Why is there a conserved quantity for every continuous global symmetry?

One of the most important theorems in physics is the theorem that states:

"To every differentiable global symmetry of a physical system there corresponds a conservation law"

This theorem is known as the first Noether's theorem, in honor of the great mathematician Emmy Noether, who proved it in 1915 in the context of classical mechanics (both relativistic and non-relativistic, but not quantum). By the way, Noether is part of the group of leading scientists and professors in her field who lost their jobs due to the intolerance of the Nazis when they came to power, as they immediately passed a law preventing Jews and Communists from working in universities and public institutions. This happened before the Holocaust and the Second World War. It is important to remember this so that it does not happen again.

This relationship between symmetries and conservation laws established by Noether is one of the most powerful ideas that human beings have ever had. Conservation laws are a very useful tool for finding out how the quantities of a physical system change over time. Knowing that there are physical quantities that do not change allows us to write equations where the unknowns are the quantities that do change. We can then use the quantities that do not change to find out how the other quantities change.

On the other hand, the symmetries of a physical system are related to its aesthetic aspect. For example, a sphere is beautiful because, no matter how you rotate it, it remains the same. Noether's theorem thus relates beauty to usefulness in physics in a certain way. Pragmatism and aesthetics go hand in hand.

However, for the physics student, it is not immediately evident that a continuous symmetry implies a conserved quantity. Apparently, they are two things that have nothing to do with each other. What is the reason for this relationship?

However, today we know that the world is not classical, but quantum, and that classical mechanics is nothing more than an approximation of the behavior of physical systems in a certain limit. Therefore, Noether's original proof does not serve us for the fundamental laws of nature. Does Noether's theorem still hold in quantum mechanics?

These are the two questions we are going to answer in this post.

Do all chlorine atoms have the same mass?

Mathematical induction is a mathematical proof technique. It is essentially used to prove that a property P(n) holds for every natural number n, i.e. for n = 1, 2, 3, and so on. The method of induction requires two cases to be proved:

1. The first case, called the base case, proves that the property holds for the number 1.
2. The second case, called the induction step, proves that, if the property holds for one natural number n, then it holds for the next natural number n + 1.

These two steps establish the property P(n) for every natural number n = 1, 2, 3, ...

Mathematical induction can be informally illustrated by reference to the sequential effect of falling dominoes. (via GIPHY)

Do all chlorine atoms have the same mass?

1. First we establish a base case for one chlorine atom (n=1). The case with just one chlorine atom is trivial. If there was only one chlorine atom in the universe, then clearly all chlorine atoms in that universe would have the same mass.
2. We then prove that if in any random set of n chlorine atoms every atom had the same mass, then in any random set of n+1 chlorine atoms every atom must also have the same mass. First, exclude the last chlorine atom and look only at the first n chlorine atoms; all these have the same mass since n chlorine atoms always have the same mass. Likewise, exclude the first chlorine atom and look only at the last n chlorine atoms. These too, must also have the same mass. Therefore, the first chlorine atom in the group has the same mass as the chlorine atoms in the middle, who in turn have the same mass as the last chlorine atom. Hence the first chlorine atom, middle chlorine atoms, and last chlorine atom have all the same mass, and we have proven that: If n chlorine atoms have the same mass, then n+1 chlorine atoms will also have the same mass.

We already saw in the base case that the rule ("all chlorine atoms have the same mass") was valid for n=1. The inductive step showed that since the rule is valid for n=1, it must also be valid for n=2, which in turn implies that the rule is valid for n=3 and so on. Thus in any group of chlorine atoms, all chlorine atoms must have the same mass. We have proof that in any universe, no matter how many chlorine atoms exist, all chlorine atoms must have the same mass.

Nevertheless, it is an experimental fact that it is false that all the atoms of the same element have the same mass. For instance, chlorine's atomic mass of 35.5 a.m.u. is an average of the masses of the different isotopes of chlorine. This is calculated by working out the relative abundance of each isotope. For example, in any sample of Chlorine 25% will be Cl-37 and 75% Cl-35, so there are chlorine atoms with different (35 a.m.u. and 37 a.m.u.) masses.

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook, Twitter or Instagram to give your answers.

Can an atom have a half-integer number of neutrons?

