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  • MASS ENERGY EQIVALENCE is the concept that the mass[1] of a body is a measure of its energy content. What we ordinarily call the mass of a body is always equal to the total energy inside, up to a factor that changes the units. Or: where E is energy, m is relativistic mass, and c is the speed of light in a vacuum, which is 299,792,458 meters per second.

    Expressed in words: energy equals mass multiplied by the speed of light squared. Because the speed of light is very large in common units, the formula implies that any small amount of matter contains a very large amount of energy. Some of this energy may be released as heat and light by nuclear transformations.

    Mass–energy equivalence was proposed in Albert Einstein's 1905 paper, "Does the inertia of a body depend upon its energy-content?", one of his Annus Mirabilis ("Miraculous Year") Papers.[2] Einstein was not the first to propose a mass–energy relationship, and various similar formulas appeared before Einstein's theory with incorrect numerical coefficients and an incomplete interpretation. Einstein was the first to propose the simple formula and the first to interpret it correctly: as a general principle which follows from the relativistic symmetries of space and time.

    In the formula, c2 is the conversion factor required to convert from units of mass to units of energy. The formula does not depend on a specific system of units. Using the International System of Units, joules are used to measure energy, kilograms for mass, meters per sec Neutron moderators To improve Pfission and enable a chain reaction, uranium-fueled reactors must include a neutron moderator that interacts with newly produced fast neutrons from fission events to reduce their kinetic energy from several MeV to several eV, making them more likely to induce fission. This is because 235U is much more likely to undergo fission when struck by one of these thermal neutrons than by a freshly-produced neutron from fission.

    Neutron moderators are materials that interact weakly with the neutrons but absorb kinetic energy from them. Most moderators rely on either weakly bound hydrogen or a loose crystal structure of another light element such as carbon to transfer kinetic energy from the fast-moving neutrons.

    Hydrogen moderators include water (H2O), heavy water(D2O), and zirconium hydride (ZrH2), all of which work because a hydrogen nucleus has nearly the same mass as a free neutron: neutron-H2O or neutron-ZrH2 impacts excite rotational modes of the molecules (spinning them around). Deuterium nuclei (in heavy water) absorb kinetic energy less well than do light hydrogen nuclei, but they are much less likely to absorb the impacting neutron. Water or heavy water have the advantage of being transparent liquids, so that, in addition to shielding and moderating a reactor core, they permit direct viewing of the core in operation and can also serve as a working fluid for heat transfer.

    Crystal structure moderators rely on a floppy crystal matrix to absorb phonons from neutron-crystal impacts. Graphite is the most common example of such a moderator. It was used in Chicago Pile-1, the world's first man-made critical assembly, and was commonplace in early reactor designs including the Soviet RBMK nuclear power plants, of which the Chernobyl plant was one.



    Moderators and reactor design The amount and nature of neutron moderation affects reactor controllability and hence safety. Because moderators both slow and absorb neutrons, there is an optimum amount of moderator to include in a given geometry of reactor core. Less moderation reduces the effectiveness by reducing the Pfission term in the evolution equation, and more moderation reduces the effectiveness by increasing the Pescape term.

    Most moderators become less effective with increasing temperature, so under-moderated reactors are stable against changes in temperature in the reactor core: if the core overheats, then the quality of the moderator is reduced and the reaction tends to slow down (there is a "negative temperature coefficient" in the reactivity of the core). Water is an extreme case: in extreme heat, it can boil, producing effective voids in the reactor core without destroying the physical structure of the core; this tends to shut down the reaction and reduce the possibility of a fuel meltdown. Over-moderated reactors are unstable against changes in temperature (there is a "positive temperature coefficient" in the reactivity of the core), and so are less inherently safe than under-moderated cores.

    Most reactors in use today use a combination of moderator materials. For example, TRIGA type research reactors use ZrH2 moderator mixed with the 235U fuel, an H2O-filled core, and C (graphite) moderator and reflector blocks around the periphery of the core.

    /**/ Anti-spam check. Do NOT fill this in! ==Neutron moderators== To improve P_{fission} and enable a chain reaction, uranium-fueled reactors must include a [[neutron moderator]] that interacts with newly produced [[fast neutrons]] from fission events to reduce their kinetic energy from several [[MeV]] to several [[eV]], making them more likely to induce fission. This is because 235U is much more likely to undergo fission when struck by one of these [[thermal neutron]]s than by a freshly-produced neutron from fission. Neutron moderators are materials that interact weakly with the neutrons but absorb kinetic energy from them. Most moderators rely on either weakly bound [[hydrogen]] or a loose crystal structure of another light element such as [[carbon]] to transfer kinetic energy from the fast-moving neutrons. Hydrogen moderators include [[water]] (H2O), [[heavy water]]([[deuterium|D]]2O), and [[zirconium hydride]] (ZrH2), all of which work because a hydrogen nucleus has nearly the same mass as a free neutron: neutron-H2O or neutron-ZrH2 impacts excite [[rotational mode]]s of the molecules (spinning them around). [[Deuterium]] nuclei (in heavy water) absorb kinetic energy less well than do light hydrogen nuclei, but they are much less likely to absorb the impacting neutron. Water or heavy water have the advantage of being [[Transparency (optics)|transparent]] [[liquid]]s, so that, in addition to shielding and moderating a reactor core, they permit direct viewing of the core in operation and can also serve as a working fluid for heat transfer. Crystal structure moderators rely on a floppy crystal matrix to absorb [[phonons]] from neutron-crystal impacts. [[Graphite]] is the most common example of such a moderator. It was used in [[Chicago Pile-1]], the world's first man-made critical assembly, and was commonplace in early reactor designs including the [[Soviet]] [[RBMK]] [[nuclear power plant]]s, of which the [[Chernobyl accident|Chernobyl]] plant was one. ===Moderators and reactor design=== The amount and nature of neutron moderation affects reactor controllability and hence safety. Because moderators both slow and absorb neutrons, there is an optimum amount of moderator to include in a given geometry of reactor core. Less moderation reduces the effectiveness by C (graphite) moderator and [[neutron reflector|reflector]] blocks around the periphery of the core. Content that violates any copyrights will be



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    [edit] Fast-moving objects and systems of objects If you push on an object in the direction of motion, it gains momentum and it gains energy. But if the object is already travelling near the speed of light, it can't move much faster, no matter how much energy it absorbs. Its momentum and energy continue to increase, but its speed approaches a constant value—the speed of light. This means that in relativity the momentum of an object cannot be a constant times the velocity, nor is the kinetic energy given by 1⁄2mv2.

    The relativistic mass is defined as the ratio of the momentum of an object to its velocity, and it depends on the motion of the object. If the object is moving slowly, the relativistic mass is nearly equal to the rest mass and both are nearly equal to the usual Newtonian mass. If the object is moving quickly, the relativistic mass is greater than the rest mass. As the object approaches the speed of light, the relativistic mass becomes infinite, because the momentum becomes infinite.

    The relativistic mass is always equal to the total energy divided by c2. Because the relativistic mass is exactly proportional to the energy, relativistic mass and relativistic energy are nearly synonyms; the only difference between them is the units. If length and time are measured in natural units, the speed of light is equal to 1, and even this difference disappears. Then mass and energy have the same units and are always equal, so it is redundant to speak about relativistic mass, because it is just another name for the energy. This is why physicists usually reserve the useful short word "mass" to mean rest-mass.

    For things made up of many parts, like a nucleus, planet, or star, the relativistic mass is the sum of the relativistic masses of the parts, because energy adds up. In some cases, however, the parts include fields of force, and if the fields are attractive, they contribute a negative amount to the mass–energy. For example, the mass of an atomic nucleus is less than the total mass of the protons and neutrons that make it up. The amount by which it is smaller is the energy required to break up the nucleus into individual protons and neutrons. Similarly, the mass of the solar system is slightly less than the masses of sun and planets individually, since the gravitational field is attractive.

    The relativistic mass of a moving object is bigger than the relativistic mass of an object that isn't moving, because a moving object has extra kinetic energy. The rest mass of an object is defined as the mass of an object when it is at rest, so that the rest mass is always the same independent of the motion of the observer: it is the same in all inertial frames.

    For a system of particles going off in different directions, the invariant mass is the analog of the rest mass, defined as the total energy (divided by c2) in the center of mass frame, where the total momentum is zero.



    [edit] Meanings of the mass–energy equivalence formula The mass–energy equivalence formula was displayed on Taipei 101 during the event of the World Year of Physics 2005. Mass–energy equivalence states that any object has a certain energy, even when it isn't moving. In Newtonian mechanics, a motionless body has no kinetic energy, and it may or may not have other amounts of internal stored energy, like chemical energy or thermal energy, in addition to any potential energy it may have from its position in a field of force. In Newtonian mechanics, all of these energies are much smaller than the mass of the object times the speed of light squared, and none of these energies have anything to do with mass.

