query: when is mass conserved according to the law of conservation of mass

pos: Conservation of mass Conservation of mass The law of conservation of mass or principle of mass conservation states that for any system closed to all transfers of matter and energy, the mass of the system must remain constant over time, as system's mass cannot change, so quantity cannot be added nor removed. Hence, the quantity of mass is conserved over time. The law implies that mass can neither be created nor destroyed, although it may be rearranged in space, or the entities associated with it may be changed in form. For example, in chemical reactions, the mass of the chemical components before the

pos: Conservation of mass Conservation of mass The law of conservation of mass or principle of mass conservation states that for any system closed to all transfers of matter and energy, the mass of the system must remain constant over time, as system's mass cannot change, so quantity cannot be added nor removed. Hence, the quantity of mass is conserved over time. The law implies that mass can neither be created nor destroyed, although it may be rearranged in space, or the entities associated with it may be changed in form. For example, in chemical reactions, the mass of the chemical components before the

pos: Conservation of energy one will get the exact decrease of chemical energy in the combustion of the dynamite. Classically, conservation of energy was distinct from conservation of mass; however, special relativity showed that mass is related to energy and vice versa by "E = mc", and science now takes the view that mass–energy is conserved. Conservation of energy can be rigorously proven by Noether's theorem as a consequence of continuous time translation symmetry; that is, from the fact that the laws of physics do not change over time. A consequence of the law of conservation of energy is that a perpetual motion machine

pos: Mass in special relativity the system are conserved over time for any observer in any single inertial frame, though their "absolute values" will vary, according to different observers in different inertial frames. The invariant mass of the system is also conserved, but does "not" change with different observers. This is also the familiar situation with single particles: all observers calculate "the same" particle rest mass (a special case of the invariant mass) no matter how they move (what inertial frame they choose), but different observers see different total energies and momenta for the same particle. Conservation of invariant mass also requires the system to

pos: Recoil being another term for mass. That force, applied to a mass, creates an acceleration, which when applied over time, changes the velocity of a mass. According to Newton's second law, the law of momentum -- changing the velocity of the mass changes its momentum, (mass multiplied by velocity). It is important to understand at this point that velocity is not simply speed. Velocity is the speed of a mass in a particular direction. In a very technical sense, speed is a scalar (mathematics), a magnitude, and velocity is a vector (physics), magnitude and direction. Newton's third law, known as conservation

pos: Conservation law Conservation law In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, conservation of angular momentum, and conservation of electric charge. There are also many approximate conservation laws, which apply to such quantities as mass, parity, lepton number, baryon number, strangeness, hypercharge, etc. These quantities are conserved in certain classes of physics processes, but not in all. A local conservation law is usually expressed mathematically as a continuity equation, a partial differential equation which

pos: Mass in special relativity require a single observer and a single inertial frame. In general, for isolated systems and single observers, relativistic mass is conserved (each observer sees it constant over time), but is not invariant (that is, different observers see different values). Invariant mass, however, is both conserved "and" invariant (all single observers see the same value, which does not change over time). The relativistic mass corresponds to the energy, so conservation of energy automatically means that relativistic mass is conserved for any given observer and inertial frame. However, this quantity, like the total energy of a particle, is not invariant. This means

pos: Conservation of mass general relativity, the total invariant mass of photons in an expanding volume of space will decrease, due to the red shift of such an expansions. The conservation of both mass and energy therefore depends on various corrections made to energy in the theory, due to the changing gravitational potential energy of such systems. Conservation of mass The law of conservation of mass or principle of mass conservation states that for any system closed to all transfers of matter and energy, the mass of the system must remain constant over time, as system's mass cannot change, so quantity cannot be added

pos: Mass in special relativity system's invariant mass. In special relativity, mass is not "converted" to energy, for all types of energy still retain their associated mass. Neither energy nor invariant mass can be destroyed in special relativity, and each is separately conserved over time in closed systems. Thus, a system's invariant mass may change "only" because invariant mass is allowed to escape, perhaps as light or heat. Thus, when reactions (whether chemical or nuclear) release energy in the form of heat and light, if the heat and light is "not" allowed to escape (the system is closed and isolated), the energy will continue to