William Prout, by Henry Wyndham Phillips, 1820 - 1868 - From a miniature by Henry Wyndham Phillips, Public Domain, Link

Prout's hypothesis was an early 19th-century attempt to explain the existence of the various chemical elements through a hypothesis regarding the internal structure of the atom. In 1815 and 1816, the English chemist William Prout published two papers in which he observed that the atomic weights that had been measured for the elements known at that time appeared to be whole multiples of the atomic weight of hydrogen. He then hypothesized that the hydrogen atom was the only truly fundamental object, which he called "protyle", and that the atoms of other elements were actually groupings of various numbers of hydrogen atoms.

Prout's hypothesis was an influence on Ernest Rutherford, and that is the reason why he suggested in 1920 the name "protons" por the positive particles that live in the atomic nuclei. The name "proton" comes from the suffix "-on" for particles, added to the stem of Prout's word "protyle". Later, the English physicist Sir James Chadwick discovered the neutron. Both particles, proton and neutron, have almost the same mass (1 a.m.u.), which is much bigger that the electron mass, and that is why the elements known at Prout's time were measured to be whole multiples of the atomic mass of hydrogen, which is 1 a.m.u.

Nevertheless, nowadays we know that chlorine's atomic mass is 35.5 a.m.u. Since we know that each chlorine atom has 17 protons inside its nucleus, does it mean that chlorine atoms have a half-integer number of neutrons?

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook, Twitter or Instagram to give your answers.

Why do nonmetals have both positive and negative oxidation numbers?

The chemical elements can be broadly divided into metals and nonmetals according to their tendency to loose or gain electrons:

• Atoms that belong to metallic elements tend to loose electrons. When they loose electrons, they become cations, positive ions with a charge that equals the number of electrons they have lost. That number is given by the oxidation number. For instance, sodium's oxidation number is +1, while calcium's oxidation number is +2.
• On the other hand, atoms that belong to nonmetallic elements tend to gain electrons, so they become anions, ions with a negative charge that equals the number of electrons they have gain. For instance, fluorine tends to gain one electron and becomes F^-. That is why it has oxidation number -1. 

But, as we can see in the following periodic table, most nonmetals have both positive and negative oxidation number:

Why do nonmetals have both positive and negative oxidation number if they always tend to gain electrons?

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

Why do metals seem colder although they have the same temperature?

By Anonimski - Own work, CC BY-SA 3.0, Link

All the objects that have been inside your room for more than one hour are at room temperature. That is because heat flows from hotter to colder objects, so if you put a cold object inside the room, heat will flow to it until it reaches room temperature.

If we touch a piece of metal that is inside the room it feels cold. But when we touch the other objects of the room they don't feel as cold. Why is that? Why do metals seem colder than the other objects in the room although they have the same temperature?

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

What is the difference between a mixture and a compound?

According to most textbooks:

• a compound is an entity consisting of two or more atoms, commonly from different chemical elements, which associate via chemical bonds.
• On the other hand, a mixture is a material made up of two or more different substances which are mixed but are not combined chemically.

So the difference is that a mixture refers a physical combination of substances, whereas a compound refers to a chemical combination.

By Chris 73 / Wikimedia Commons, CC BY-SA 3.0, Link

But these definitions do not say anything unless we establish the difference between a chemical and a physical combination and, according to the same textbooks:

• a chemical process is a method or means of somehow changing one or more compounds,
• whereas physical changes are changes affecting a substance, but not its chemical composition, because they do not change chemical bonding.

As you probably have already realized, the definitions are circular! We put two substances together. If the process is not chemical, what we obtain is a mixture, not a compound. But we defined a non-chemical process as the one where the compounds are still the same compounds, but mixed. Who is Alice? She is Bob's cousin. And who is Bob? Alice's cousin. We still do not know who is Alice!

Are you able to give a definition of compounds and mixtures that is not circular? How can we define chemical process without saying that a chemical process is different from a physical one in that compounds change?

Notice the distinction cannot come from the physical properties if the substance, because the physical properties of a mixture may differ from those of the components. In addition, evolved or absorbed heat cannot be the solution because, both in chemical reactions and in mixtures, heat is either evolved (an exothermic process) or absorbed (an endothermic process).

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers. 

Why don't protons in a nucleus repel each other?

The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom.

A model of the atomic nucleus showing it as a compact bundle of the two types of nucleons:

protons (red) and neutrons (blue).