    In relativity, all of the energy that moves along with an object adds up to the total mass of the body, which measures how much it resists deflection. Each potential and kinetic energy makes a proportional contribution to the mass. Even a single photon traveling in empty space has a relativistic mass, which is its energy divided by c2. If a box of ideal mirrors contains light, the mass of the box is increased by the energy of the light, since the total energy of the box is its mass.

    In relativity, removing energy is removing mass, and the formula m = E/c2 tells you how much mass is lost when energy is removed. In a chemical or nuclear reaction, the mass of the atoms that come out is less than the mass of the atoms that go in, and the difference in mass shows up as heat and light with the same relativistic mass. In this case, the E in the formula is the energy released and removed, and the mass m is how much the mass goes down. In the same way, when any kind of energy is added, the increase in the mass is equal to the added energy divided by c2. For example, When water is heated in a microwave oven, the oven adds about 1.11×10−17 kg of mass for every joule of heat added to the water.

    An object moves with different speed in different frames, depending on the motion of the observer, so the kinetic energy in both Newtonian mechanics and relativity is frame dependent. This means that the amount of energy, and therefore the amount of relativistic mass, that an object is measured to have depends on the observer. The rest mass is defined as the mass that an object has when it isn't moving. This is the smallest possible value of the mass of the object.

    The rest mass is almost never additive: the rest mass of an object is not the sum of the rest masses of its parts. The rest mass of an object is the total energy of all the parts, including kinetic energy, as measured by an observer that sees the center of the mass of the object to be standing still. The rest mass adds up only if the parts are standing still and don't attract or repel, so that they don't have any extra kinetic or potential energy. The other possibility is that they have a positive kinetic energy and a negative potential energy that exactly cancels.

    The difference between the rest mass of a bound system and of the unbound parts is exactly proportional to the binding energy of the system. A water molecule weighs a little less than two free hydrogen atoms and an oxygen atom; the minuscule mass difference is the energy that is needed to split the molecule into three individual atoms (divided by c2). Likewise, a stick of dynamite weighs a little bit more than the fragments after the explosion; the mass difference is the energy that is released when the dynamite explodes. The change in mass only happens when the system is open, and the energy escapes. If a stick of dynamite is blown up in a hermetically sealed chamber, the mass of the chamber and fragments, the heat, sound, and light is equal to the original mass of the chamber and dynamite.



    [edit] Massless particles In relativity, all energy moving along with a body adds up to the total energy, which is exactly proportional to the relativistic mass. Even a single photon, graviton, or neutrino traveling in empty space has a relativistic mass, which is its energy divided by c2. But the rest mass of a photon is slightly subtler to define in terms of physical measurements, because a photon is always moving at the speed of light—it is never at rest.

    If you run away from a photon, having it chase you, by moving fast enough in the same direction, when the photon catches up to you the photon would be seen as having less energy, and even less the faster you were traveling when it caught you. As you approach the speed of light, the photon looks redder and redder, by doppler shift (although for a photon the Doppler shift is relativistic), and the energy of a very long-wavelength photon approaches zero. This is why a photon is massless; this means that the rest mass of a photon is zero. A massless particle in relativity is the limit of a particle with very small mass, but which is moving so close to the speed of light, so that it has a non-negligible total energy.

    Two photons moving in different directions can't both be made to have arbitrarily small total energy by changing frames, by chasing them. The reason is that in a two-photon system, the energy of one photon is decreased by chasing it, but the energy of the other will increase. Two photons not moving in the same direction still have an inertial frame where the combined energy is smallest, but not zero. This is called the center of mass frame or the center of momentum frame; these terms are almost synonyms (the center of mass frame is the special case of a center of momentum frame where the center of mass is put at the origin). If you move at the same direction and speed as the center of mass of the two photons, the total momentum of the photons is zero. Their combined energy E in this frame gives them, as a system, a mass equal to the energy divided by c2. This mass is called the invariant mass of the pair of photons together.

    If the photons formed by the collision of a particle and an antiparticle, the invariant mass is the same as the total energy of the particle and antiparticle (their rest energy plus the kinetic energy), in the center of mass frame where they are moving in equal and opposite directions. If the photons are formed by the disintegration of a single particle with a well-defined rest mass, like the neutral pion, the invariant mass of the photons is equal to rest mass of the pion. In this case, the center of mass frame for the pion is just the frame where the pion is at rest, and the center of mass doesn't change. After the two photons are formed, their center of mass is still moving the same way the pion did, and their total energy in this frame adds up to the mass energy of the pion. So the invariant mass of the photons is equal to the pion's rest energy. So by calculating the invariant mass of pairs of photons in a particle detector, pairs can be identified which were probably produced by pion disintegration.



    [edit] Are photons massless? The photon might not be a strictly massless particle, in which case, it would not move at the exact speed of light. Relativity would be unaffected by this, the "speed of light", c, would then not be the actual speed at which light moves, but a constant of nature which is the maximum speed that any object could theoretically attain[3]. It would still be the speed of gravitons, but it would not be the speed of photons.

    The photon is currently believed to be strictly massless, like the graviton, but this is an experimental question. The current bound on the photon mass is that it is no greater than 10−51 g, or 10−32 eV. [4][5][6]



    [edit] Consequences for nuclear physics Max Planck pointed out that the mass–energy equivalence formula implied that bound systems would have a mass less than the sum of their constituents, once the binding energy had been allowed to escape. However, Planck was thinking about chemical reactions, where the binding energy is too small to measure. Einstein suggested that radioactive materials such as radium would provide a test of the theory, but even though a large amount of energy is released per atom, only a small fraction of the atoms decay.

    Once the nucleus was discovered, experimenters realized that the very high binding energies of the atomic nuclei should allow calculation of their binding energies from mass differences. But it was not until the discovery of the neutron in 1932, and the measurement of its mass, that this calculation could actually be performed (see nuclear binding energy for example calculation). A little while later, the first transmutation reactions (such as 7Li + p → 2 4He) verified Einstein's formula to an accuracy of ±0.5%.

    The mass–energy equivalence formula was used in the development of the atomic bomb. By measuring the mass of different atomic nuclei and subtracting from that number the total mass of the protons and neutrons as they would weigh separately, one gets the exact binding energy available in an atomic nucleus. This is used to calculate the energy released in any nuclear reaction, as the difference in the total mass of the nuclei that enter and exit the reaction.

    In quantum chromodynamics the modern theory of the nuclear force, most of the mass of the proton and the neutron is explained by special relativity. The mass of the proton is about eighty times greater than the sum of the rest masses of the quarks that make it up, while the gluons have zero rest mass. The extra energy of the quarks and gluons in a region within a proton, as compared to the energy of the quarks and gluons in the QCD vacuum, accounts for over 98% of the mass.

    The internal dynamics of the proton are complicated, because they are determined by the quarks exchanging gluons, and interacting with various vacuum condensates. Lattice QCD provides a way of calculating the mass of the proton directly from the theory to any accuracy, in principle. The most recent calculations[7][8] claim that the mass is determined to better than 4% accuracy, arguably accurate to 1% (see Figure S5 in Dürr et al.[8]). These claims are still controversial, because the calculations cannot yet be done with quarks as light as they are in the real world. This means that the predictions are found by a process of extrapolation, which can introduce systematic errors.[9] It is hard to tell whether these errors are controlled properly, because the quantities that are compared to experiment are the masses of the hadrons, which are known in advance.

    These recent calculations are performed by massive supercomputers, and, as noted by Boffi and Pasquini: “a detailed description of the nucleon structure is still missing because ... long-distance behavior requires a nonperturbative and/or numerical treatment..." [10] More conceptual approaches to the structure of the proton are: the topological soliton approach originally due to Tony Skyrme and the more accurate AdS/QCD approach which extends it to include a string theory of gluons, various QCD inspired models like the bag model and the constituent quark model, which were popular in the 1980s, and the SVZ sum rules which allow for rough approximate mass calculations. These methods don't have the same accuracy as the more brute force lattice QCD methods, at least not yet.

    But all these methods are consistent with special relativity, and so calculate the mass of the proton from its total energy.



    [edit] Practical examples Einstein used the CGS system of units (centimeters, grams, seconds, dynes, and ergs), but the formula is independent of the system of units. In natural units, the speed of light is defined to equal 1, and the formula expresses an identity: E = m. In the SI system (expressing the ratio E / m in joules per kilogram using the value of c in meters per second):

    E / m = c2 = (299,792,458 m/s)2 = 89,875,517,873,681,764 J/kg (≈9.0 × 1016 joules per kilogram) So one gram of mass is equivalent to the following amounts of energy:

    89.9 terajoules24.9 million kilowatt-hours (≈25 GW·h)21.5 billion kilocalories (≈21 Tcal)[11]21.5 kilotons of TNT-equivalent energy (≈21 kt)[11]85.2 billion BTUs[11] Any time energy is generated, the process can be evaluated from an E = mc2 perspective. For instance, the "Gadget"-style bomb used in the Trinity test and the bombing of Nagasaki had an explosive yield equivalent to 21 kt of TNT. About 1 kg of the approximately 6.15 kg of plutonium in each of these bombs fissioned into lighter elements totaling almost exactly one gram less, after cooling [The heat, light, and electromagnetic radiation released in this explosion carried the missing one gram of mass.][12] This occurs because nuclear binding energy is released whenever elements with more than 62 nucleons fission.