neg: Conservation of mass modern natural science of chemistry. The conservation of mass only holds approximately and is considered part of a series of assumptions coming from classical mechanics. The law has to be modified to comply with the laws of quantum mechanics and special relativity under the principle of mass-energy equivalence, which states that energy and mass form one conserved quantity. For very energetic systems the conservation of mass-only is shown not to hold, as is the case in nuclear reactions and particle-antiparticle annihilation in particle physics. Mass is also not generally conserved in open systems. Such is the case when various forms

neg: Chemical law Chemical law Chemical laws are those laws of nature relevant to chemistry. The most fundamental concept in chemistry is the law of conservation of mass, which states that there is no detectable change in the quantity of matter during an ordinary chemical reaction. Modern physics shows that it is actually energy that is conserved, and that energy and mass are related; a concept which becomes important in nuclear chemistry. Conservation of energy leads to the important concepts of equilibrium, thermodynamics, and kinetics. Additional laws of chemistry elaborate on the law of conservation of mass. Joseph Proust's law of definite composition

neg: Conservation of mass The conservation of both relativistic and invariant mass applies even to systems of particles created by pair production, where energy for new particles may come from kinetic energy of other particles, or from one or more photons as part of a system that includes other particles besides a photon. Again, neither the relativistic nor the invariant mass of totally closed (that is, isolated) systems changes when new particles are created. However, different inertial observers will disagree on the value of this conserved mass, if it is the relativistic mass (i.e., relativistic mass is conserved but not invariant). However, all observers

neg: Conservation of mass retained within a system, this energy exhibits mass. Also, mass must be differentiated from matter (see below), since matter may "not" be perfectly conserved in isolated systems, even though mass is always conserved in such systems. However, matter is so nearly conserved in chemistry that violations of matter conservation were not measured until the nuclear age, and the assumption of matter conservation remains an important practical concept in most systems in chemistry and other studies that do not involve the high energies typical of radioactivity and nuclear reactions. The change in mass of certain kinds of open systems where atoms

neg: Mass transfer heat, and mass transfer. The molecular transfer equations of Newton's law for fluid momentum at low Reynolds number (Stokes flow), Fourier's law for heat, and Fick's law for mass are very similar, since they are all linear approximations to transport of conserved quantities in a flow field. At higher Reynolds number, the analogy between mass and heat transfer and momentum transfer becomes less useful due to the nonlinearity of the Navier-Stokes equation (or more fundamentally, the general momentum conservation equation), but the analogy between heat and mass transfer remains good. A great deal of effort has been devoted to developing

neg: Scientific law energy that is conserved, and that energy and mass are related; a concept which becomes important in nuclear chemistry. Conservation of energy leads to the important concepts of equilibrium, thermodynamics, and kinetics. Additional laws of chemistry elaborate on the law of conservation of mass. Joseph Proust's law of definite composition says that pure chemicals are composed of elements in a definite formulation; we now know that the structural arrangement of these elements is also important. Dalton's law of multiple proportions says that these chemicals will present themselves in proportions that are small whole numbers (i.e. 1:2 for Oxygen:Hydrogen ratio in

neg: Conservation of mass masses, called a mass defect, is a measure of the binding energy in bound systems – in other words, the energy needed to break the system apart. The greater the mass defect, the larger the binding energy. The binding energy (which itself has mass) must be released (as light or heat) when the parts combine to form the bound system, and this is the reason the mass of the bound system decreases when the energy leaves the system. The total invariant mass is actually conserved, when the mass of the binding energy that has escaped, is taken into account. In

neg: Continuity equation Continuity equation A continuity equation in physics is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described using continuity equations. Continuity equations are a stronger, local form of conservation laws. For example, a weak version of the law of conservation of energy states that energy can neither be created nor destroyed—i.e., the

neg: Radioactive decay and, later, the thermal energy of the surrounding matter, all contribute to the invariant mass of the system. Thus, while the sum of the rest masses of the particles is not conserved in radioactive decay, the "system" mass and system invariant mass (and also the system total energy) is conserved throughout any decay process. This is a restatement of the equivalent laws of conservation of energy and conservation of mass. The "decay rate", or "activity", of a radioactive substance is characterized by: Constant quantities: Although these are constants, they are associated with the statistical behavior of populations of atoms. In

neg: Conservation of mass agree on the value of the conserved mass if the mass being measured is the invariant mass (i.e., invariant mass is both conserved and invariant). The mass-energy equivalence formula gives a different prediction in non-isolated systems, since if energy is allowed to escape a system, both relativistic mass and invariant mass will escape also. In this case, the mass-energy equivalence formula predicts that the "change" in mass of a system is associated with the "change" in its energy due to energy being added or subtracted: formula_12 This form involving changes was the form in which this famous equation was originally