By Marekich - Own work (vector version of PNG image), CC BY-SA 3.0, Link

The atomic number Z of a chemical element is the number of protons found in the nucleus of an atom. Since neutrons are neutral while protons are positively charged with a charge that is, in absolute value, equal to the electron charge, the atomic number is identical to the charge number of the nucleus. In an uncharged atom, the atomic number is also equal to the number of electrons, and that is why the atomic number uniquely identifies a chemical element. For instance, Z=1 is called hydrogen and Z=6 carbon.

The world nuclide is referred to a 'species of nucleus', characterized by its number of protons Z, its number of neutrons N, and its nuclear energy state. Identical nuclei belong to one nuclide, for example each nucleus of the carbon-13 nuclide is composed of 6 protons and 7 neutrons. On the other hand, the members of the group of all the nuclides of the same elements are called isotopes. That is, the nuclides with equal atomic number, i.e., of the same chemical element but different neutron numbers, are called isotopes of that element.

Stable nuclides are nuclides that are not radioactive and so (unlike radionuclides) do not spontaneously undergo radioactive decay. It has been found that there are 80 elements with one or more stable isotope. For instance, carbon has 15 known isotopes, from carbon-8 to carbon-22, of which carbon-12 and carbon-13 are stable.

But, because of the fact all the protons have the same charge and are very closed to one another in the nucleus, one expects that they repel each other by a strong electric force, a force that must be much stronger than the force acting between the nucleus and the surrounding electrons. This force should make the nucleus explode. Therefore, no nuclide with more than one proton should be stable, that is, the only stable element should be hydrogen!

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers. 

Why do astronauts float weightless in the International Space Station?

By NASA/Crew of STS-132 - http://spaceflight.nasa.gov/gallery/images/shuttle/sts-132/hires/s132e012208.jpg (http://spaceflight.nasa.gov/gallery/images/shuttle/sts-132/html/s132e012208.html), Public Domain, Link

We have seen on TV that astronauts float weightless in the International Space Station (ISS), and during spacewalks. In fact, the ISS serves as a microgravity research laboratory in which crew members conduct experiments in biology, physics and other fields. Microgravity is more or less a synonym of weightlessness and zero-g (zero gravitational field strength).

Nevertheless, the ISS maintains an orbit with an altitude h of between 330 and 435 km so, according to Newton's law of universal gravitation, the gravitational field strength g in the ISS is:

where G is Newton's constant, M mass of Earth and R is the radius of Earth.

Taking into account that the gravitational field strength at the surface of the Earth is g=9.8 m/s^2, we conclude that things and astronauts inside the ISS weigh only a 10% less than they do on Earth! The weight is almost the same! How is this possible?

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

Preparation for the International Physics Olympiad (IPhO): Electromagnetism

The IPhO syllabus includes:

2.3 Electromagnetic fields

• 2.3.1 Basic concepts: Concepts of charge and current; charge conservation and Kirchhoff's current law. Coulomb force; electrostatic field as a potential field; Kirchhoff's voltage law. Mag­netic B-field; Lorentz force; Ampere's force; Biot-Savart law and B-field on the axis of a circular current loop and for simple symmetric systems like straight wire, circular loop and long solenoid.
• 2.3.2 Integral forms of Maxwell's equations: Gauss' law (for E- and B-fields); Ampere's law; Faraday's law; using these laws for the calculation of fields when the integrand is almost piecewise constant. Boundary conditions for the electric field (or electrostatic potential) at the surface of conductors and at infinity; concept of grounded conductors. Superposition principle for elec­tric and magnetic fields; uniqueness of solution to well- posed problems; method of image charges.
• 2.3.3 Interaction of matter with electric and magnetic fields; Resistivity and conductivity; differential form of Ohm's law. Dielectric and magnetic permeability; relative per­mittivity and permeability of electric and magnetic ma­terials; energy density of electric and magnetic fields; fer­romagnetic materials; hysteresis and dissipation; eddy currents; Lenz's law. Charges in magnetic field: helicoidal motion, cyclotron frequency, drift in crossed E- and B-fields. Energy of a magnetic dipole in a magnetic field; dipole moment of a current loop.
• 2.3.4 Circuits: Linear resistors and Ohm's law; Joule's law; work done by an electromotive force; ideal and non-ideal batter­ies, constant current sources, ammeters, voltmeters and ohmmeters. Nonlinear elements of given V-I charac­teristic. Capacitors and capacitance (also for a single electrode with respect to infinity); self-induction and in­ductance; energy of capacitors and inductors; mutual in­ductance; time constants for RL and RC circuits. AC circuits: complex amplitude; impedance of resistors, in­ductors, capacitors, and combination circuits; phasor di­agrams; current and voltage resonance; active power.