    Another example is hydroelectric generation. The electrical energy produced by Grand Coulee Dam’s turbines every 3.7 hours represents one gram of mass. This mass passes to the electrical devices which are powered by the generators (such as lights in cities), where it appears as a gram of heat and light.[13] Turbine designers look at their equations in terms of pressure, torque, and RPM. However, Einstein’s equations show that all energy has mass, and thus the electrical energy produced by a dam's generators, and the heat and light which result from it, all retain their mass, which is equivalent to the energy. The potential energy—and equivalent mass—represented by the waters of the Columbia River as it descends to the Pacific Ocean would be converted to heat due to viscous friction and the turbulence of white water rapids and waterfalls were it not for the dam and its generators. This heat would remain as mass on site at the water, were it not for the equipment which converted some of this potential and kinetic energy into electrical energy, which can be moved from place to place (taking mass with it).

    Whenever energy is added to a system, the system gains mass.

    A spring's mass increases whenever it is put into compression or tension. Its added mass arises from the added potential energy stored within it, which is bound in the stretched chemical (electron) bonds linking the atoms within the spring.Raising the temperature of an object (increasing its heat energy) increases its mass. If the temperature of the platinum/iridium "international prototype" of the kilogram—the world’s primary mass standard—is allowed to change by 1°C, its mass will change by 1.5 picograms (1 pg = 1 × 10-12 g).[14]
    A spinning ball will weigh more than a ball that is not spinning. Note that no net mass or energy is really created or lost in any of these scenarios. Mass/energy simply moves from one place to another. These are some examples of the transfer of energy and mass in accordance with the principle of mass–energy conservation.

    Note further that in accordance with Einstein’s Strong Equivalence Principle (SEP), all forms of mass and energy produce a gravitational field in the same way.[15] So all radiated and transmitted energy retains its mass. Not only does the matter comprising Earth create gravity, but the gravitational field itself has mass, and that mass contributes to the field too. This effect is accounted for in ultra-precise laser ranging to the Moon as the Earth orbits the Sun when testing Einstein’s general theory of relativity.[15]

    According to E=mc2, no closed system (any system treated and observed as a whole) ever loses mass, even when rest mass is converted to energy. This statement is more than an abstraction based on the principle of equivalence—it is a real-world effect.

    All types of energy contribute to mass, including potential energies. In relativity, interaction potentials are always due to local fields, not to direct nonlocal interactions, because signals can't travel faster than light. The field energy is stored in field gradients or, in some cases (for massive fields), where the field has a nonzero value. The mass associated with the potential energy is the mass–energy of the field energy. The mass associated with field energy can be detected, in principle, by gravitational experiments, by checking how the field attracts other objects gravitationally. [16]

    The energy in the gravitational field itself is different. There are several consistent ways to define the location of the energy in a gravitational field, all of which agree on the total energy when space is mostly flat and empty. But because the gravitational field can be made to vanish locally by choosing a free-falling frame, it is hard to avoid making the location dependent on the observer's frame of reference. The gravitational field energy is the familiar Newtonian gravitational potential energy in the Newtonian limit.



    [edit] Efficiency In nuclear reactions, typically only a small fraction of the total mass–energy is converted into heat, light, radiation and motion, into a form which can be used. When an atom fissions, it loses only about 0.1% of its mass, and in a bomb or reactor not all the atoms can fission. In a fission based atomic bomb, the efficiency is only 40%, so only 40% of the fissionable atoms actually fission, and only 0.04% of the total mass appears as energy in the end. In nuclear fusion, more of the mass is released as usable energy, roughly 0.3%. But in a fusion bomb (see nuclear weapon yield), the bomb mass is partly casing and non-reacting components, so that again only about 0.03% of the total mass is released as usable energy.

    In theory, it should be possible to convert all the mass in matter into heat and light, but none of the theoretically known methods are practical. One way to convert all rest-mass into usable energy is to annihilate matter with antimatter. But antimatter is rare in our universe, and must be made first. Making the antimatter requires more energy than would be released.

    Since most of the mass of ordinary objects is in protons and neutrons, in order to convert all the mass in ordinary matter to useful energy, the protons and neutrons must be converted to lighter particles. In the standard model of particle physics, the number of protons plus neutrons is nearly exactly conserved. Still, Gerardus 't Hooft showed that there is a process which will convert protons and neutrons to antielectrons and neutrinos.[17] This is the weak SU(2) instanton proposed by Belavin Polyakov Schwarz and Tyupkin.[18] This process, can in principle convert all the mass of matter into neutrinos and usable energy, but it is normally extraordinarily slow. Later it became clear that this process will happen at a fast rate at very high temperatures,[19] since then instanton-like configurations will be copiously produced from thermal fluctuations. The temperature required is so high that it would only have been reached shortly after the big bang.

    Many extensions of the standard model contain magnetic monopoles, and in some models of grand unification, these monopoles catalyze proton decay, a process known as the Callan–Rubakov effect.[20] This process would be an efficient mass–energy conversion at ordinary temperatures, but it requires making monopoles and anti-monopoles first. The energy required to produce monopoles is believed to be enormous, but magnetic charge is conserved, so that the lightest monopole is stable. All these properties are deduced in theoretical models—magnetic monopoles have never been observed, nor have they been produced in any experiment so far.

    The third known method of total mass–energy conversion is using gravity, specifically black holes. Stephen Hawking theorized[21] that black holes radiate thermally with no regard to how they are formed. So it is theoretically possible to throw matter into a black hole and use the emitted heat to generate power. According to the theory of Hawking radiation, however, the black hole used will radiate at a higher rate the smaller it is, producing usable powers at only small black hole masses, where usable may for example be something greater than the local background radiation. It is also worth noting that the ambient irradiated power would change with the mass of the black hole, increasing as the mass of the black hole decreases, or decreasing as the mass increases, at a rate where power is proportional to the inverse square of the mass. In a "practical" scenario, mass and energy could be dumped into the black hole to regulate this growth, or keep its size, and thus power output, near constant.



    [edit] Background E = mc2 where m stands for rest mass (invariant mass) m0, applies most simply to single particles viewed in an inertial frame where they have no momentum. But it also applies to ordinary objects composed of many particles so long as the particles are moving in different directions so the "net" or total momentum is zero. The rest mass of the object includes contributions from heat and sound, chemical binding energies, and trapped radiation. Familiar examples are a tank of gas, or a hot poker. The kinetic energy of their particles, the heat motion and radiation, contribute to their weight on a scale according to E = mc2.

    The formula is the special case of the relativistic energy–momentum relationship:

    This equation gives the rest mass of an object which has an arbitrary amount of momentum and energy. The interpretation of this equation is that the rest mass is the relativistic length of the energy–momentum four-vector.

    If the equation E = mc2 is used with the rest mass or invariant mass of the object, the E given by the equation will be the rest energy of the object, and will change according to the object's internal energy, heat and sound and chemical binding energies (all of which must be added or subtracted from the object), but will not change with the object's overall motion (in the case of systems, the motion of its center of mass). However, if a system is closed, its invariant mass does not vary between different inertial observers (different inertial frames), and is also constant, and conserved.

    If the equation E = mc2 is used with the relativistic mass of the object, the energy will be the total energy of the object, which is also conserved so long as no energy is added to or subtracted from the object, However, like the kinetic energy, this total energy will depend on the velocity of the object, and is different in different inertial frames. Thus, this quantity is not invariant between different inertial observers, even though it is constant over time for any single observer. As in the case of rest energy, these relationships for total energy are also true for systems of objects, so long as the system is closed.

    Mass–Velocity Relationship In developing special relativity, Einstein found that the kinetic energy of a moving body is

    with v the velocity, and m0 the rest mass.

    He included the second term on the right to make sure that for small velocities, the energy would be the same as in classical mechanics:

    Without this second term, there would be an additional contribution in the energy when the particle is not moving.

    Einstein found that the total momentum of a moving particle is:

    and it is this quantity which is conserved in collisions. The ratio of the momentum to the velocity is the relativistic mass, m.

    And the relativistic mass and the relativistic kinetic energy are related by the formula:

    Einstein wanted to omit the unnatural second term on the right-hand side, whose only purpose is to make the energy at rest zero, and to declare that the particle has a total energy which obeys:

    which is a sum of the rest energy m0c2 and the kinetic energy. This total energy is mathematically more elegant, and fits better with the momentum in relativity. But to come to this conclusion, Einstein needed to think carefully about collisions. This expression for the energy implied that matter at rest has a huge amount of energy, and it is not clear whether this energy is physically real, or just a mathematical artifact with no physical meaning.

    In a collision process where all the rest-masses are the same at the beginning as at the end, either expression for the energy is conserved. The two expressions only differ by a constant which is the same at the beginning and at the end of the collision. Still, by analyzing the situation where particles are thrown off a heavy central particle, it is easy to see that the inertia of the central particle is reduced by the total energy emitted. This allowed Einstein to conclude that the inertia of a heavy particle is increased or diminished according to the energy it absorbs or emits.



