Most of us have wondered why rainbow forms in the sky, yet few have thought of how to create a rainbow using an alternative method. Quanta will hold a physics session (Lecture and practical) at 4:30pm on Week 5 Tuesday, July 26^{th}, featuring the science behind rainbows. In the coming session, our club members will introduce basic theories in modern optics which explains physics behind various types of rainbows. The theories, including dispersion, interference, birefringence and interaction between light and matter, can also be applied in many other real-life problems. Though all these may sound bizarre or trial to some, you will learn something new whichever level you are in.

On top of that, there will be a special hands-on time for you to explore these theories in RI(JC) lab on your own, with all experiments unseen in normal classroom settings.

This session will be conducted at the JC campus and detailed venue will be sent to you via e-mail as soon as the sign-ups are confirmed. Drop a message at quanta.clubautomatica@gmail.com should you have any queries.

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**Putt-putt -oops**

However, as it turns out, that explanation leaves out another reason which is simpler and probably more significant! This is based on the conservation of momentum in the first stage when water is being sucked in and the second stage when the water is being blown out.

In the first stage, the boat sucks in water from the rear. In the lab frame, the velocity of the boat is v0 (taking the left as the positive direction) and the velocity of the water outside the boat is 0. When the boat has sucked in a volume of water and its pipes are full, a perfectly inelastic collision has taken place, as the water is now contained inside the boat, and moving at the same speed as the boat. Depending on the relative masses of the water and the boat, the final velocity of the boat filled with water is a fraction of its initial velocity.

Fig. 1: Conservation of momentum during the sucking-in stage. The mass M1 represents the boat and the mass M2 represents the water that is sucked into the boat. |

In the second stage, the boat expels the mass of water with an exhaust velocity of v_{d}, which propels it forward (v_{d} has a negative value since it is in the direction of the right). As shown in the last equation in Fig. 2, the final velocity v_{2}, can be very high, as long as v_{d} is negative enough. Since the water is expelled from the boat very rapidly as the water in the diaphragm boils, it is quite likely that v_{d} is negative enough for the boat to accelerate forward each cycle.

Fig. 2: Conservation of momentum in the blowing-out stage. Vd is negative. |

In fact, as long as -v_{d }> v_{0}, the final velocity will be greater than the initial velocity v_{0} and the boat will accelerate in each cycle.

This is what we should expect anyway, because if one looks at the first and last stages only, ignoring the intermediate stage when the two masses are joined, the condition -v_{d }> v_{0} is necessary for the mass m_{2} to have a more negative velocity after the whole cycle is over.

**Reverse sprinkler**

It turns out that this situation is related to a problem discussed by Feynman and other physicists in the 1940s, dubbed the ‘reverse sprinkler’. A normal s-shaped garden sprinkler expels water from its two ends and hence experiences a torque opposite to the direction of water flow. But what happens if the sprinkler is submerged in water and made to suck in water instead?

The answer is that the reverse sprinkler does not accelerate when water is flowing into it! This phenomenon, which may be counterintuitive, is because of two effects that oppose each other.

(By the way, the reverse sprinkler is different from the putt-putt boat in that the sprinkler continuously sucks in water, whereas the boat only sucks in a finite volume of water then stops.)

The first effect is due to the difference in pressure between the inside of the sprinkler and the outside. The pressure inside the sprinkler is lower than the external pressure. If the sprinkler’s openings were capped, as in Fig. 3a, there is no net torque or force on the sprinkler as all the forces on the surfaces of the sprinkler due to the external pressures cancel out. This makes sense: a body lying passively in a fluid should not experience a net force or torque due to hydrostatic pressure alone. But if the caps are removed and water is allowed to flow into the sprinkler, then there is a torque on the sprinkler due to the pressures exerted at the areas marked out by coloured arrows shown in Fig. 3b.

Fig. 3a: Forces exerted by external pressure (blue arrows) cancel out when the sprinkler is capped at both ends. |
Fig. 3b: Due to the pressure difference between the inside of the sprinkler and the outside, there is a net clockwise torque when water is allowed to flow into the sprinkler. |

This pressure effect would cause the sprinkler to accelerate in the clockwise direction, towards the incoming water. However, this does not actually occur due to an opposing effect.

The incoming water is made of many tiny volume elements that were initally at rest, somewhere out there in the surrounding water where the pressure is p2. These little chunks of water were accelerated along the pressure gradient until they entered the mouth of the sprinkler, where the pressure is p1 (Fig.4). Along this path, the difference in pressure between one side of the volume element and the other side is dP, so the total force exerted on the group of volume elements is AdP, where A is the total area of the group of volume elements when they occupy a thin disk across the cross-section of the sprinkler. Since the force is the rate of change of momentum, at each moment in time during the acceleration of all the small volume elements, the total rate of momentum change experienced by the whole group of volume elements is AdP. Hence, the total rate of change of momentum of the group of volume elements when they travel from far away to the opening of the sprinkler is A(P2-P1).