Members of the Spanish National Team can download the course notes here:

• Campo electrostático: 
• Corriente eléctrica
• Campo magnetostático
• Campo electromagnético

Some cognitive conflicts involving Electromagnetism:

• Cognitive Dissonances: Electromagnetism

It is also useful to follow the IPhO's Study Guide by Jaan Kalda

• Study guide: Circuits

Here you can find the solutions to some of the problems:

• Pr 3:  [IPhO1996Theory1] Problem Solution
• Pr 69: [IPhO1994Theory2] Problem Solution
• [IPhO1983Theory2] Problem Solution
• [IPhO2010Theory1] Problem Solution
• [IPhO2001Theory1c] Problem Solution

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Preparation for the International Physics Olympiad (IPhO): Relativity

The IPhO Syllabus includes:

• 2.5 Relativity: Principle of relativity and Lorentz transformations for the time and spatial coordinate, and for the energy and momentum; mass-energy equivalence; invariance of the spacetime interval and of the rest mass. Addition of par­allel velocities; time dilation; length contraction; relativ­ity of simultaneity; energy and momentum of photons and relativistic Doppler effect; relativistic equation of motion; conservation of energy and momentum for elas­tic and non-elastic interaction of particles.

Members of the Spanish National Team can download the notes here:

• Introducción a la relatividad especial

To know more:

• Taller de Relatividad para estudiantes de bachillerato.

Some cognitive conflicts involving Relativity:

• Cognitive Dissonances: Relativity

It is also useful to follow the IPhO's Study Guide by Siim Ainsaar:

• A Glimpse into the Special Theory of Relativity

Here you can find the solutions to some of the problems:

• [IPho2006Theory2] Problem Solution
• [IPhO2003Theory3] Problem Solution
• [IPhO1994Theory1] Problem Solution

Back to the main page

Preparation for the International Physics Olympiad (IPhO): Thermodynamics and Statistical Physics

The IPhO Syllabus includes:

2.7 Thermodynamics and statistical physics

• 2.7.1 Classical thermodynamics: Concepts of thermal equilibrium and reversible pro­cesses; internal energy, work and heat; Kelvin's tem­perature scale; entropy; open, closed, isolated systems; first and second laws of thermodynamics. Kinetic the­ory of ideal gases: Avogadro number, Boltzmann factor and gas constant; translational motion of molecules and pressure; ideal gas law; translational, rotational and os­cillatory degrees of freedom; equipartition theorem; in­ternal energy of ideal gases; root-mean-square speed of molecules. Isothermal, isobaric, isochoric, and adiabatic processes; specific heat for isobaric and isochoric pro­cesses; forward and reverse Carnot cycle on ideal gas and its efficiency; efficiency of non-ideal heat engines.
• 2.7.2 Heat transfer and phase transitions: Phase transitions (boiling, evaporation, melting, subli­mation) and latent heat; saturated vapour pressure, rel­ative humidity; boiling; Dalton's law; concept of heat conductivity; continuity of heat flux.
• 2.7.3 Statistical physics: Planck's law (explained qualitatively, does not need to be remembered), Wien's displacement law; the Stefan- Boltzmann law.

Members of the Spanish National Team can download the notes here:

• Introducción a la termodinámica y a la física estadística

Some cognitive conflicts involving Thermodynamis and Statistical Physics:

• Cognitive Dissonances: Thermodynamics and Statistical Physics

It is also useful to follow the IPhO's Study Guide by Jaan Kalda:

• Study guide for IPhO-s: Thermodynamics

Here you can find the solutions to some of the problems:

• Pr 1 & 7: [IPhO1996Theory1] Problem Solution
• Pr 6 & 34: [IPhO1992Theory3] Problem Solution
• Pr 24: [IPhO1989Theory1] Problem Solution
• [IPhO2007Theory] Problem Solution

Back to the main page

Why do not the sizes of Venus and Mars as viewed from Earth change during the course of the year?