    [edit] Relativistic mass Main article: Mass in special relativity After Einstein first made his proposal, it became clear that the word mass can have two different meanings. The rest mass is what Einstein called m, but others defined the relativistic mass with an explicit index:

    This mass is the ratio of momentum to velocity, and it is also the relativistic energy divided by c2 (it is not Lorentz-invariant, in contrast to m0). The equation E = mrelc2 holds for moving objects. When the velocity is small, the relativistic mass and the rest mass are almost exactly the same.

    • E = mc2 either means E = m0c2 for an object at rest, or E = mrelc2 when the object is moving.
    Also Einstein (following Hendrik Lorentz and Max Abraham) used velocity—and direction-dependent mass concepts (longitudinal and transverse mass) in his 1905 electrodynamics paper and in another paper in 1906.[22] [23] However, in his first paper on E = mc2 (1905) he treated m as what would now be called the rest mass.[2] Some claim that (in later years) he did not like the idea of "relativistic mass."[24] When modern physicists say "mass", they are usually talking about rest mass, since if they meant "relativistic mass", they would just say "energy".

    Considerable debate has ensued over the use of the concept "relativistic mass" and the connection of "mass" in relativity to "mass" in Newtonian dynamics. For example, one view is that only rest mass is a viable concept and is a property of the particle; while relativistic mass is a conglomeration of particle properties and properties of spacetime. A perspective that avoids this debate, due to Kjell Vøyenli, is that the Newtonian concept of mass as a particle property and the relativistic concept of mass have to be viewed as embedded in their own theories and as having no precise connection.[25][26]



    [edit] Low-speed expansion We can rewrite the expression E = γm0c2 as a Taylor series:

    For speeds much smaller than the speed of light, higher-order terms in this expression get smaller and smaller because v/c is small. For low speeds we can ignore all but the first two terms:

    The total energy is a sum of the rest energy and the Newtonian kinetic energy.

    The classical energy equation ignores both the m0c2 part, and the high-speed corrections. This is appropriate, because all the high-order corrections are small. Since only changes in energy affect the behavior of objects, whether we include the m0c2 part makes no difference, since it is constant. For the same reason, it is possible to subtract the rest energy from the total energy in relativity. By considering the emission of energy in different frames, Einstein could show that the rest energy has a real physical meaning.

    The higher-order terms are extra correction to Newtonian mechanics which become important at higher speeds. The Newtonian equation is only a low-speed approximation, but an extraordinarily good one. All of the calculations used in putting astronauts on the moon, for example, could have been done using Newton's equations without any of the higher-order corrections.



    [edit] History While Einstein was the first to have correctly deduced the mass–energy equivalence formula, he was not the first to have related energy with mass. But nearly all previous authors thought that the energy which contributes to mass comes only from electromagnetic fields.[27][28][29][30]



    [edit] Newton: Matter and light In 1717 Isaac Newton speculated that light particles and matter particles were inter-convertible in "Query 30" of the Opticks, where he asks:

    Are not the gross bodies and light convertible into one another, and may not bodies receive much of their activity from the particles of light which enter their composition?

    Since Newton did not understand light as the motion of a field, he was not speculating about the conversion of motion into matter. Since he did not know about energy, he could not have understood that converting light to matter is turning work into mass.



    [edit] Electromagnetic rest mass There were many attempts in the 19th and the beginning of the 20th century—like those of J. J. Thomson (1881), Oliver Heaviside (1888), and George Frederick Charles Searle (1897)—to understand how the mass of a charged object depends on the electrostatic field.[27][28] Because the electromagnetic field carries part of the momentum of a moving charge, it was also suspected that the mass of an electron would vary with velocity near the speed of light. Searle calculated that it is impossible for a charged object to supersede the velocity of light because this would require an infinite amount of energy. [31] [32] [33]

    Following Thomson and Searle (1896), Wilhelm Wien (1900), Max Abraham (1902), and Hendrik Lorentz (1904) argued that this relation applies to the complete mass of bodies, because all inertial mass is electromagnetic in origin. The formula of the mass–energy-relation given by them was m = (4 / 3)E / c2.[27] Wien went on by stating, that if it is assumed that gravitation is an electromagnetic effect too, then there has to be a strict proportionality between (electromagnetic) inertial mass and (electromagnetic) gravitational mass. This interpretation is in the now discredited electromagnetic worldview, and the formulas that they discovered always included a factor of 4/3 in the proportionality. For example, the formulas given by Lorentz in 1904 for the pre-relativistic longitudinal and transverse masses were (in modern notation): [34] [35] [36]

    , where In July 1905 (published 1906), nearly at the same time when Einstein found the simple relation from relativity, Poincaré was able to explain the reason that the electromagnetic mass calculations always had a factor of 4/3. In order for a particle consisting of positive or negative charge to be stable, there must be some sort of attractive force of non-electrical nature which keeps it together. If the mass–energy of this force field is included in a way which is consistent with relativity theory, the attractive contribution adds an amount − (1 / 3)E / c2 to the energy of the bodies, and this explains the discrepancy between the pure electromagnetic theory and relativity. [37]



    [edit] Inertia of energy and radiation James Clerk Maxwell (1874) and Adolfo Bartoli (1876) found out that the existence of tensions in the ether like the radiation pressure follows from the electromagnetic theory. However, Lorentz (1895) recognized that this led to a conflict between the action/reaction principle and Lorentz's ether theory. [38][39][40]

    Poincaré In 1900 Henri Poincaré studied this conflict and tried to determine whether the center of gravity still moves with a uniform velocity when electromagnetic fields are included.[30] He noticed that the action/reaction principle does not hold for matter alone, but that the electromagnetic field has its own momentum. The electromagnetic field energy behaves like a fictitious fluid ("fluide fictif") with a mass density of E / c2 (in other words m = E/c2). If the center of mass frame is defined by both the mass of matter and the mass of the fictitious fluid, and if the fictitious fluid is indestructible—it is neither created or destroyed—then the motion of the center of mass frame remains uniform. But electromagnetic energy can be converted into other forms of energy. So Poincaré assumed that there exists a non-electric energy fluid at each point of space, into which electromagnetic energy can be transformed and which also carries a mass proportional to the energy. In this way, the motion of the center of mass remains uniform. Poincaré said that one should not be too surprised by these assumptions, since they are only mathematical fictions. [41]

    But Poincaré's resolution led to a paradox when changing frames: if a Hertzian oscillator radiates in a certain direction, it will suffer a recoil from the inertia of the fictitious fluid. In the framework of Lorentz ether theory Poincaré performed a Lorentz boost to the frame of the moving source. He noted that energy conservation holds in both frames, but that the law of conservation of momentum is violated. This would allow a perpetuum mobile, a notion which he abhorred. The laws of nature would have to be different in the frames of reference, and the relativity principle would not hold. Poincaré's paradox was resolved[30] by Einstein's insight that a body losing energy as radiation or heat was losing a mass of the amount m = E / c2. The Hertzian oscillator loses mass in the emission process, and momentum is conserved in any frame. Einstein noted in 1906 that Poincaré's solution to the center of mass problem and his own were mathematically equivalent (see below).

    Poincaré came back to this topic in "Science and Hypothesis" (1902) and "The Value of Science" (1905). This time he rejected the possibility that energy carries mass: "... [the recoil] is contrary to the principle of Newton since our projectile here has no mass, it is not matter, it is energy". He also discussed two other unexplained effects: (1) non-conservation of mass implied by Lorentz's variable mass γm, Abraham's theory of variable mass and Kaufmann's experiments on the mass of fast moving electrons and (2) the non-conservation of energy in the radium experiments of Madame Curie. [42]

    Abraham and Hasenöhrl Following Poincaré, Max Abraham in 1902 introduced the term "electromagnetic momentum" to maintain the action/reaction principle.[29] Poincaré's result was verified by him, whereby the field density of momentum per cm3 is E / c2 and E / c per cm2. [43]

    In 1904, Friedrich Hasenöhrl specifically associated inertia with radiation in a paper, which was according to his own words very similar to some papers of Abraham.[29] Hasenöhrl suggested that part of the mass of a body (which he called apparent mass) can be thought of as radiation bouncing around a cavity. The apparent mass of radiation depends on the temperature (because every heated body emits radiation) and is proportional to its energy, and he first concluded that m = (8 / 3)E / c2. However, in 1905 Hasenöhrl published a summary of a letter, which was written by Abraham to him. Abraham concluded that Hasenöhrl's formula of the apparent mass of radiation is not correct, and based on his definition of electromagnetic momentum and longitudinal electromagnetic mass Abraham changed it to m = (4 / 3)E / c2, the same value for the electromagnetic mass for a body at rest. Hasenöhrl re-calculated his own derivation and verified Abraham's result. He also noticed the similarity between the apparent mass and the electromagnetic mass. However, Hasenöhrl stated that this energy–apparent-mass relation only holds as long a body radiates, i.e. if the temperature of a body is greater than 0 K. [44] [45]

    However, Hasenöhrl did not include the pressure of the radiation on the cavity shell. If he had included the shell pressure and inertia as it would be included in the theory of relativity, the factor would have been equal to 1 or m = E / c2. This calculation assumes that the shell properties are consistent with relativity, otherwise the mechanical properties of the shell including the mass and tension would not have the same transformation laws as those for the radiation.[46] Nobel Prize-winner and Hitler advisor Philipp Lenard claimed that the mass–energy equivalence formula needed to be credited to Hasenöhrl to make it an Aryan creation.[47]



    [edit] Einstein: Mass–energy equivalence Albert Einstein did not formulate exactly the formula E = mc2 in his 1905 Annus Mirabilis paper "Does the Inertia of a Body Depend Upon Its Energy Content?";[2] rather, the paper states that if a body gives off the energy L in the form of radiation, its mass diminishes by L/c2. (Here, "radiation" means electromagnetic radiation, or light, and mass means the ordinary Newtonian mass of a slow-moving object.) This formulation relates a only a change Δm in mass to a change L in energy.