Now, when the group of water volume elements collides with the bend in the sprinkler and makes a right angled turn, their momentum in the horizontal direction becomes zero. This change in momentum is due to a force exerted by the sprinkler on the water, which has reduced its momentum to zero. The magnitude of this force is P2-P1.

This nicely means that the total force exerted on the inner surface of the sprinkler wall is the hydrostatic pressure P1 plus the additional force required to change the momentum of the incoming water P2-P1. As a result there is no net force exerted on the sprinkler walls and no acceleration of the sprinkler occurs.

**References:**

http://web.mit.edu/Edgerton/www/FeynmanSprinkler.html

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.137.1700&rep=rep1&type=pdf

http://www.copernicusproject.ucr.edu/ssi/2007PhysicsRes/feynman-inverse-sprinkler1.pdf

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The Earth may seem all bumpy and uneven, with molehills, mountains, trenches, buildings and what nots. But really if you shrank it to the size of a billiard ball, it’s smoother! The standard set by The World Pool-Billiard Association requires a billiard ball to have humps or bumps no more than 0.002 times the diameter of the ball. For a 5.72cm ball, that’s about 0.1mm. The Earth is roughly 12, 500km. The deepest point on Earth is the Mariana Trench, about 11km deep. That’s deeper than the height of the tallest point on Earth – Mount Everest at around 9km. So… The deepest bump on Earth is roughly 11/12500 = 0.0009 times the diameter of Earth. Less than half the largest allowable bump:diameter ratio for a competition-standard billiard ball! Regrettably, the Earth is probably not spherical enough.

Taken from: http://blogs.discovermagazine.com/badastronomy/2008/09/08/ten-things-you-dont-know-about-the-earth/

Many people claim that Bernoulli’s Principle is the main factor contributing to lift in an airplane. For those unfamiliar with the principle, it states that a fluid (like air) that flows faster exerts a lower pressure than a fluid that flows slower. The wings of airplanes are usually curved at the top surface and flat at the bottom. As it cuts through air, it forces the air at the top surface to move at a higher velocity than the air at the bottom surface. This is so since the air on top has to “catch up and join back” with the air at the bottom, by the time both flows reach the end of the wing. Therefore, higher pressure on the underside of the wings, lower pressure on the top side, results in lift. (See here for explanation of Bernoulli’s Principle with diagrams: http://www.grc.nasa.gov/WWW/k-12/airplane/wrong1.html) Some physics teachers/texts aerospace museums cite this as the reason why planes fly. This was Fact, till Jiahuang (who has the ability to change Facts) asked during Mr Wee’s lesson “how then do planes fly upside down then?” That got us all thinking. Well, turns out this very popular and very pervasive reasoning (called theory of Equal Transit Time) on how planes fly is wrong! Xkcd did up an interesting comic strip summarizing everything that I just said (and everything I’m about to say):

In the first place, nobody ever said that the top flow and bottom flow must meet at the end of the wing (which was the assumption we made when we said the air flow at the top is faster than the air flow at the bottom). Fact is they don’t.

Sounds like the Bernoulli’s argument is simply wrong. But actually it’s not so simple (as the comic suggests). It turns out that the earlier argument was wrong on 2 fronts, but correct on one. Wrong firstly that “Bernoulli’s Principle is the main reason for lift”. Wrong secondly that “the airflow meets again at the end of the wings”. From the video, the airflow clearly does not meet again. However, the theory is right when it says that “Bernoulli’s Principle generates lift”. From the video, clearly the speed of the top flow is much higher than the speed of the bottom flow (in fact much higher than required for the two flows to meet again at the end of the wings). So, the lift generated due to the pressure difference would actually be more substantial than what the wrong textbooks would have calculated. BUT, it is still nowhere near enough to lift a plane. It might suffice to end off with “So there, the most common and most appealing (probably because it’s the simplest) myth on how planes fly debunked.” But then the entire conclusion would be based on the claim that “BUT, (the lift generated by pressure differences between the top and bottom of the wing) is still nowhere near enough to lift a plane.” Not very convincing eh? Considering I didn’t provide any figures. Haha if you want figures based on simple maths can just go google/estimate yourself.

A more convincing way to disprove this myth would be to ask what Jiahuang and the xkcd person asked “how then, do planes fly upside down?” If the myth were true, then the wings flown upside down would definitely produce a force downwards, plunging the plane to the ground right? Er… I wasn’t joking when I said that the xkcd comic strip summarized everything I’m about to say – It’s really not that simple. A normal wing flown upside down CAN still generate lift. According to some dude from the xkcd forum, the point where the front of the wing cuts the air changes when you turn the wing upside down. The result is that the airflow on the underside (now the top of the upside down wing) of the wing is still faster than the airflow on the other side. Thus, lift is still generated by the wings even when the plane is flying upside down. So… The answer to the question “how then, do planes fly upside down?” really does not provide a convincing argument against the myth of Equal Transit Time.