Venus, by NASA or Ricardo Nunes - http://www.astrosurf.com/nunes/explor/explor_m10.htm, Public Domain, https://commons.wikimedia.org/w/index.php?curid=338424

Just before his death, in 1543, Nicolaus Copernicus published in his book On the Revolutions of the Celestial Spheres a Heliocentric model of the universe, that is, a model of the universe that placed the Sun rather than the Earth at the center of the universe.  This is considered a major event in the history of science, triggering the Copernican Revolution and making an important contribution to the Scientific Revolution.

By Original image by Niko Lang SVG version by User:Booyabazooka - Own work, CC BY-SA 2.5, https://commons.wikimedia.org/w/index.php?curid=5502912

According to Copernicus' model, since the Earth circulates the Sun in an orbit outside that of Venus and inside that of Mars, the apparent size of both Venus and Mars should change appreciably during the course of the year. This is because when the Earth is around the same side of the sun as one of those planets it is relatively close to it, whereas when it is on the opposite side of the sun to one of them it is relatively distant from it. When the matter is considered quantitatively, as it can be within Copernicus's own version of his theory, the effect is a sizeable one, with a predicted change in apparent diameter by a factor of about eight in the case of Mars and about six in the case of Venus.

On the other hand, according to the Ptolemaic system (the Geocentric model) Venus and Mars should not change appreciably during the course of the year because its epicyclical motion implies only a small change in distance from the Earth.

However, when the planets are observed carefully with the naked eye, no change in size can be detected for Venus, and Mars changes in size by no more than a factor of two. This gives us strong evidence for the Geocentric model and refutes the Heliocentric model! How is this possible?

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

Something happens with the light

If we split a collimated light beam by using a half-silvered mirror, then the two resulting beams (A and B) have exactly the same intensity. Since the light is made of photons, that means that half of the photons go through path A, and the other half through path B.

If we now reflect both beams by a mirror and the two beams then pass a second half-silvered mirror and enter two detectors as explained in the picture:

then we expect the A beam to be split into two beams. We will call them A1 and A2. A1 goes to dectector 1, while A2 goes to detector 2. Each one contains 50% of A-photons, that is, 25% of the photons of the original light beam:

On the other hand, we also expect the B beam to be split into two beams. We will call them B1 and B2. B1 goes to dectector 1, while B2 goes to detector 2. Each one contains 50% of B-photons, that is, 25% of the photons of the original light beam:

So the amount of photons that should arrive to detector 1 is 25% + 25% = 50%, and the same for detector 2:

Nevertheless, once we have carried out the experiment, what we found is that 100% of photons arrive to detector 2  and no photon arrives to detector 2!

Moreover, what is even more puzzling, if we obstruct channel A (or B, it does not matter), then we detect the same number of photons in detector 2 as the number detected in detector 1. Are you able to figure it out? Try it!

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

Why do ice cubes melt faster in fresh water than in salt water?

The melting point of a solid is the temperature at which it changes state from solid to liquid at atmospheric pressure. When considered as the temperature of the reverse change from liquid to solid, it is referred to as the freezing point.

The freezing point of a solvent is depressed when another compound is added, meaning that a solution has a lower freezing point than a pure solvent. This phenomenon is used in technical applications to avoid freezing, for instance by adding salt or ethylene glycol to water. If you live in a place that has lots of snow and ice in the winter, then you have probably seen the highway department spreading salt on the road to melt the ice.

Now, let us consider the following experiment:

1. Make two almost identical ice cubes.
2. Mix 1 teaspoon of salt in an 8 oz. cup of water. This will be our salt water cup.
3. Fill a 8 oz. cup with water, but with no salt added. This will be our fresh water cup
4. Place one ice cube into each cup simultaneously. Which ice cube do you predict would melt the fastest?

Naively, one would think that, according to the previous information, since salt lowers the freezing/melting point of water, the ice cube in the salt water cup should melt the fastest.

Nevertheless, if you carry out the experiment, it leaves no doubt. The ice cube in the fresh water cup melts faster!

Why do ice cubes melt slower in salt water? Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

I will give you a clue: repeat the experiment, but this time, after you place the ice cubes in the cups, wait 30 seconds and add a couple of drops of food coloring to each cup without disturbing the water in the cups.

