent.

Tumlirz noted that the water must not flow with a speed not higher than about 1 mm per minute.

Edit, Oct 2, 2013: See Ernst Mach’s original account of Tumlirz’ experiment (who was Mach’s student)

Edit, June 3, 2015: Actually somebody did that careful experiment now – and observed the tiny effect just due to the Coriolis force, as Tumlirz. Follow-up post here.

Edit, Augist 2016: Stumbled upon another reference to an experiment done in 1962 and published in Nature (and filmed) – link added to the post from 2015.

Intuition and the Magic of the Gyroscope – Reloaded

I am baffled by the fact that my article The Spinning Gyroscope and Intuition in Physics is the top article on this blog so far.

So I believe I owe you, dear readers, an update.

In the previous article I have summarized the textbook explanation, some more intuitive comments in Feynman’s Physics Lectures, and a new paper by Eugene Butikov.

But there is an explanation of the gyroscope’s motion that might become my new favorite:

Gyroscope Physics by Cleon Teunissen

It is not an accident that Cleon is also the main author of the Wikipedia article on the Coriolis flow meter, as his ingenious take on explaining the gyroscope’s precession is closely related to his explanation of the flow meter.

The Coriolis force is a so-called pseudo-force you “feel” in a rotating frame of reference: Imagine yourself walking across a rotating disk, or rolling a ball soaked in white color rolling across a black disk and watch its trace. In the center of the rotating disk, there is no centrifugal force. But you would still feel being dragged to the right if the disk is rotating counter-clockwise (viewed from the top).

This force dragging you to the right or making the path of the ball bend even in the center – this is the Coriolis force. It also makes tubes bend in the following way when a liquid flows through them and allows for determining the flow velocity from the extent of bending:

Vibration pattern of the tubes during mass flow. Credits: Cleon Teunissen.

Vibration pattern of the tubes during mass flow. Credits: Cleon Teunissen, used with permission – Coriolis flow meter

The “loop” formed by the tubes rotates about an axis which is parallel to the direction of the flow whose speed should be measured (though not the full 360°). Now the Coriolis force always drags a moving particle “to the right” – same with the volume elements in the liquid. Note that the force is always directed perpendicular to the axis of rotation and perpendicular to the velocity of the flowing volume element (mind the example of the rolling ball).

In the flow meter, the liquid moves away from the axis in one arm, but to the axis in the other arm. Thus the forces acting on each arm are antiparallel to each other and a torque is exerted on the “loop” that consists of the two arms. The loop is flexible and thus bent by the torque in the way shown in the figure above.

Now imagine a gyroscope:

Gyroscope with Coriolis force per quadrant indicated. Credits: Cleon Teunissen.

Gyroscope with Coriolis force per quadrant indicated. Credits: Cleon Teunissen – Gyroscope Physics

Gravity (acting on the weight mounted on the gyroscope’s axis) tries to make the gyroscope pitch. Cleon now shows why precession results in an “upward pitch” that compensates for that downward pitch and thus finally keeps the gyroscope stable.

The clue is to consider the 4 quadrants the gyroscope wheel consist of separately – in a way similar to evaluating the Coriolis force acting on each of the arms of the flow meter:

The tangential velocity associated with the rotation about the symmetry axis of the gyroscope (“roll”, spinning of the wheel) is equivalent to the velocity of the flowing liquid. In each quadrant, mass is moving – “flowing” – away from or to the “swivel axis” – the axis of precession (indicated in black, parallel to gravity).

The per-quadrant Coriolis force is again perpendicular to the swivel axis and perpendicular to the “flow velocity”. Imaging yourself sitting on the blue wheel and looking into the direction of the tangential velocity: Again you are “dragged to the right” if precession in counter-clockwise. As right is defined in relation to the tangential velocity, the direction of the force is reversed in the two lower quadrants.

A torque tries to pitch the wheel “upward”.

.

If you want to play with gyroscopes yourself: I have stumped upon a nice shop selling gyroscopes – incl. a steampunk version, miniature Stirling engines, combustion engines (… and strange materials such as ferrofluids).