    Objects with zero mass presumably have zero energy, so the extension that all mass is proportional to energy is obvious from this result. In 1905, even the hypothesis that changes in energy are accompanied by changes in mass was untested. Not until the discovery of the first type of antimatter (the positron in 1932) was it found that all of the mass of pairs of resting particles could be converted to radiation.

    First correct derivation (1905) Einstein considered a body at rest with mass M. If the body is examined in a frame moving with nonrelativistic velocity v, it is no longer at rest and in the moving frame it has momentum P = Mv.

    Einstein supposed the body emits two pulses of light to the left and to the right, each carrying an equal amount of energy E/2. It its rest frame, the object remains at rest after the emission since the two beams are equal in strength and carry opposite momentum.

    But if the same process is considered in a frame moving with velocity v to the left, the pulse moving to the left will be redshifted while the pulse moving to the right will be blue shifted. The blue light carries more momentum than the red light, so that the momentum of the light in the moving frame is not balanced: the light is carrying some net momentum to the right.

    The object hasn't changed its velocity before or after the emission. Yet in this frame it has lost some right-momentum to the light. The only way it could have lost momentum is by losing mass. This also solves Poincaré's radiation paradox, discussed above.

    The velocity is small, so the right moving light is blueshifted by an amount equal to the nonrelativistic Doppler shift factor 1 − v/c. The momentum of the light is its energy divided by c, and it is increased by a factor of v/c. So the right moving light is carrying an extra momentum ΔP given by:

    The left-moving light carries a little less momentum, by the same amount ΔP. So the total right-momentum in the light is twice ΔP. This is the right-momentum that the object lost.

    The momentum of the object in the moving frame after the emission is reduced by this amount:

    So the change in the object's mass is equal to the total energy lost divided by c2. Since any emission of energy can be carried out by a two step process, where first the energy is emitted as light and then the light is converted to some other form of energy, any emission of energy is accompanied by a loss of mass. Similarly, by considering absorption, a gain in energy is accompanied by a gain in mass. Einstein concludes that all the mass of a body is a measure of its energy content.

    1906—Relativistic center-of-mass theorem Like Poincaré, Einstein concluded in 1906 that the inertia of electromagnetic energy is a necessary condition for the center-of-mass theorem to hold. On this occasion, Einstein referred to Poincaré's 1900 paper and wrote:[48]

    Although the merely formal considerations, which we will need for the proof, are already mostly contained in a work by H. Poincaré2, for the sake of clarity I will not rely on that work.[49]

    In Einstein's more physical, as opposed to formal or mathematical, point of view, there was no need for fictitious masses. He could avoid the perpetuum mobile problem, because based on the mass–energy equivalence he could show that the transport of inertia which accompanies the emission and absorption of radiation solves the problem. Poincaré's rejection of the principle of action–reaction can be avoided through Einstein's E = mc2, because mass conservation appears as a special case of the energy conservation law.



    [edit] Others During the nineteenth century there were several speculative attempts to show that mass and energy were proportional in various discredited ether theories.[50] In particular, the writings of Samuel Tolver Preston,[51][52] and a 1903 paper by Olinto De Pretto,[46][53] presented a mass–energy relation. De Pretto's paper received recent press coverage when Umberto Bartocci discovered that there were only three degrees of separation linking De Pretto to Einstein, leading Bartocci to conclude that Einstein was probably aware of De Pretto's work.[54][55]

    Preston and De Pretto, following Le Sage, imagined that the universe was filled with an ether of tiny particles which are always moving at speed c. Each of these particles have a kinetic energy of mc2 up to a small numerical factor. The nonrelativistic kinetic energy formula did not always include the traditional factor of 1/2, since Leibniz introduced kinetic energy without it, and the 1/2 is largely conventional in prerelativistic physics.[56] By assuming that every particle has a mass which is the sum of the masses of the ether particles, the authors would conclude that all matter contains an amount of kinetic energy either given by E = mc2 or 2E = mc2 depending on the convention. A particle ether was usually considered unacceptably speculative science at the time,[57] and since these authors didn't formulate relativity, their reasoning is completely different from that of Einstein, who used relativity to change frames.

    Independently, Gustave Le Bon in 1905 speculated that atoms could release large amounts of latent energy, reasoning from an all-encompassing qualitative philosophy of physics.[58][59]



    [edit] Radioactivity and nuclear energy It was quickly noted after the discovery of radioactivity in 1897, that the total energy due to radioactive processes is about one million times greater than that involved in any known molecular change. However, it raised the question where this energy is coming from. After eliminating the idea of absorption and emission of some sort of Lesagian ether particles, the existence of a huge amount of latent energy, stored within matter, was proposed by Ernest Rutherford and Frederick Soddy in 1903. Rutherford also suggested that this internal energy is stored within normal matter as well. He went on to speculate in 1904:[60][61]

    If it were ever found possible to control at will the rate of disintegration of the radio-elements, an enormous amount of energy could be obtained from a small quantity of matter.

    Einstein mentions in his 1905 paper that mass–energy equivalence might perhaps be tested with radioactive decay, which releases enough energy (the quantitative amount known roughly even by 1905) to possibly be "weighed," when missing. But the idea that great amounts of usable energy could be liberated from matter, however, proved initially difficult to substantiate in a practical fashion. Because it had been used as the basis of much speculation, Rutherford himself, rejecting his ideas of 1904, was once reported in the 1930s to have said that: "Anyone who expects a source of power from the transformation of the atom is talking moonshine."

    The popular connection between Einstein, E = mc2, and the atomic bomb was prominently indicated on the cover of Time magazine in July 1946 by the writing of the equation on the mushroom cloud itself. This changed dramatically after the demonstration of energy released from nuclear fission after the atomic bombings of Hiroshima and Nagasaki in 1945. The equation E = mc2 became directly linked in the public eye with the power and peril of nuclear weapons. The equation was featured as early as page 2 of the Smyth Report, the official 1945 release by the US government on the development of the atomic bomb, and by 1946 the equation was linked closely enough with Einstein's work that the cover of Time magazine prominently featured a picture of Einstein next to an image of a mushroom cloud emblazoned with the equation.[62] Einstein himself had only a minor role in the Manhattan Project: he had cosigned a letter to the U.S. President in 1939 urging funding for research into atomic energy, warning that an atomic bomb was theoretically possible. The letter persuaded Roosevelt to devote a significant portion of the wartime budget to atomic research. Without a security clearance, Einstein's only scientific contribution was an analysis of an isotope separation method based on the rate of molecular diffusion through pores, a now-obsolete process that was then competitive and contributed a fraction of the enriched uranium used in the project.[63]

    While E = mc2 is useful for understanding the amount of energy released in a fission reaction, it was not strictly necessary to develop the weapon. As the physicist and Manhattan Project participant Robert Serber put it: "Somehow the popular notion took hold long ago that Einstein's theory of relativity, in particular his famous equation E = mc2, plays some essential role in the theory of fission. Albert Einstein had a part in alerting the United States government to the possibility of building an atomic bomb, but his theory of relativity is not required in discussing fission. The theory of fission is what physicists call a non-relativistic theory, meaning that relativistic effects are too small to affect the dynamics of the fission process significantly."[64] However the association between E = mc2 and nuclear energy has since stuck, and because of this association, and its simple expression of the ideas of Albert Einstein himself, it has become "the world's most famous equation".[65]

    While Serber's view of the strict lack of need to use mass–energy equivalence in designing the atomic bomb is correct, it does not take into account the pivotal role which this relationship played in making the fundamental leap to the initial hypothesis that large atoms could split into approximately equal halves. In late 1938, while on the winter walk on which they solved the meaning of Hahn's experimental results and introduced the idea that would be called atomic fission, Lise Meitner and Otto Robert Frisch made direct use of Einstein's equation to help them understand the quantitative energetics of the reaction which overcame the "surface tension-like" forces holding the nucleus together, and allowed the fission fragments to separate to a configuration from which their charges could force them into an energetic "fission." To do this, they made use of "packing fraction," or nuclear binding energy values for elements, which Mitner had memorized. These, together with use of E = mc2 allowed them to realize on the spot that the basic fission process was energetically possible:

    ...We walked up and down in the snow, I on skis and she on foot. ...and gradually the idea took shape... explained by Bohr's idea that the nucleus is like a liquid drop; such a drop might elongate and divide itself... We knew there were strong forces that would resist, ..just as surface tension. But nuclei differed from ordinary drops. At this point we both sat down on a tree trunk and started to calculate on scraps of paper. ...the Uranium nucleus might indeed be an unstable drop, ready to divide itself... But, ...when the two drops separated they would be driven apart by electrical repulsion, about 200 MeV in all. Fortunately Lise Meitner remembered how to compute the masses of nuclei... and worked out that the two nuclei formed... would be lighter by about one-fifth the mass of a proton. Now whenever mass disappears energy is created, according to Einstein's formula E = mc2, and... the mass was just equivalent to 200 MeV; it all fitted! [66]



    [edit] See also

    [edit] References
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    12. ^ The 6.2 kg core comprised 0.8% gallium by weight. Also, about 20% of the Gadget’s yield was due to fast fissioning in its natural uranium tamper. This resulted in 4.1 moles of Pu fissioning with 180 MeV per atom actually contributing prompt kinetic energy to the explosion. Note too that the term "Gadget"-style is used here instead of "Fat Man" because this general design of bomb was very rapidly upgraded to a more efficient one requiring only 5 kg of the Pu/gallium alloy.
    13. ^ Assuming the dam is generating at its peak capacity of 6,809 MW.
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    62. ^ Cover. Time magazine, July 1, 1946.
    63. ^ Isaacson, Einstein: His Life and Universe.
    64. ^ Robert Serber, The Los Alamos Primer: The First Lectures on How to Build an Atomic Bomb (University of California Press, 1992), page 7. Note that the quotation is taken from Serber's 1992 version, and is not in the original 1943 Los Alamos Primer of the same name.
    65. ^ David Bodanis, E = mc2: A Biography of the World's Most Famous Equation (New York: Walker, 2000).
    66. ^ http://homepage.mac.com/dtrapp/people/Meitnerium.html A quote from Frisch about the discovery day. Accesssed April 4, 2009.


    [edit] External links Wikisource has original text related to this article: Relativity: The Special and General Theory

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Posted by tansyclopedia at 7:08 AM 0 comments google_protectAndRun("ads_core.google_render_ad", google_handleError, google_render_ad); euclids division lemma:

 What is a dividend? Let us understand it with the help of a simple example.

Can you divide 14 by 6?



After division, we get 2 as the quotient and 2 as the remainder.

Thus, we can also write 14 as 6 × 2 + 2.

A dividend can thus be written as:

Dividend = Divisor × Quotient + Remainder

Can you think of any other number which, when multiplied with 6, gives 14 as the dividend and 2 as the remainder?

Let us try it out with some other sets of dividends and divisors.

(1) Divide 100 by 20: 100 = 20 × 5 + 0

(2) Divide 117 by 15: 117 = 15 × 7 + 12

(3) Divide 67 by 17: 67 = 17 × 3 + 16

Thus, if we have a dividend and a divisor, then there will be a unique pair of a quotient and a remainder that will fit into the above equation.

This brings us to Euclid’s division lemma.

If a and b are positive integers, then there exist two unique integers, q and r,

such that a = bq + r

This lemma is very useful for finding the H.C.F. of large numbers where breaking them into factors is difficult. This method is known as Euclid’s Division Algorithm.

To understand the method, look at the following video.

Let us look at some more examples.

Example 1:

Find the H.C.F. of 4032 and 262 using Euclid’s division algorithm.

Solution:

Step 1:

First, apply Euclid’s division lemma on 4032 and 262.

4032 = 262 × 15 + 102

Step 2:

As the remainder is non-zero, we apply Euclid’s division lemma on 262 and 102.

262 = 102 × 2 + 58

Step 3:

Apply Euclid’s division lemma on 102 and 58.

102 = 58 × 1 + 44

Step 4:

Apply Euclid’s division lemma on 58 and 44.

58 = 44 × 1 + 14

Step 5:

Apply Euclid’s division lemma on 44 and 14.

44 = 14 × 3 + 2

Step 6:

Apply Euclid’s division lemma on 14 and 2.

14 = 2 × 7 + 0

In the problem given above, to obtain 0 as the remainder, the divisor has to be taken as 2. Hence, 2 is the H.C.F. of 4032 and 262.

Note that Euclid’s division algorithm can be applied to polynomials also.

Example 2:

A rectangular garden of dimensions 190 m × 60 m is to be divided in square blocks to plant different flowers in each block. Into how many blocks can this garden be divided so that no land is wasted?

Solution:

If we do not want to waste any land, we need to find the largest number that completely divides both 190 and 60 and gives the remainder 0, i.e., the H.C.F. of (190, 60).

To find the H.C.F., let us apply Euclid’s algorithm.

190 = 60 × 3 + 10

60 = 10 × 6 + 0

Therefore, the H.C.F. of 190 and 60 is 10.

Therefore, there will be = 19 square blocks along the length of the garden and = 6 blocks along its breadth.

Hence, the total number of blocks in the garden will be 19 × 6 = 114.

Example 3:

Find the H.C.F. of 336 and 90 using Euclid’s division algorithm.

Solution:

As 336 > 90, we apply the division lemma to 336 and 90.

336 = 90 × 3 + 66

Applying Euclid’s division lemma to 90 and 66:

90 = 66 × 1 + 24

Applying Euclid’s division lemma to 66 and 24:

66 = 24 × 2 + 18

Applying Euclid’s division lemma to 24 and 18:

24 = 18 × 1 + 6

Applying Euclid’s division lemma to 18 and 6:

18 = 6 × 3 + 0

As the remainder is zero, we need not apply Euclid’s division lemma anymore. The divisor (6) is the required H.C.F.

Example 4:

Find the H.C.F. of 45, 81, and 117 using Euclid’s division algorithm.

Solution:

Let us begin by choosing any two out of the three given numbers, say 45 and 81.

As 81 > 45, we apply Euclid’s division lemma to 81 and 45.

81 = 45 × 1 + 36

Applying Euclid’s division lemma to 45 and 36:

45 = 36 × 1 + 9

Applying Euclid’s division lemma to 36 and 9:

36 = 9 × 4 + 0

As the remainder is zero, the H.C.F. of 45 and 81 is 9.

Now, we again need to apply Euclid’s division algorithm on the H.C.F. of the two numbers and the remaining number.

Since the H.C.F. of 45 and 81 is 9 and the third number is 117, we apply Euclid’s division lemma to 117 and 9.

117 = 9 × 13 + 0

As the remainder is zero, the H.C.F. of 9 and 117 is 9.

Here, the second H.C.F. (the H.C.F. of the H.C.F. of the first two numbers and the third number) is the H.C.F. of the three numbers.

Thus, we can say that the H.C.F. of the three numbers is 9.

Example 5:

In an inter-school essay writing competition, the numbers of participants from schools A, B, and C are 20, 16, and 28 respectively. If the participants in each room are from the same school, then find the minimum number of rooms required such that each room has the same number of participants.

Solution:

If we need to find the minimum number of rooms, then we need to keep the maximum number of participants in each room i.e., we need to find the largest number that completely divides 20, 16, and 28.

Thus, we start by choosing any two out of the given three numbers, say 20 and 16.

Applying Euclid’s division lemma to 20 and 16:

20 = 16 × 1 + 4

Applying Euclid’s division lemma to 16 and 4:

16 = 4 × 4 + 0

Hence, 4 is the H.C.F. of 20 and 16.

Applying Euclid’s division algorithm to 28 and 4:

28 = 4 × 7 + 0

Hence, 4 is the H.C.F. of 28 and 4.

∴H.C.F. of 20, 16, 28 = 4

Therefore, each room would have 4 participants.

The number of rooms in which the participants from schools A, B, and C can be accommodated is = 5, = 4, and = 7 respectively.

Therefore, a total of 5 + 4 + 7 =16 rooms are required.
Posted by tansyclopedia at 7:06 AM 0 comments art of solvingproblems
Welcome to the Art of Problem Solving bookstore! Books for eager students of mathematics in grades 4-12 Art of Problem Solving (AoPS) texts are designed for outstanding math students and present a broader and deeper mathematics education than the standard curriculum. The AoPS texts have been used by thousands of top students in contests such as MATHCOUNTS and the AMC. Introduction to Algebra by Richard Rusczyk Learn the basics of algebra from former USA Mathematical Olympiad winner and Art of Problem Solving founder Richard Rusczyk. Topics covered in the book include linear equations, ratios, quadratic equations, special factorizations, complex numbers, graphing linear and quadratic equations, linear and quadratic inequalities, functions, polynomials, exponents and logarithms, absolute value, sequences and series, and much more!