Why does the death of a living being affect the decay of carbon-14?

Carbon-14 is a radioactive isotope of carbon with an atomic nucleus containing 6 protons and 8 neutrons. Carbon-14 decays into nitrogen-14 through beta decay:

By emitting a beta particle (an electron, e^-) and an electron antineutrino (ν_e), one of the neutrons in the carbon-14 nucleus changes to a proton and the carbon-14 nucleus becomes the stable (non-radioactive) isotope nitrogen-14.

The equation governing the decay of a radioactive isotope is

$$ N=N_0 e^{-\frac{t}{\tau}}$$

where N_o is the number of atoms of the isotope in the original sample (at time t = 0, when the organism from which the sample was taken died), and N is the number of atoms left after time t. On the other hand, the mean-life τ is the average or expected time a given atom will survive before undergoing radioactive decay.

Since the amount of carbon-14 inside a piece of wood or a fragment of bone decrease as the carbon-14 undergoes radioactive decay, measuring the amount of carbon-14 in a sample provides information that can be used to calculate when the animal or plant died. The mean-life of carbon-14 is 8267 years, so the equation above can be rewritten as:

Nevertheless, radioactive decay is a process that takes place inside the nucleus, so nor a change of temperature neither chemical reactions affect radioactive decay. Carbon-14 atoms inside a living being are decaying after and before the living being dies. So why is this method used efficiently to measure when the living being died? How do we know N_o, the amount of carbon-14 the living being had at the moment it died, if carbon-14 was also decaying when the plant or the animal was alive?

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

Does the expansion of space apply in the solar system?

The prevailing cosmological model for the universe accounts for the fact that the universe expanded from a very high density and high temperature state, and that nowadays the expansion is even accelerating. This is an expansion of space, that is, the increase of the distance between two distant parts of the universe with time. It is an intrinsic expansion whereby the scale of space itself changes. This is different from other examples of expansions and explosions in that, as far as observations can ascertain, it is a property of the entirety of the universe rather than a phenomenon that can be contained and observed from the outside.

By NASA/WMAP Science Team - Original version: NASA; modified by Ryan Kaldari, Public Domain, https://commons.wikimedia.org/w/index.php?curid=11885244

Since it is an intrinsic expansion, it is natural to think that the planets in our solar system are expanding with time, as universe is. Moreover, our measurement devices should be expanding too. But, taking into account that a measurement is the assignment of a number to a characteristic of an object by comparing with other objects, why were we able to measure the expansion of the universe if our devices are expanding too?

Does the expansion of space apply to the objects inside our solar system?

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

Why are cosmic ray muons decaying more slowly than predicted?

Muons are unstable elementary particles. They are heavier than electrons and neutrinos but lighter than all other matter particles. They decay via the weak interaction. A muon decays most commonly to an electron, an electron antineutrino, and a muon neutrino:

The mean lifetime, τ = 1/Γ, of the muon is (2.1969811±0.0000022 ) µs. That means that every 2.19698 µs the population of muons is reduced by a factor e=2.71828.

An experiment compared the population of cosmic-ray-produced muons at the top of a mountain, whose height is 2 km, to that observed at sea level. Those muons were traveling at 0.95c, where c is the speed of light, so they arrive to the sea level t=7 µs later. At the top of the mountain the measured population was No=563 muons per hour. Therefore, according to the decay law, the expected population of muons at the sea level should be:

$$ N=N_0 e^{-\frac{t}{\tau}}=23$$

muons. Nevertheless, 413 muons where measured, so the muon sample at the sea level was only moderately reduced! The muons were decaying about 10 times slower!

Are you able to explain this anomaly? Try it!

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

Why did the hunter wound the bear?

A hunter is located 100 meters due South from a bear. Then he goes East 100 meters. After that, he looks to the North and he shoots to the North, wounding the bear. Nevertheless, we know that the bear did not move!

You will be even more puzzled after realizing that the question is:

What color is the bear?

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

The Moon is getting further away from Earth. Where does this extra energy come from?