neg: Conserved current which are required to define a foliation and thus an integrable system. The conservation law is expressed as the vanishing of a 4-divergence, where the Noether charge forms the zeroth component of the 4-current. The "conservation of charge", for example, in the notation of Maxwell's equations, where: ρ is the "free" electric charge density (in units of C/m³) J is the current density: v is the velocity of the charges. The equation would apply equally to masses (or other conserved quantities), where the word "mass" is substituted for the words "electric charge" above. Conserved current In physics a conserved current

neg: Momentum For example, in a collision of two particles, a reference frame can be chosen, where, one particle begins at rest. Another, commonly used reference frame, is the center of mass frame – one that is moving with the center of mass. In this frame, the total momentum is zero. By itself, the law of conservation of momentum is not enough to determine the motion of particles after a collision. Another property of the motion, kinetic energy, must be known. This is not necessarily conserved. If it is conserved, the collision is called an "elastic collision"; if not, it is an

neg: Conservation of mass the system, which remains both conserved and invariant (unchanging), even for different observers who view the entire system. Invariant mass is a system combination of energy and momentum, which is invariant for any observer, because in any inertial frame, the energies and momenta of the various particles always add to the same quantity (the momentum may be negative, so the addition amounts to a subtraction). The invariant mass is the relativistic mass of the system when viewed in the center of momentum frame. It is the minimum mass which a system may exhibit, as viewed from all possible inertial frames.

neg: Scientific law transformations reduce to the Galilean transformations for low velocities much less than the speed of light "c". The magnitudes of 4-vectors are invariants - "not" "conserved", but the same for all inertial frames (i.e. every observer in an inertial frame will agree on the same value), in particular if "A" is the four-momentum, the magnitude can derive the famous invariant equation for mass-energy and momentum conservation (see invariant mass): in which the (more famous) mass-energy equivalence "E" = "mc" is a special case. General relativity is governed by the Einstein field equations, which describe the curvature of space-time due to

neg: Conservation of mass For moving massive particles in a system, examining the rest masses of the various particles also amounts to introducing many different inertial observation frames (which is prohibited if total system energy and momentum are to be conserved), and also when in the rest frame of one particle, this procedure ignores the momenta of other particles, which affect the system mass if the other particles are in motion in this frame. For the special type of mass called invariant mass, changing the inertial frame of observation for a whole closed system has no effect on the measure of invariant mass of

neg: Conservation of mass The law of conservation of mass can only be formulated in classical mechanics when the energy scales associated to an isolated system are much smaller than formula_1, where formula_2is the mass of a typical object in the system, measured in the frame of reference where the object is at rest, and formula_3 is the speed of light. The law can be formulated mathematically in the fields of fluid mechanics and continuum mechanics, where the conservation of mass is usually expressed using the continuity equation, given in differential form asformula_4where formula_5 is the density (mass per unit volume), formula_6 is the

neg: Mass–energy equivalence title of his article intended as an explanation for a general reader by analogy. Mass and energy can be seen as two names (and two measurement units) for the same underlying, conserved physical quantity. Thus, the laws of conservation of energy and conservation of (total) mass are equivalent and both hold true. Einstein elaborated in a 1946 essay that "the principle of the conservation of mass [...] proved inadequate in the face of the special theory of relativity. It was therefore merged with the energy conservation principle—just as, about 60 years before, the principle of the conservation of mechanical energy

neg: Mass–energy equivalence notable consequences of relativity. It has no counterpart in classical Newtonian physics, in which radiation, light, heat, and kinetic energy never exhibit weighable mass under any circumstances. Just as the relativistic mass of an isolated system is conserved through time, so also is its invariant mass.This property allows the conservation of all types of mass in systems, and also conservation of all types of mass in reactions where matter is destroyed (annihilated), leaving behind the energy that was associated with it (which is now in non-material form, rather than material form). Matter may appear and disappear in various reactions, but

neg: Conservation of mass of energy and matter are allowed into, or out of, the system. However, unless radioactivity or nuclear reactions are involved, the amount of energy escaping (or entering) such systems as heat, mechanical work, or electromagnetic radiation is usually too small to be measured as a decrease (or increase) in the mass of the system. For systems where large gravitational fields are involved, general relativity has to be taken into account, where mass-energy conservation becomes a more complex concept, subject to different definitions, and neither mass nor energy is as strictly and simply conserved as is the case in special relativity.