Physics / Math Puzzle: Where Is the Center of Mass?

On randomly searching for physics puzzles I have come across QuantumBoffin‘s site.

The puzzle is about how to determine the center of mass of a body using the plumb line method, given there is some uncertainty due to experimental errors.

You have found three plumb lines not intersecting in a single point; thus the three intersections of two lines each form a triangle. The question is:

What is the probability that the center of mass is actually located within the triangle?

The following assumption should be used:

The probability of finding the center of mass on either side of each line is 50%

If you want to entertain yourself with trying to solve the puzzle on your own, do not scroll down. I did not find a published solution, so the following is just my proposal. Thus there is a chance that I make a fool of myself. But I enjoy those deceptively simple science puzzles, and there is always a good chance so-called intuition might lead you astray.

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I have tackled the problem as follows:

If there is a single line, it divides the body into two pieces, the center of mass (COM) is found with 50% probability in either piece.

If there are two lines, they would intersect in a point, and the body would be divided into four quadrants. The COM is located in one of these quadrants with a probability of 25%. I’d like to stress that the reason is that the body is divided into 4 equivalent pieces, each of them is located either to the left or to the right of each line. In this case the result is the same as 0,5 times 0,5.

However, if there are three lines, the probability is not simply equal to 0,53 = 1/8 (as confirmed by QuantumBoffin)

I propose: The solution is 1/7.[*]

The three lines divide the piece into 7 (not 8!) parts – the triangle in the center and 6 sections adjacent to the triangle: Each of these parts is located either ‘to the left’ or ‘to the right’ of each line, I call these the ‘+’ and ‘-‘ parts (halves) of the body:

Center of mass, physics puzzle.

Proposed solution: The body is divided into 7 parts; the center of mass being located in either with equal probability. (Image Credits: Mine)

The seven parts are equivalent. This may sound a bit awkward as you might expect the probability to find the COM on the periphery lower than in the middle. But this is due to the assumption that should be made – the assumption (50%) should be replaced by a probability distribution. So in a sense, this is more a math puzzle than a physics puzzle.

[*] Edit as per Nov. 2013:  I think I have found a flaw in my argument as the sum of probabilities for areas on either side of a line would not add up to 0,5 if I assign the same probability to each of the 7 areas depicted below (It is: 4/7 on one side versus 3/7 on the other side). I am now searching for a system of equations that let me determining the probabilities for the three different types of areas: the triangle, the corner areas (wedges) and the open trapezoids.
You can find my updated proposal here. I keep the rest of the post unchanged as I consider my random musings about curved spaces (on the bottom) correct.

I did not use the number of 0,5 in a calculation, but as a justification of the equivalence of the 7 pieces.

Now there is a remaining – mathematical – puzzle that still baffles me: There are 8 (23) possibilities of combining + and -: Which is missing and why?

-+- is the missing combination.

The picture exhibits some nice symmetry, but note that the definition of + and – as such is arbitrary. So we could have ended up with other missing combinations.

  • +-+ in the center. The missing combination is the ‘negation’ of the combination reflecting the triangle.
  • +++ and — opposite to each other (opposite = one attached to a corner of the triangle and the other to the side of the triangle not connected to this corner)
  • Other pairs / opponents:
    –+ and ++-
    -++ and +–

Imagine the negative of the triangle now: To the left of the red line, to the right of the blue and above the green line. Obviously there is no intersection between these, and this is the only combination that does not result in any intersection at all. I suppose there is a better way to state that in stricter mathematical terms. And if we put the figure on a sphere (Non-Euclidian geometry), there would be an intersection – actually the figure would collapse onto 4 equivalent triangles (a ‘bulged tetrahedron’).

In case of two lines only, all combinations can be ‘realized’ in flat space, therefore the solution is simply equal to 1 over the total number of combinations (4).

This stuff is addictive – finally I understand why Feynman was fascinated by flexagons.