At the end of the day, xkcd is correct. How planes fly is not that simple. What’s 100% wrong is that Bernoulli’s Principle is the main reason for the lift. There are many reasons that generate lift. But they are too chim for me to understand, thus I will not attempt to touch on them. (Thus the title of this section).

Interesting side fact: A B2 stealth bomber is basically a flying wing. No tail to stabilize, no fuselage. It’s so unstable that it’s not flyable by a human. It requires 4 computers to constantly make adjustments every split second in order to keep that flying wing in the air. So it’s really the computers doing the flying, not the pilot. Well, such technology for the price of twice its weight in gold.

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Mirages are formed by refraction, which requires a varying refractive index. For the sake of formality, I’ll remind the reader what the refractive index of a material is:

$n=\frac{c}{v}$

where *c* is the speed of light in vacuum and *v* is the (phase) velocity of light in that material. The greater the refractive index, the slower light travels through the material.

All mirages are formed because the refractive index of air varies with temperature. The higher the temperature, the lower the refractive index is. This is intuitively correct, as higher temperatures lead to reduced density air, and thus reduced optically density too. However it is to be noted that although density and optical density are related in the case of air, they are two distinct concepts in most other physical situations. Now let us find a more quantitative relationship between the refractive index and temperature.

Another way to define the refractive index is , where and are the relative permittivity and permeability of the material respectively. In air, and variations in accounts for nearly all the changes in *n*. The refractive index of air is very close, but not exactly equal, to 1. In fact . This value, however, is not constant, but varies with temperature. From theoretical analysis, it is known that

Where *N* is the number density of air particles and is the molecular polarizability. As a consequence of the gas laws, , where *T* is the absolute temperature. If we model air as an ideal gas, it’s not hard to arrive at the result that:

where is the atmospheric pressure and is the Boltzmann’s constant. From this equation we can see that the refractive index decreases as temperature increases, as mentioned before.

Now let us move on explore how mirages actually form. Figure 2 shows the typical mirage that is often associated with the apocryphal oasis. In this situation, we have a hot pavement that is heated up by the sun, resulting in a vertical temperature gradient, and thus a refractive index gradient. Light from the sky undergoes a series of infinitesimal refractions before entering the eyes of the observer from below, thus appearing to have originated from somewhere on the ground in front of the observer. The fictitious oasis seen by the observer is actually blue light from the sky.

One way to think about the situation is to apply Snell’s Law to every infinitesimal refraction. Since , we observe that is a conserved quantity, allowing us to do some mathematics. However, that is rather tedious and doesn’t give a good intuitive feel of why the light bends. A more intuitive way to make sense of the situation is to apply **Fermat’s principle of least time*. Suppose our light ray starts in the sky and ends up in the observer’s eye. What is the path of least time given these starting and ending points? If the refractive index were uniform, then it would just be a straight line. However, now the air at the bottom is hotter, thus light could travel faster the nearer it went to the ground. It makes sense, then, for the light to travel closer to the ground for a while. It cannot overdo this though, since going too far down would make the path too long. If you work out the maths, the optimal balance would arise when light follows Snell’s law, as expected.

Normal mirages that form over the surface of a hot road or desert can be called inferior mirages, because the light appears to emerge from below the horizon. On the other hand, superior mirages are formed over a surface that is colder than the surround air,

such as over a chilly ocean. This gave rise to images of ships floating in mid air, albeit upside down (figure 3). This appears to be rather well documented (figure 4), and has even been proposed to be the origins of the Flying Dutchman.

A even more mysterious form of superior mirage is known as the Fata Morgana. The name is an Italian phrase derived from the vulgar Latin for “fairy” and the Arthurian sorcerer Morgan le Fay, from a belief that the mirage, often seen in theStraitofMessina, were fairy castles in the air or false land designed to lure sailors to their death created by her witchcraft. Fata Morgana mirages arise due to different layers of air having different temperatures, and are thus extremely complex. The mirage typically distorts the objects on which it is based, often to the extent that the original objects become unrecognizable. Because of it depends on a delicate multilayered atmospheric condition, the mirage is often rapidly changing too. A video of a Fata Morgana based on a boat can be found here:

http://en.wikipedia.org/wiki/File:Fata_Morgana_is_changing_

shape_of_a_distant_boat_.OGG

Note:

*Fermat’s Principle of least time: It states that the path taken between two points by a ray of light is the path that can be traversed in the extreme value of time (meaning least, most, and stationary value, corresponding to zero-valued first derivative in mathematics). Fermat’s Principal of Least Time, together with Lagrange’s variational priciple, provides a basis for the study of the geometry of space and it is widely used in General Theory of Relativity. (Added by Qiushi when editing)

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This week’s fact of the week will seek to provide an answer to those fascinating “what if” questions in physics.