neg: Boltzmann equation is conserved in a collision. Assume also that the force formula_40 is a function of position only, and that "f" is zero for formula_41. Multiplying the Boltzmann equation by "A" and integrating over momentum yields four terms, which, using integration by parts, can be expressed as where the last term is zero, since "A" is conserved in a collision. Letting formula_46, the mass of the particle, the integrated Boltzmann equation becomes the conservation of mass equation: where formula_48 is the mass density, and formula_49 is the average fluid velocity. Letting formula_50, the momentum of the particle, the integrated Boltzmann equation

neg: ADM formalism theorem then implies that the ADM energy is conserved. According to general relativity, the conservation law for the total energy does not hold in more general, time-dependent backgrounds – for example, it is completely violated in physical cosmology. Cosmic inflation in particular is able to produce energy (and mass) from "nothing" because the vacuum energy density is roughly constant, but the volume of the Universe grows exponentially. By using the ADM decomposition and introducing extra auxiliary fields, in 2009 Deruelle et al. found a method to find the Gibbons–Hawking–York boundary term for modified gravity theories "whose Lagrangian is an arbitrary

neg: Spacetime Looking at the events of this scenario in reverse sequence, we see that non-conservation of mass is a common occurrence: when an unstable elementary particle spontaneously decays into two lighter particles, total energy is conserved, but the mass is not. Part of the mass is converted into kinetic energy. The freedom to choose any frame in which to perform an analysis allows us to pick one which may be particularly convenient. For analysis of momentum and energy problems, the most convenient frame is usually the "center-of-momentum frame" (also called the zero-momentum frame, or COM frame). This is the frame in

neg: Mass momentum p by the relativistic energy-momentum equation: So long as the system is closed with respect to mass and energy, both kinds of mass are conserved in any given frame of reference. The conservation of mass holds even as some types of particles are converted to others. Matter particles (such as atoms) may be converted to non-matter particles (such as photons of light), but this does not affect the total amount of mass or energy. Although things like heat may not be matter, all types of energy still continue to exhibit mass. Thus, mass and energy do not change into

neg: Mass in special relativity to make measurements of the mass of particles like the Z boson or the top quark. Total energy is an additive conserved quantity (for single observers) in systems and in reactions between particles, but rest mass (in the sense of being a sum of particle rest masses) may not be conserved through an event in which rest masses of particles are converted to other types of energy, such as kinetic energy. Finding the sum of individual particle rest masses would require multiple observers, one for each particle rest inertial frame, and these observers ignore individual particle kinetic energy. Conservation laws

neg: Mass–energy equivalence had been combined with the principle of the conservation of heat [thermal energy]. We might say that the principle of the conservation of energy, having previously swallowed up that of the conservation of heat, now proceeded to swallow that of the conservation of mass—and holds the field alone." If the conservation of mass law is interpreted as conservation of "rest" mass, it does not hold true in special relativity. The "rest" energy (equivalently, rest mass) of a particle can be converted, not "to energy" (it already "is" energy (mass)), but rather to "other" forms of energy (mass) that require motion,

neg: Conservation of energy recognized in twentieth century experience, rest mass is not conserved, unlike the "total" mass or "total" energy. All forms of energy contribute to the total mass and total energy. For example, an electron and a positron each have rest mass. They can perish together, converting their combined rest energy into photons having electromagnetic radiant energy, but no rest mass. If this occurs within an isolated system that does not release the photons or their energy into the external surroundings, then neither the total "mass" nor the total "energy" of the system will change. The produced electromagnetic radiant energy contributes just

neg: Henri Poincaré of mass implied by Lorentz's variable mass formula_9, 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. It was Albert Einstein's concept of mass–energy equivalence (1905) that a body losing energy as radiation or heat was losing mass of amount "m" = "E"/"c" that resolved Poincaré's paradox, without using any compensating mechanism within the ether. The Hertzian oscillator loses mass in the emission process, and momentum is conserved in any frame. However, concerning Poincaré's solution of the Center of Gravity problem,