As you'll see in the excerpts below, the text is structured to inspire the reader to explore and develop new ideas. Each section starts with problems, giving the student a chance to solve them without help before proceeding. The text then includes solutions to these problems, through which algebraic techniques are taught. Important facts and powerful problem solving approaches are highlighted throughout the text. In addition to the instructional material, the book contains well over 1000 problems. The solutions manual contains full solutions to all of the problems, not just answers.

This book can serve as a complete Algebra I course, and also includes many concepts covered in Algebra II. Middle school students preparing for MATHCOUNTS, high school students preparing for the AMC, and other students seeking to master the fundamentals of algebra will find this book an instrumental part of their mathematics libraries.

About the author: Richard Rusczyk is a co-author of Art of Problem Solving, Volumes 1 and 2, the author of Art of Problem Solving's Introduction to Geometry, and the founder of www.artofproblemsolving.com. He was a national MATHCOUNTS participant, a USA Math Olympiad winner, and is currently director of the USA Mathematical Talent Search.

ISBN: 978-1-934124-01-7 Text: 656 pages. Solutions: 312 pages. Paperback. 10 7/8 x 8 3/8 x 1 3/16 inches. The '''Regional Mathematical Olympiad''' or '''Regional Mathematics Olympiad''' or '''RMO''' is a set of regional-level Olympiads held in India as a qualifying round for the [[Indian National Mathematical Olympiad]], which in turn is a qualifying round for the [[International Mathematical Olympiad Training Camp]] (IMOTC), where students are selected for representing India at the [[International Mathematical Olympiad]].

== Regions ==
Currently, the RMO is held in eighteen regions:

* Andhra Pradesh
* Bihar
* Delhi
* Gujarat
* Karnataka
* Kerala
* Maharashtra (except Mumbai) + Goa
* Mumbai
* Madhya Pradesh
* North Eastern States (Assam, Meghalaya, Tripura, Manipur)
* Orissa
* Punjab (Includes Haryana, Jammu and Kashmir, Himachal Pradesh and the Union territory of Chandigarh)
* Rajasthan
* Tamil Nadu
* Uttar Pradesh
* West Bengal
* Kendriya Vidyalay Schools
* Navodaya Schools

==Rules ==
All students who are still in school are eligible for the RMO. A total of thirty students is selected from each state, based on performance at the RMO, to sit for the INMO. The MO Cell has imposed a restriction that each region is allowed to send at most six students from Class 12 for the INMO.

The [[Mathematical Olympiad Cell]] (MO Cell) (which is responsible for conducting the [[Indian National Mathematical Olympiad]] and the IMOTC) prepares a paper for the Regional Mathematical Olympiad. The co-ordinator of every region has the option of either following the MO Cell's paper or preparing his/her own paper. If the co-ordinator chooses to use the MO Cell's paper, then the examination must be conducted on a fixed date and time as prescribed by the MO Cell. This is usually the first Sunday of December, 1:00 - 4:00 p.m.

===Level of the problems===

The RMO has the same topics as the International Mathematical Olympiad: [[algebra]], [[geometry]], [[number theory]] and [[combinatorics]]. However, the paper prepared by the MO Cell usually does not assume strong prerequisites beyond what is covered up to the eleventh standard syllabus. Individual regions that have greater expectations from students and that have instituted training programmes, may set papers with more prerequisites.

==See also==
* [[Indian National Mathematical Olympiad]]
* [[International Mathematical Olympiad Training Camp]]
* [[International Mathematical Olympiad]]

==External links==
* [http://www.bprim.org/rmoinfon.php Information on RMO and INMO provided by Bhaskaracharya Pratishthana]
* [http://www.kalva.demon.co.uk/indian.html Indian Mathematical Olympiad problems]
* [http://www.isid.ac.in/~rbb/olympiads.html Page of INMO coordinator on Olympiads]
* [http://www.nbhm.dae.gov.in/olympiad.html NBHM official page on Olympiads]
* [http://www.iisc.ernet.in/mocell/ Official page of the MO cell]
* [http://www.simoonline.org/ South Indian Mathematical Olympiad Foundation]
* [http://www.iaams.in Integral Association of Amateur Mathematicians and Scientists]

[[Category:Mathematical Olympiads in India]]
[[Category:International Mathematical Olympiad]] Posted by tansyclopedia at 7:02 AM 0 comments Subscribe to: Posts (Atom) google_protectAndRun("ads_core.google_render_ad", google_handleError, google_render_ad); google_protectAndRun("ads_core.google_render_ad", google_handleError, google_render_ad); Followers if (!window.google || !google.friendconnect) { document.write('' + ''); } if (!window.registeredBloggerCallbacks) { window.registeredBloggerCallbacks = true; var registeredGadgets = []; gadgets.rpc.register('registerGadgetForRpcs', function(gadgetDomain, iframeName) { // Trim the gadget domain from a random url (w/ query params) // down to just a top level domain. var startIndex = 0; var protocolMarker = "://"; // Find the start of the host name if (gadgetDomain.indexOf(protocolMarker) != -1) { startIndex = gadgetDomain.indexOf(protocolMarker) + protocolMarker.length; } // Now find the start of the path var pathIndex = gadgetDomain.indexOf("/", startIndex); // Now extract just the hostname if (pathIndex != -1) { gadgetDomain = gadgetDomain.substring(0, pathIndex); } gadgets.rpc.setRelayUrl(iframeName, gadgetDomain + "/ps/rpc_relay.html"); // Just return some random stuff so the gadget can tell when // we're done. return "callback"; }); gadgets.rpc.register('getBlogUrls', function() { var holder = {}; holder.postFeed = "http://www.blogger.com/feeds/4392267737997162189/posts/default"; holder.commentFeed = "http://www.blogger.com/feeds/4392267737997162189/comments/default"; return holder; }); gadgets.rpc.register('requestReload', function() { document.location.reload(); }); gadgets.rpc.register('requestSignOut', function(siteId) { google.friendconnect.container.openSocialSiteId = siteId; google.friendconnect.requestSignOut(); }); } var skin = {}; skin['FACE_SIZE'] = '32'; skin['HEIGHT'] = "260"; skin['TITLE'] = "Followers"; skin['BORDER_COLOR'] = "transparent"; skin['ENDCAP_BG_COLOR'] = "transparent"; skin['ENDCAP_TEXT_COLOR'] = "#000000"; skin['ENDCAP_LINK_COLOR'] = "#333333"; skin['ALTERNATE_BG_COLOR'] = "transparent"; skin['CONTENT_BG_COLOR'] = "transparent"; skin['CONTENT_LINK_COLOR'] = "#cc0000"; skin['CONTENT_TEXT_COLOR'] = "#ff0000"; skin['CONTENT_SECONDARY_LINK_COLOR'] = "#000099"; skin['CONTENT_SECONDARY_TEXT_COLOR'] = "#3366ff"; skin['CONTENT_HEADLINE_COLOR'] = "#ff0000"; skin['FONT_FACE'] = "normal normal 100% \x27Trebuchet MS\x27,Trebuchet,Verdana,Sans-serif"; google.friendconnect.container.setParentUrl("/"); google.friendconnect.container["renderMembersGadget"]( {id: "div-auv127a73xb0", height: 260, site: "04241582173159459578", useLightBoxForCanvas: false, locale: 'en' }, skin); Blog Archive Blog Archive September (3) About Me tansyclopediahyderabad, andhra pradeshhai iam tarun the founder of teem titans View my complete profile  
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test?

Remaining Time   1/25 Area of quadrilateral formed by the vertices (-1, 6), (-3, -9), (5,-8) and (3,9) is
   a). 96 sq units b). 18 sq units c). 50 sq units d). 25 sq unitsOne end of diameter of a circle is (2,3) and the centre is (-2,5). The coordinates of the other end is __________
Answered Questions    a). (-6,7) b). (6,-7) c). (6,7) d). None
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history:
            
In early times, astronomy only comprised the observation and predictions of the motions of objects visible to the naked eye. In some locations, such as Stonehenge, early cultures assembled massive artifacts that likely had some astronomical purpose. In addition to their ceremonial uses, these observatories could be employed to determine the seasons, an important factor in knowing when to plant crops, as well as in understanding the length of the year.[8]

Before tools such as the telescope were invented early study of the stars had to be conducted from the only vantage points available, namely tall buildings and high ground using the bare eye. As civilizations developed, most notably in Mesopotamia, Greece, Egypt, Persia, Maya, India, China, Nubia[9] and the Islamic world, astronomical observatories were assembled, and ideas on the nature of the universe began to be explored. Most of early astronomy actually consisted of mapping the positions of the stars and planets, a science now referred to as astrometry. From these observations, early ideas about the motions of the planets were formed, and the nature of the Sun, Moon and the Earth in the universe were explored philosophically. The Earth was believed to be the center of the universe with the Sun, the Moon and the stars rotating around it. This is known as the geocentric model of the universe.