What if Planck constant was 6.63 X 10^{30}, how would this change the world?

These issues are very tricky. Again, the only thing that makes physical sense is to change *dimensionless* quantities. For instance, change the fine structure constant alpha. The reason for that is that we could work in natural Planck units where h = 1, c = 1, G = 1. All of physics can be expressed that way, and there it doesn’t make sense to ask what will happen when h changes value!

You would first have to say how the other constants, like the lightspeed, the mass of the electron and so on would alter. But let us assume that we keep the lightspeed and the mass of the electron constant. In that case, you’ve just re-defined the units “second” and “meter”, and kept the universe the way it is.

The reason is that the planck constant gives you a definition of what means 1 J s = 1 kg m^{2 }s^{-1}: it says essentially that 1 kg m^{2 }s^{-1}, is times the spin angular momentum of an electron.

Lightspeed essentially says that 1 m s^{-1} = 1/(3.00 X 10^{8}) = 3.34 X 10^{-9} times the velocity of light.

The electron mass essentially says that 1 kg = 1/(9.109 X 10^{-31}) = 1.098 X 10^{30} times the mass of an electron.

Let’s define the new units kg’, m’ and s’

If we say now that Planck’s constant equals 6.63 x 10^{30}, then this means that 1 kg’ m’^{2} s’^{-1 }= times the spin angular momentum of an electron.

Keeping lightspeed, it means that 1 m’ s’^{-1 }= 1/(3.00 X 10^{8}) = 3.34 X 10^{-9 } the speed of light.

Keeping the electron mass means that 1 kg’ = 1/(9.109 X 10^{-31}) = 1.098 X 10^{30 } times the mass of an electron.

Now, you can say that you want to KEEP 1 kg = 1 kg’, 1m = 1m’ and 1s = 1s’. However, each unit is defined in a particular way, as a certain number of times a physical quantity. By keeping the mass of the electron, we’ve used the mass of the electron as our definition of the unit kg.

The lightspeed is just a definition of the ratio of the unit of length over the unit of time: if we keep it numerically the same, both units are now in fixed relation. In the same way, the planck constant relates the combination of the unit of mass, of length (square) and of time so that it is a certain number times the angular momentum of the electron. All this means that if we change that number, we’ve just changed the unit of length (and, because of the fixed ratio (lightspeed!), the unit of time).

In fact, 1 kg = 1 kg’, and 1m’ = (6.6 X 10^{-34})/(6.63 X 10^{30}) = 10^{-64} m and 1s’ = 10^{-64 }s.

When Planck’s constant becomes very big, then all physical laws will tend to become quantum laws. In such a case we would have a world exclusively governed by quantum physics, with everything proceeding through quantum jumps.

If the electromagnetic force becomes greater, then the strong force will no longer have that sufficient energy to bind nucleons together and our world will collapse into protons and neutrons.

A high-temperature superconductor levitating above a magnet

Diagram of the Meissner effect. Magnetic field lines, represented as arrows, are excluded from a superconductor when it is below its critical temperature.

What if high T super-conductors are discovered? Sadly right now we do not have all those magnificent technology available to realize its full potential. Even the unobtainium of electronics-super high T superconductors are discovered accidentally these days, they may not have applications beyond the boundaries of labs in a few decades to follow.

Superconductors are not perfect—they are far from what we expect them to be. Though you can manipulate to create the bizarre anti-gravitational effect via causing diamagnetic materials to float in air and even straight out of Earth, the energy you spend to hold the superconductor in position will be exactly the same as you use a rocket to propel it up into cloud. Currently developed high T superconductors are all defected either for their brittleness or the inability to hold much charge and their superconductivity is highly sensitive to external electric and magnetic field, they must be well protected and cooled in liquid nitrogen. And these limitations will hamper our effort to try to utilize the superconductors as a effective electric cell to store electricity in excess and many industrial applications may only be a smoke in the air.

The Meissner effect was given a phenomenological explanation by the brothers Fritz and Heinz London, who showed that the electromagnetic free energy in a superconductor is minimized provided. where H is the magnetic field and λ is the London penetration depth.

This equation, which is known as the London equation, predicts that the magnetic field in a superconductor decays exponentially from whatever value it possesses at the surface.

A superconductor with little or no magnetic field within it is said to be in the Meissner state. The Meissner state breaks down when the applied magnetic field is too large. Superconductors can be divided into two classes according to how this breakdown occurs. In Type I superconductors, superconductivity is abruptly destroyed when the strength of the applied field rises above a critical value Hc. Depending on the geometry of the sample, one may obtain an intermediate state[8] consisting of a baroque pattern[9] of regions of normal material carrying a magnetic field mixed with regions of superconducting material containing no field. In Type II superconductors, raising the applied field past a critical value Hc1 leads to a mixed state (also known as the vortex state) in which an increasing amount of magnetic flux penetrates the material, but there remains no resistance to the flow of electric current as long as the current is not too large. At a second critical field strength Hc2, superconductivity is destroyed.