neg: Conservation form weak solutions and severe numerical difficulties in simulations of such equations. An example of a set of equations written in conservation form are the Euler equations of fluid flow: Each of these represents the conservation of mass, momentum and energy, respectively. Conservation form Conservation form or "Eulerian form" refers to an arrangement of an equation or system of equations, usually representing a hyperbolic system, that emphasizes that a property represented is conserved, i.e. a type of continuity equation. The term is usually used in the context of continuum mechanics. Equations in conservation form take the form for any conserved quantity

neg: Conservation law density form: In a space with more than one dimension the former definition can be extended to an equation that can be put into the form: where the "conserved quantity" is "y(r,t)", "formula_11" denotes the scalar product, "∇" is the nabla operator, here indicating a gradient, and "a(y)" is a vector of current coefficients, analogously corresponding to the divergence of a vector c.d. associated to the c.q. j(y): This is the case for the continuity equation: Here the conserved quantity is the mass, with density "ρ"(r,t) and current density "ρ"u, identical to the momentum density, while u(r,t) is the flow

neg: Mass balance Mass balance A mass balance, also called a material balance, is an application of conservation of mass to the analysis of physical systems. By accounting for material entering and leaving a system, mass flows can be identified which might have been unknown, or difficult to measure without this technique. The exact conservation law used in the analysis of the system depends on the context of the problem, but all revolve around mass conservation, i.e. that matter cannot disappear or be created spontaneously. Therefore, mass balances are used widely in engineering and environmental analyses. For example, mass balance theory is used

neg: Radiation stress ∇"h", as well the forcing by the wind and the bed friction. In terms of the horizontal momentum M the above equations become: In a Cartesian coordinate system, the mass conservation equation becomes: with and respectively the "x" and "y" components of the mass transport velocity . The horizontal momentum equations are: For an inviscid flow the mean mechanical energy of the total flow – that is the sum of the energy of the mean flow and the fluctuating motion – is conserved. However, the mean energy of the fluctuating motion itself is not conserved, nor is the energy of

neg: Conservation of energy and quantum theory (including QED) in the flat space-time. With the discovery of special relativity by Henri Poincaré and Albert Einstein, energy was proposed to be one component of an energy-momentum 4-vector. Each of the four components (one of energy and three of momentum) of this vector is separately conserved across time, in any closed system, as seen from any given inertial reference frame. Also conserved is the vector length (Minkowski norm), which is the rest mass for single particles, and the invariant mass for systems of particles (where momenta and energy are separately summed before the length is calculated—see

neg: Fractional calculus of mass equation is needed to model fluid flow when the control volume is not large enough compared to the scale of heterogeneity and when the flux within the control volume is non-linear. In the referenced paper, the fractional conservation of mass equation for fluid flow is: In 2013–2014 Atangana "et al." described some groundwater flow problems using the concept of derivative with fractional order. In these works, The classical Darcy law is generalized by regarding the water flow as a function of a non-integer order derivative of the piezometric head. This generalized law and the law of conservation of

neg: First law of thermodynamics (fluid mechanics) per unit mass equals proper energy per unit mass plus pressure times volume per unit mass (reciprocal of mass density). First law of thermodynamics (fluid mechanics) In physics, the first law of thermodynamics is an expression of the conservation of total energy of a system. The increase of the energy of a system is equal to the sum of work done on the system and the heat added to that system: where In fluid mechanics, the first law of thermodynamics takes the following form: where Because it expresses conservation of total energy, this is sometimes referred to as the energy

neg: Conservation of energy Naturalis Principia Mathematica". This is now regarded as an example of Whig history. Matter is composed of such things as atoms, electrons, neutrons, and protons. It has "intrinsic" or "rest" mass. In the limited range of recognized experience of the nineteenth century it was found that such rest mass is conserved. Einstein's 1905 theory of special relativity showed that it corresponds to an equivalent amount of "rest energy". This means that it can be converted to or from equivalent amounts of "other" (non-material) forms of energy, for example kinetic energy, potential energy, and electromagnetic radiant energy. When this happens, as

neg: Conservation of mass enable their mass-change to be measured, once the energy of the reaction had been removed from the system. This later indeed proved to be possible, although it was eventually to be the first artificial nuclear transmutation reaction in 1932, demonstrated by Cockcroft and Walton, that proved the first successful test of Einstein's theory regarding mass-loss with energy-loss. The law conservation of mass and the analogous law of conservation of energy were finally overruled by a more general principle known as the mass–energy equivalence. Special relativity also redefines the concept of mass and energy, which can be used interchangeably and are