All bounded orbits where the gravity of a central body dominates are elliptical in nature. In the case of the Moon orbiting the Earth, the eccentricity of the ellipse is so small (0.055) that it is almost a circle:

Therefore, the gravitational force F_g that the Earth exerts on the Moon is perpendicular to Moon's velocity v, so it is a centripetal force F_c, making the trajectory of the Moon bend:
$$ F_{g}=F_{c} \\
\frac{GMm}{r^2}=\frac{mv^2}{r} $$ where G is Newton's constant, M is Earth's mass, m is Moon's mass and r is the radius of the orbit.

This implies that the kinetic energy of the Moon is
$$
K=\frac{1}{2}mv^2=\frac{GMm}{2r}
$$ which is smaller than the absolute value of the potential energy
$$
U=-\frac{GMm}{r}
$$ So the mechanical energy of the Moon is
$$
E=-\frac{GMm}{2r}
$$

We know that at the time of its formation, the Moon sat much closer to the Earth, a mere 22,500 km away, compared with the 402,336 km between the Earth and the Moon today. So the Moon is getting further away from Earth, now at the rate of 3.78 cm per year. Nevertheless, according to the last equation, a larger r means that the Moon has more energy every year. Is its energy non conserved? Who is giving energy to the Moon?

Can an object exceed the speed of light if we push it for enough time?

The second Newton's law of motion establishes that, in an inertial reference frame, the vector sum of the forces F on an object is equal to the mass m of that object multiplied by the acceleration vector a of the object. That means that if we apply a constant force on a body without friction, then the object will move with constant acceleration, increasing its speed by the same amount every second.

The acceleration is proportional to the force, so if the net force is 100 Newtons and the mass is 2 kilograms, the acceleration will be 50 meters per second every second. But if the force is 2 N, the body will increase its speed by 1 m/s every second. Notice that this is not a huge acceleration. Nevertheless, if we keep pushing and wait for 300000000 seconds (9.5 years) the object will move faster that light.

But we know that nothing can exceed the speed of light. This is a well-established law of nature whose confirmation has become routine in current particle accelerators.

Try to find the solution to this contradiction!

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

Is hydrogen atom in bifluoride connected by two covalent bonds?

In chemistry, a valence electron is an electron that is associated with an atom, and that can participate in the formation of a chemical bond. In a single covalent bond, both atoms in the bond contribute one valence electron in order to form a shared pair. For instance, hydrogen atoms have one valence electron, while oxygen atoms have two. That is why a water molecule contains one oxygen and two hydrogen atoms that are connected by singles covalent bonds:

By Dbc334 (first version); Jynto (second version) - File:Water-3D-vdW.png, Public Domain, https://commons.wikimedia.org/w/index.php?curid=1332739

So the Lewis structure of water is H-O-H.

But let us consider a more complex example: bifluoride. Bifluoride is an inorganic anion with the chemical formula [HF₂]⁻.

Public Domain, https://commons.wikimedia.org/w/index.php?curid=1496287

It is not a strange anion. Some [HF₂]^- salts are common, examples include potassium bifluoride (KHF2) and ammonium bifluoride ([NH4][HF2]).

As shown in the figure above, the structure of the anion is symmetric, with the hydrogen situated in the mid-point of the F-F distance. In addition, the H-F contacts in this ion are very short, like in covalent bonds, and the corresponding H.F interactions are also strong enough to be classified as covalent bonds. So the corresponding Lewis structure of the anion should be [F-H-F]^-. But, how is this possible? Hydrogen atoms have one valence electron. They cannot be connected by two covalent bond!

Are you able to give an explanation? Accept the challenge!

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

Is expansion of gases a violation of inertia law?

The first Newton's law (also called inertia law) says that, when viewed in an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a net force.

Let us suppose that we have a box with a wall splitting it in two halves. The left half contains a rest gas (no wind inside), while the right half is empty. Suddenly, we remove the wall. Because of the fact that gases expand to fill their containers, we know that the gas of the left half will move to the right side in order to fill the whole box:

But there is no net force acting on the gas, so why has it moved to the right half of the box? Do gases violate the inertia law?

Are you able to give an explanation? Accept the challenge!

Can we obtain extra surface by assembling the pieces in a different way?

Picture A and B show different arrangements made of similar shapes in slightly different configurations:

By Krauss (Own work) [CC BY-SA 4.0 (http://creativecommons.org/licenses/by-sa/4.0)], via Wikimedia Commons

So the question is: where does the extra one unit square come from?

By Trekky0623 at English Wikipedia (Transferred from en.wikipedia to Commons.) [Public domain], via Wikimedia Commons

Are you able to solve this puzzle?