In a weak applied field, a superconductor “expels” nearly all magnetic flux. It does this by setting up electric currents near its surface. The magnetic field of these surface currents cancels the applied magnetic field within the bulk of the superconductor. As the field expulsion, or cancellation, does not change with time, the currents producing this effect (called persistent currents) do not decay with time. Therefore the conductivity can be

thought of as infinite: a superconductor.

Superconductors in the Meissner state exhibit perfect diamagnetism, or superdiamagnetism, meaning that the total magnetic field is very close to zero deep inside them (many penetration depths from the surface). This means that their magnetic susceptibility, χv = −1.

Since the late 1980s there have been several attempts to investigate the possibility of harvesting energy from lightning. While a single bolt of lightning carries a relatively large amount of energy, this energy is concentrated in a small location and is passed during an extremely short period of time (milliseconds); therefore, extremely high electrical power is involved. It has been proposed that the energy contained in lightning be used to generate hydrogen from water, or to harness the energy from rapid heating of water due to lightning.

A technology capable of harvesting lightning energy would need to be able to capture rapidly the high power involved in a lightning bolt. Several schemes have been proposed, but the high energy involved in each lightning bolt render lightning power harvesting from ground based rods impractical. According to Northeastern University physicists Stephen Reucroft and John Swain, a lightning bolt carries a few million joules of energy, enough to power a 100-watt bulb for 5.5 hours. Additionally, lightning is sporadic, and therefore energy would have to be collected and stored; it is difficult to convert high-voltage electrical power to the lower-voltage power that can be stored.

In the summer of 2007, an alternative energy company called Alternate Energy Holdings, Inc. (AEHI) tested a method for capturing the energy in lightning bolts. The design for the system had been purchased from an Illinois inventor named Steve LeRoy, who had reportedly been able to power a 60-watt light bulb for 20 minutes using the energy captured from a small flash of artificial lightning. The method involved a tower, a means of shunting off a large portion of the incoming energy, and a capacitor to store the rest. According to Donald Gillispie, CEO of AEHI, they “couldn’t make it work,” although “given enough time and money, you could probably scale this thing up… it’s not black magic; it’s truly math and science, and it could happen.”

According to Dr. Martin A. Uman, co-director of the Lightning Research Laboratory at the University of Florida and a leading authority on lightning, a single lightning strike, while fast and bright, contains very little energy, and dozens of lighting towers like those used in the system tested by AEHI would be needed to operate five 100-watt light bulbs for the course of a year. When interviewed by The New York Times, he stated that the energy in a thunderstorm is comparable to that of an atomic bomb, but trying to harvest the energy of lightning from the ground is “hopeless”.

A relatively easy method is the direct harvesting of atmospheric charge before it turns into lightning. At a small scale, it was done a few times with the most known example being Benjamin Franklin’s experiment with his kite. However, to collect reasonable amounts of energy very large constructions are required, and it is relatively hard to utilize the resulting extremely high voltage with reasonable efficiency.

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A magnetorheological fluid (MR fluid) is another type of smart fluid, usually a type of oil. It contains particles much larger than that of ferrofluid, at the micrometre scale. When a magnetic field is applied, the viscosity of the fluid greatly increases, and the fluid becomes a viscoelastic solid. By varying the strength of the magnetic field, the yield stress of the fluid can be controlled accurately.

**How it works**

Under normal conditions (with no magnetic field), the magnetic particles in the fluid are suspended within the medium in a random way as shown below:

However, when a magnetic field is applied the magnetic particles align themselves in the direction of the magnetic flux. When the magnetic field is applied across the sides of the container, the particles form chains linking between the two sides of the container, restricting the movement of the fluid in the direction perpendicular to the direction of the magnetic flux. Hence its viscosity increases:

**Behaviour of the Material**

To better understand the behaviour of this fluid, let us model it mathematically.

As mentioned above, MR fluids have a low viscosity in the absence of an applied magnetic field, but become quasi-solid when such a field is applied. The properties of the quasi-solid is in fact, comparable to a solid, unless shearing occurs. The yield stress (or apparent yield stress) is the maximum stress before deformation occurs. It is dependent on the magnetic field applied, but reaches a maximum when the fluid is magnetically saturated, after which increasing the magnetic flux density yields no changes. Hence, MR fluids can be considered similar to a Bingham plastic.

Thus we can model MR fluid as:

where τ = shear stress; τ* _{y}* = yield stress; H = Magnetic field intensity η = Newtonian viscosity; is the velocity gradient in the z-direction.