A number of notable astronomical discoveries were made prior to the application of the telescope. For example, the obliquity of the ecliptic was estimated as early as 1000 BC by Chinese astronomers. The Chaldeans discovered that lunar eclipses recurred in a repeating cycle known as a saros.[10] In the 2nd century BC, the size and distance of the Moon were estimated by Hipparchus[11] and later Arabic astronomers. The Andromeda Galaxy, the nearest galaxy to the Milky Way, was discovered in 964 by the Persian astronomer Azophi and first described in his Book of Fixed Stars.[12] The SN 1006 supernova, the brightest apparent magnitude stellar event in recorded history, was observed by the Egyptian Arabic astronomer Ali ibn Ridwan and the Chinese astronomers in 1006.

The earliest known astronomical device was the Antikythera mechanism, an ancient Greek device for calculating the movements of planets, that dates from about 150-80 BC, and was the earliest ancestor of an astronomical analog computer. Similar astronomical analog computing devices were later constructed by Arabic astronomers and then European astronomers.

During the Middle Ages, observational astronomy was mostly stagnant in medieval Europe, at least until the 13th century. However, astronomy flourished in the Islamic world and other parts of the world. Some of the prominent Arabic astronomers who made significant contributions to the science include Al-Battani, Thebit, Azophi, Albumasar, Biruni, Arzachel, the Maragha school, Qushji, Al-Birjandi, Taqi al-Din, and others. Astronomers during that time introduced many Arabic names now used for individual stars.[13][14] It is also believed that the ruins at Great Zimbabwe and Timbuktu[15] may have housed an astronomical observatory.[16] Europeans had previously believed that there had been no astronomical observation in pre-colonial Middle Ages sub-Saharan Africa but modern discoveries show otherwise.[17][18][19]
INFORMATION:
                    
                                  n astronomy, information is mainly received from the detection and analysis of visible light or other regions of the electromagnetic radiation.[20] Observational astronomy may be divided according to the observed region of the electromagnetic spectrum. Some parts of the spectrum can be observed from the Earth's surface, while other parts are only observable from either high altitudes or space. Specific information on these subfields is given below.
Use of terms "astronomy" and "astrophysics":                               Generally, either the term "astronomy" or "astrophysics" may be used to refer to this subject.[2][3][4] Based on strict dictionary definitions, "astronomy" refers to "the study of objects and matter outside the Earth's atmosphere and of their physical and chemical properties"[5] and "astrophysics" refers to the branch of astronomy dealing with "the behavior, physical properties, and dynamic processes of celestial objects and phenomena".[6] In some cases, as in the introduction of the introductory textbook The Physical Universe by Frank Shu, "astronomy" may be used to describe the qualitative study of the subject, whereas "astrophysics" is used to describe the physics-oriented version of the subject.[7] However, since most modern astronomical research deals with subjects related to physics, modern astronomy could actually be called astrophysics.[2] Various departments that research this subject may use "astronomy" and "astrophysics", partly depending on whether the department is historically affiliated with a physics department,[3] and many professional astronomers actually have physics degrees.[4] One of the leading scientific journals in the field is named Astronomy and Astrophysics.
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RADIOASTROLOGY:
                               
Radio astronomy studies radiation with wavelengths greater than approximately one millimeter.[21] Radio astronomy is different from most other forms of observational astronomy in that the observed radio waves can be treated as waves rather than as discrete photons. Hence, it is relatively easier to measure both the amplitude and phase of radio waves, whereas this is not as easily done at shorter wavelengths.[21]

Though some radio waves are produced by astronomical objects in the form of thermal emission, most of the radio emission that is observed from Earth is seen in the form of synchrotron radiation, which is produced when electrons oscillate around magnetic fields.[21] Additionally, a number of spectral lines produced by interstellar gas, notably the hydrogen spectral line at 21 cm, are observable at radio wavelengths.[7][21]

A wide variety of objects are observable at radio wavelengths, including supernovae, interstellar gas, pulsars, and active galactic nuclei.[7][21
HUNS DYNASTY:
        
       
The Han Dynasty (206 BCE – 220 CE) of ancient China experienced contrasting periods of economic prosperity and decline. It is normally divided into three periods: Western Han (206 BCE – 9 CE), the Xin Dynasty (9–23 CE), and Eastern Han (25–220 CE). The Xin Dynasty, established by the former regent Wang Mang, formed a brief interregnum between lengthy periods of Han rule. Following the fall of Wang Mang, the Han capital was moved eastward from Chang'an to Luoyang. In consequence, historians have named the succeeding eras Western Han and Eastern Han respectively.[1]

The Han economy was defined by significant population growth, increasing urbanization, unprecedented growth of industry and trade and government experimentation with nationalization. In this era, the levels of minting and circulation of coin currency grew significantly, forming the foundation of a stable monetary system. The Silk Road facilitated the establishment of trade and tributary exchanges with foreign countries across Eurasia, many of which were previously unknown to the people of ancient China. The imperial capitals of both Western-Han (Chang'an), and of Eastern-Han (Luoyang), were among the largest cities in the world at the time, in both population and area. Here, government workshops manufactured furnishings for the palaces of the emperor and produced goods for the common people. The government oversaw the construction of roads and bridges, which facilitated official government business and encouraged commercial growth. Under Han rule, industrialists, wholesalers and merchants—from minor shopkeepers to wealthy businessmen—could engage in a wide range of enterprises and trade in the domestic, public, and even military spheres.

In the early Han period, rural peasant farmers were largely self-sufficient, but they began to rely more heavily upon commercial exchanges with the wealthy landowners of large agricultural estates. Many peasants fell into debt and were forced to become either hired laborers or rent-paying tenants of the land-owning classes. The Han government continually strove to provide economic aid to poor farmers, who had to compete with powerful and influential nobles, landowners, and merchants. The government tried to limit the power of these wealthy groups through heavy taxation and bureaucratic regulation. Emperor Wu's (r. 141–87 BCE) government even nationalized the iron and salt industries; however, these government monopolies were repealed during Eastern Han. Increasing government intervention in the private economy during the late 2nd century BCE severely weakened the commercial merchant class. This allowed wealthy landowners to increase their power and to ensure the continuation of an agrarian-dominated economy. The wealthy landlords eventually dominated commercial activities as well, maintaining control over the rural peasants—upon whom the government relied for tax revenues—military manpower, and public works labor. By the 180s CE, economic and political crises had caused the Han government to become heavily decentralized, while the great landowners became increasingly independent and powerful in their communities.
HYSTORY:
     
During the Warring States Period (403–221 BCE), the development of private commerce, new trade routes, handicraft industries, and a money economy led to the growth of new urban centers. These centers were markedly different from the older cities, which had merely served as power bases for the nobility.[2] The use of a standardized, nationwide currency during the Qin Dynasty (221–206 BCE) facilitated long-distance trade between cities. [3] Many Han cities grew large: the Western Han capital, Chang'an, had approximately 250,000 inhabitants, while the Eastern Han capital, Luoyang, had approximately 500,000 inhabitants.[4] The population of the Han Empire, recorded in the tax census of 2 CE, was 57.6 million people in 12,366,470 households.[5] The majority of commoners who populated the cities lived in extended urban and suburban areas outside the city walls and gatehouses.[6] The total urban area of Western-Han Chang'an—including the extensions outside the walls—was 36 km2 (13.9 mi2). The total urban area of Eastern-Han Luoyang—including the extensions outside the walls—was 24.5 km2 (9.4 mi2).[7] Both Chang'an and Luoyang had two prominent marketplaces; each market had a two-story government office demarcated by a flag and drum at the top.[8] Market officials were charged with maintaining order, collecting commercial taxes, setting standard commodity prices on a monthly basis, and authorizing contracts between merchants and customers.[8]

[edit] Variations in currency Further information: History of the Han Dynasty and Government of the Han Dynasty During the early Western Han period, founding Emperor Gaozu of Han (r. 202–195 BCE) closed government mints in favor of coin currency produced by the private sector.[9] Gaozu's widow Empress Lü Zhi, as grand empress dowager, abolished private minting in 186 BCE. She first issued a government-minted bronze coin weighing 5.7 g (0.2 oz), but issued another, weighing 1.5 g (0.05 oz), in 182 BCE.[9] The change to the lighter coin caused widespread inflation, so in 175 BCE Emperor Wen of Han (r. 180–157 BCE) lifted the ban on private minting; private mints were required to mint coins weighing exactly 2.6 g (0.09 oz).[9] Private minting was again abolished in 144 BCE during the end of Emperor Jing of Han's (r. 157–141 BCE) reign. Despite this, the 2.6 g (0.09 oz) bronze coin was issued by both central and local commandery governments until 120 BCE, when for one year it was replaced with a coin weighing 1.9 g (0.067 oz).[10] Other currencies were introduced around this time. Token money notes made of embroidered white deerskin, with a face value of 400,000 coins, were used to collect government revenues.[10] Emperor Wu also introduced three tin-silver alloy coins worth 3,000, 500, and 300 bronze coins respectively; all of these weighed less than 120 g (4.23 oz).[10]

   
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