If nuclear fusion is the reverse of fission, why is energy released in both processes?

Nuclear power is the use of nuclear reactions that release nuclear energy to generate heat. There are basically two ways to release energy from nuclei:

• nuclear fission, which is either a nuclear reaction or a radioactive decay process in which the nucleus of an atom splits into smaller parts (lighter nuclei).  This released energy is the one that is frequently used in steam turbines to produce electricity in nuclear power plants.

Public Domain, https://commons.wikimedia.org/w/index.php?curid=486924

• nuclear fusion, which is a reaction in which two atomic nuclei fuse to form a heavier nucleus. Nuclear fusion reactors are not yet economically viable, but this technology is currently under research and it could become viable in a few decades.

By Wykis - Own work, based on w:File:D-t-fusion.png, Public Domain, https://commons.wikimedia.org/w/index.php?curid=2069575

If splitting a nucleus into two smaller nuclei releases energy, it seems that combining two smaller nuclei into one larger nucleus would require energy, not release it, because it is the inverse process. So, why can we obtain energy from both processes?

 Are you able to resolve this cognitive conflict?

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

Why are cognitive dissonances useful for teaching?

Psychologists define cognitive dissonance as the mental stress or discomfort experienced by an individual who holds two or more contradictory beliefs, ideas, or values at the same time, performs an action that is contradictory to one or more beliefs, ideas, or values, or is confronted by new information that conflicts with existing beliefs, ideas, or values. It has been found that an individual who experiences a cognitive dissonance tends to become psychologically uncomfortable, and is motivated to try to reduce this inconsistency [Festinger, L. (1957). A Theory of Cognitive Dissonance. California: Stanford University Press].

That is why psychologists have incorporated cognitive dissonance into models of basic processes of learning, notably constructivist models [Ausubel, David P.,Novak, J.D.,Hanesian, H. (1978) Educational Psychology: A Cognitive View (2ª ed.). New York: Holt, Rinehart and Winston.] [Ausubel, David.P. (2000). The Acquisition and Retention of Knowledge. Dortrecht, Netherlands: Kluwer.].

In these models teachers gather information about students’ existing ideas and use this information to design activities that foster dissonances in their minds by increasing their awareness of conflicts between students’ prior beliefs and new information (e.g., by requiring students to defend prior beliefs and confronting them with unexpected experimental results or different ideas). Then, teacher guides students to find by themselves correct explanations that resolve the conflict.

Notice that with this methodology no external reward is needed. Cognitive dissonances are enough to increase students' enthusiasm for educational activities. Some researchers have concluded that students who are forced to attribute their work to this intrinsic motivation came to find the task genuinely enjoyable [Aronson, E. (1995). The Social Animal. New York: W.H. Freeman and Co.].

Once students have found by themselves the correct explanation that resolve the cognitive dissonance, a conceptual change has happened. Now, students do not see the world with the same eyes as before, and this change will remain forever, so they keep easily what they have learned even several years before. In fact, it has been shown that teaching methodologies based on cognitive dissonances significantly increase learning in science and reading [Guzzetti, B.J.; Snyder, T.E.; Glass, G.V.; Gamas, W.S. (1993). "Promoting conceptual change in science: A comparative meta-analysis of instructional interventions from reading education and science education". Reading Research Quarterly 28: 116–159.].

Therefore, we can say that we have found a magic wand that will make the students learn science! Nevertheless, things are not so easy. The really difficult part of this process is to find the suitable cognitive dissonances that make the students learn every specific chunk of scientific knowledge. And this is a too technical task to be left to psychologists! Scientists should do it. That is why I have created this web site, to make a database of questions/interventions that could help teachers to foster dissonances in student minds. Of course, since we are looking for cognitive conflicts with students beliefs, the way we can foster these conflicts depends strongly on the students. So,

• If you are a teacher, you have to select the most suitable question/intervention in each case. You can also collaborate with this project by sharing the questions/interventions that you use to foster dissonances in your students.
• If you are a student, enjoy trying to resolve the conflicts. You can post your trials in the comment box below each puzzling question. Some weeks after each publication we will include the explanations that eliminate the inconsistency.  

I hope you found this web site useful!

Sergio Montañez
Founder of Cognitive Dissonances