Shear strength

Low shear strength acts as a obstacle for application. However, when pressure is applied in the direction of the magnetic field, the shear strength is raised from about 100kPa to 1100kPa. Shear strength can also be increased by replacing the particles with elongated particles.

Particle sedimentation

Due to the inherent density difference between the particles and the medium, the ferroparticles tend to settle out of the suspension over time. To prevent this, surfactants are used, but at the cost of the fluid’s magnetic saturation, and hence the maximum yield stress. Common surfactants include oleic acid, citric acid and tetramethylammonium hydroxide.

Surfactants have a polar head and a non-polar tail. The polar head absorbs to the nanoparticle, while the non-polar tail sticks out into the medium, forming a micelle around the particle. This increases the effective particle diameter. As the particle is apparently larger, steric repulsion between the larger particles prevents the particles from congregating during settling.

Spherical ferromagnetic nanoparticles can also be added to prevent settling. Due to the random Brownian motion of the nanoparticles, they interfere with the settling of the fluid particles to delay settling.

However, addition of other particles decreases the packing density of the fluid particles, thus decreasing the viscosity of the fluid, reducing its apparent yield stress.

**Modes of operation**

The MR fluid can operate in three main modes: the flow mode, shear mode and the squeeze-flow mode.

In the flow mode, the fluid flows as a result of pressure gradient between the two plates. This mode is commonly used in dampers and shock absorbers, where the movement is controlled by fluid through channels, across which a magnetic field is applied.

In the shear mode, the fluid moves because the two plates move relative to one another horizontally. Shear mode is used in clutches and brakes to control rotational motion.

In the squeeze-flow mode, the fluid moves due to a plate moving perpendicularly with respect to the other. It is most suitable for controlling small, millimeter-order movements but involving large forces.

**Limitations**

- High density of ferromagnetic particles such as iron makes them heavy.
- High-quality fluids are expensive.
- Fluids are subjected to thickening after prolonged use. Hence, they need to be replaced regularly.

**Applications**

- Mechanical engineering: dampers, e.g. heavy motor damping, operator seat/cab damping in construction vehicles and even seismic dampers which operate within the building’s resonance frequency, absorbing detrimental shock waves and oscillations within the structure to increase the resistance of the building to earthquakes
- Military: bullet resistant body armour
- Optics: Magnetorheological Finishing is an optical polishing method that is highly precise. It was used in the construction of the Hubble Space Telescope’s corrective lens.
- Medical: dampers to absorb shock in semi-active human prosthetic legs

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There is, however, a much more interesting way in which bubbles can form in bulk liquid. Instead of raising the liquid vapour pressure (by increasing temperature) to equal external atmospheric pressure, why not lower the local liquid pressure to a point below that of the liquid’s vapour pressure? This phenomenon, known as cavitation, is an important research area in the study of fluid dynamics, and can be used for a mindboggling range of application ranging from industrial mixing machines to water purification to the removal of kidney stones and even to catching fish!

Meet the pistol shrimp. This little creature competes with much larger animals like the Sperm Whale and Beluga Whale for the title of ‘loudest animal in the sea’. The animal snaps a specialized claw shut to create a cavitation bubble that generates acoustic pressures of up to 80 kPa at a distance of 4 cm from the claw. As it extends out from the claw, the bubble reaches speeds of 97 km/h and releases a sound reaching 218 decibels (just for comparison, a plane taking off around 100m away is only 120 decibels loud). Although the duration of the click is less than 1 millisecond, the spike in pressure is strong enough to stun and even kill small fish.

More interestingly, the snap can also produce sonoluminescence from the collapsing cavitation bubble. As it collapses, the cavitation bubble reaches temperatures of over 5000 K (the surface temperature of the sun is estimated to be around 5800 K!). And if all this is not enough, the pistol shrimp also has a bigger cousin – the mantis shrimp – whose club-like forelimbs can strike so quickly and with such force (these creatures have been known to break aquarium glass) as to induce sonoluminescent cavitation bubbles upon impact! So how does motion (e.g. of those claws you see in the picture) actually result in cavitation? The simple answer is that if the motion of a body within a fluid is fast enough such that the region behind the object is “vacated” by the object faster than water can rush in to fill its place, a region of localized low pressure develops and cavitation bubbles can form.

Cavitation was first studied by Lord Rayleigh in the late 19th century, when he considered the collapse of a spherical void within a liquid. As briefly discussed earlier, cavitation inception occurs when the local pressure falls sufficiently far below the saturated

vapour pressure, a value given by the tensile strength of the liquid. This may occur behind the blade of a rapidly rotating propeller or on any surface vibrating in the liquid with sufficient amplitude and acceleration. In order for cavitation to occur, the bubbles generally need a surface on which they can nucleate (e.g. the sides of a container, impurities in the liquid, or even small undissolved microbubbles within the liquid). It is generally accepted that hydrophobic surfaces stabilize small bubbles. These pre-existing bubbles start to grow unbounded when they are exposed to a pressure below the threshold pressure, termed Blake’s threshold.

However, physical motion of bodies in liquid is not the only means by which cavitation can occur. Acoustic cavitation occurs whenever a liquid is subjected to sufficiently intense sound or ultrasound (that is, sound with frequencies of roughly 20 kHz to 10 MHz). When sound passes through a liquid, it produces compressions and rarefactions (low pressure regions!). Hence, if the sound intensity high enough, it can cause the formation, growth, and rapid recompression of vapour bubbles in the liquid. Other ways of generating cavitation voids involve the local deposition of energy, such as an intense focused laser pulse (optic cavitation) or with an electrical discharge through a spark.

To correct our previous hand-waving explanation on the region being “vacated” faster than water can rush in, the cavitation bubble is not actually a vacuum. Vapour gases evaporate into the cavity from the surrounding medium; thus, the cavity is not a perfect vacuum, but has a relatively low gas pressure. Such a low-pressure cavitation bubble in a liquid begins to collapse due to the higher pressure of the surrounding medium. As the bubble collapses, the pressure and temperature of the vapour within it increases. The bubble eventually collapses to a minute fraction of its original size, at which point the gas within dissipates into the surrounding liquid via a rather violent mechanism, which releases a significant amount of energy in the form of an acoustic shock wave and as visible light. At the point of total collapse, the temperature of the vapour within the bubble may be several thousand Kelvin, and the pressure several hundred atmospheres.

In engineering, cavitation is undesirable in many propulsion and hydraulic systems because it produces extensive erosion of rotating blades, additional noise from the resultant knocking and vibrations, and a significant reduction of efficiency because it distorts the flow pattern. However, cavitation is also utilized in many interesting applications, such as high-power ultrasonics which utilize the inertial cavitation of microscopic vacuum bubbles for the cleaning of surfaces or homogenizing colloids such as paint mixtures or milk. Water purification devices have also been designed, in which the extreme conditions of cavitation can break down pollutants and organic molecules. Cavitation also plays an important role for the destruction of kidney stones in shock wave lithotripsy, and nitrogen cavitation is a method used in research to lyse cell membranes while leaving organelles intact. So don’t ever look at bubbles in a beaker in the same way again!

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In this week’s fact of the week, I will introduce the vector dot and cross product, which may be already familiar to you. However, some of the mathematical perspective provided here may be useful in furthering your understanding in these operations. These operations are important in solving Olympiad physics problems and reading further in physics. These operations help to differentiate vectors from scalars in physics easily and solve the problem of judging the direction of certain vector, especially in three-dimensional space where intuition may fail.

The dot product is also called the scalar product. The result is a scalar quantity. In this article, all vectors are denoted by letters with arrow-heads, and their magnitudes by letters.

As defined, dot product —— (1)，

where *θ* is the angle between and .

Alternatively, if and are both *n* dimensional vectors, i.e. they can be denoted in the form of , , then ——- (2).

Usually in physics, the vectors space will be two or three dimensional.

In the following part, I will provide a geometric perspective of dot product and derive (2) from (1) in two-dimensional space. The definition of dot product is subsequently generalised to *n* dimensional space.

Geometrically, the vector product is the product of magnitude of and the projection of on . Projection can be thought as a ‘shadow’. In the figure above, imagine if there exist parallel light beams perpendicular to shining towards , will leave a projection of magnitude on . The dot product is then the product of this scalar projection and the magnitude of itself. This is where the definition in (1) comes about.

A short digression here. In physics, this is a useful operation. For example, it avoids judging the sign of a scalar quantity obtained from calculation, like work. If we are only concerned with magnitude, . We will have to remember that *s* is the displacement in the direction of the force and if the two are in the opposite direction, *W* will be negative. However, avoids these considerations altogether. If the two are anti-parallel, from our definition, *θ*=180° and *W* will thus be the negative product of *F* and *s *(the magnitudes).

Now I will derive (2) from (1) in two dimensional plane. (I am not sure which definition arose first. Supposedly either one of the definitions can lead to the other. However, from what I read (1) came about first because the cross product was defined in two dimensional planes due to its usefulness in physics. Then with the correspondence between vectors and points in Cartesian plane, (2) is derived which is later generalised to *n* dimensional plane as a formal definition of scalar product in maths. You may want to find out more about the history, but anyway it is not our concern here.)

In the above diagram, let , . According to (1), .

Suppose OM direction is the positive *x* direction. Let and be the unit vector along positive *x* and *y* directions respectively.

We know from addition formula that

Observe that

Sub in to (3), we have

Note that this is applicable to three-dimensional (or higher) vector spaces.

Some properties are applicable to dot products:

I will now move on to cross product, which is also known as the vector product. It is denoted by . The result of cross product is a vector. Its direction is perpendicular to the plane which and lie in (note that any two intersecting lines determine a plane). The direction can be then specified by right-hand rule: curl the right hand from to . The direction in which the thumb points at is the direction of .

The magnitude is given by the formula , where *θ* is the angle between the two vectors.

I will avoid the tedious mathematical expressions that explain how to derive the result of a cross product here, but a useful expression can be used:

The way to calculate this ‘monster’ is to take the sum of product entries in the top-left-to-bottom-right diagonals minus the sum of product of entries in the top-right-to-bottom-left diagonals. If the diagonal is not complete (less than three entries), take the entries that are two skewed column from the column you are looking at.

If this looks abstract to you, consider the extended matrix as following:

Now the top-left-to-bottom-right diagonals and the top-right-to-bottom-left diagonals become pretty clear.

For those of you who are familiar with matrix, you can use the cofactor expansion along the first row as well.

Note that distributive rule still applies to cross product, but generally commutative property no longer holds. In fact, from definition, .

Vector product is very useful in electromagnetism. Where Fleming’s right-hand rule is a general rule of thumb, it may sometimes be confusing especially if the vectors are in three-dimensional space. Also, once is not orthogonal (perpendicular) to , it may be difficult to judge the direction. For example, in electromagnetism, if the field is not perpendicular to the length vector , it might not be so easy to figure out which angle to use or to determine the direction of the force. Writing equations in cross product provides a convenient way to avoid such confusion. In this case, by using , we can get the magnitude and the direction of force at one shot.

There are some interesting results related to operations of cross and dot products. I list some here. You may want to read more about their derivation or about more properties if you are interested.

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In the now-classic Call of Duty 4 Modern Warfare, the only time you get to lay your hands on the sleek .50 Barrett M82 is during this scene – One shot, One Kill. Now, if you are familiar with the plot, there is this guy called Imran Zakhaev whom you are supposed to put a hole through from a mile away. If you do not know what I am talking about, watch this video:

One of the clichés in contemporary FPS cinematic is to cite the Coriolis Effect. For the movies, we have the 2007 film Shooter. But what indeed is this “Coriolis Effect”? And does it really affect the shot in the above situation? (Oh btw, you were scripted to fail. Zakhaev’s arm gets blown off even if you shoot his head)

The Coriolis force that we are concerned with presently is the fictitious force which is seen when the observer is in a rotating frame. The following diagram may help illustrate this point. Given that there is a stationary planar disc, on which I put a particle. The particle is now going to move with constant speed from the center to the circumference.

Now imagine that you rotate your vision at a constant angular velocity. The disc seems fine – you can’t tell if it’s moving or not. But the particle seems to follow a curved path, because relative to the disc, it is not moving tangentially. As it moves out in the radial direction, the apparent tangential velocity increases, and we need some force to account for this tangential acceleration.

This is the fictitious Coriolis force. It is not really there. We just added it because you are rotating your head, and the particle appears to move in such a way. Quantitatively, it is given by:

Up to this point, all you need to remember is that the Coriolis Effect is the changes in trajectory of a particle when one goes into a rotating frame. We can now go on to analyze a rather simplistic view of our initial problem. How significant is it in shooting Zakhaev?

Prypiat, Ukraine is located at approximated 51.405556 deg North. Assuming Earth is a sphere of radius 6370km, spinning at 7.27×10^{-5} rad/s. Also assume that the shot is directed in the NS direction. Since angle shot upwards is small, we can assume shot was fired horizontally. (Bad assumption, but I lazy) Horizontal distance is around 1.5km.

Muzzle Velocity of .50 Caliber Barrett M82 is 853m/s. Time in flight =1.8s.

Looking from the above the North Pole, projected values are:

Velocity of Earth’s rotation at point of shot:

Velocity of Earth’s rotation at Zakhaev’s head:

Difference in velocity = 1.487573 x 10^{-3} m/s

Note that we have to account for initial velocity because when the bullet is shot out, the gun was co-rotating with Earth at the original latitude.

Finally, the deflection in the bullet’s final impact point is….. 2.68 mm. Even if the previous bad assumption about a horizontal shot wasn’t made, it’s still around the same order of magnitude. Unless you are near the equator, this is likely to be the least of your problems.

The Coriolis Effect is often one of the last few effects that a sniper’s calculations account for, together with the Eötvös effect (Go read up). In reality, numerous other factors compound and more complicated situations may result. Together, a whole list of phenomena forms the study of External Ballistics, which you may check out at Wikipedia.

And oh, if by the end of this issue, you are thinking COD4 is a bad game, I need to defend it. **It’s a damn good shooting game, together with MW2.** (Don’t like BlackOps so much)** Go play. **

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