neg: Mikhail Lomonosov from the action of pure heat. The experiments– of which I append the record in 13 pages– demonstrated that the famous Robert Boyle was deluded, for without access of air from outside the mass of the burnt metal remains the same". That is the Law of Mass Conservation in chemical reaction, which was well-known today as "in a chemical reaction, the mass of reactants is equal to the mass of the products." Lomonosov, together with Lavoisier, is regarded as the one who discovered the law of mass conservation. He stated that all matter is composed of corpuscles – molecules that

neg: Mass in special relativity condition of "closure" to mass-energy (total isolation) is violated. In this case, conservation of invariant mass of the system also will no longer hold. Such a loss of rest mass in systems when energy is removed, according to "E=mc" where "E" is the energy removed, and "m" is the change in rest mass, reflect changes of mass associated with movement of energy, not "conversion" of mass to energy. Again, in special relativity, the rest mass of a system is not required to be equal to the sum of the rest masses of the parts (a situation which would be analogous

neg: Anti-gravity mass, combined with negative inertial mass, would comply with the strong equivalence principle of general relativity theory and the Newtonian laws of conservation of linear momentum and energy. Bondi's proof yielded singularity free solutions for the relativity equations. In July 1988, Robert L. Forward presented a paper at the AIAA/ASME/SAE/ASEE 24th Joint Propulsion Conference that proposed a Bondi negative gravitational mass propulsion system. Bondi pointed out that a negative mass will fall toward (and not away from) "normal" matter, since although the gravitational force is repulsive, the negative mass (according to Newton's law, F=ma) responds by accelerating in the opposite

neg: Conservation of energy special relativity continues to hold, so long as the reference frame of the observer is unchanged. This applies to the total energy of systems, although different observers disagree as to the energy value. Also conserved, and invariant to all observers, is the invariant mass, which is the minimal system mass and energy that can be seen by any observer, and which is defined by the energy–momentum relation. In general relativity, energy–momentum conservation is not well-defined except in certain special cases. Energy-momentum is typically expressed with the aid of a stress–energy–momentum pseudotensor. However, since pseudotensors are not tensors, they do not

neg: Conservation of mass reaction is equal to the mass of the components after the reaction. Thus, during any chemical reaction and low-energy thermodynamic processes in an isolated system, the total mass of the reactants, or starting materials, must be equal to the mass of the products. The concept of mass conservation is widely used in many fields such as chemistry, mechanics, and fluid dynamics. Historically, mass conservation was demonstrated in chemical reactions independently by Mikhail Lomonosov and later rediscovered by Antoine Lavoisier in the late 18th century. The formulation of this law was of crucial importance in the progress from alchemy to the

neg: Newton's cradle the first ball. When two (or three) balls are dropped, the two (or three) balls on the opposite side swing out. Some say that this behavior demonstrates the conservation of momentum and kinetic energy in elastic collisions. However, if the colliding balls behave as described above with the same mass possessing the same velocity before and after the collisions, then any function of mass and velocity is conserved in such an event. Newton's cradle can be modeled with simple physics and minor errors if you incorrectly assume the balls always collide in pairs. If one ball strikes four stationary balls

neg: Force a system such that net momentum is never lost nor gained. In the special theory of relativity, mass and energy are equivalent (as can be seen by calculating the work required to accelerate an object). When an object's velocity increases, so does its energy and hence its mass equivalent (inertia). It thus requires more force to accelerate it the same amount than it did at a lower velocity. Newton's Second Law remains valid because it is a mathematical definition. But for relativistic momentum to be conserved, it must be redefined as: where formula_19 is the rest mass and formula_20 the

neg: Recoil of momentum, recognizes that changes in the motion of a mass, brought about by the application of forces and accelerations, does not occur in isolation; that is, other bodies of mass are found to be involved in directing those forces and accelerations. Furthermore, if all the masses and velocities involved are accounted for, the vector sum, magnitude and direction, of the momentum of all the bodies involved does not change; hence, momentum of the system is conserved. This conservation of momentum is why gun recoil occurs in the opposite direction of bullet projection -- the mass times velocity of the

neg: Spacetime velocities. Since , the momentum . If the total momentum of an interacting system of particles is observed to be conserved in one frame, it will likewise be observed to be conserved in any other frame. Conservation of momentum in the COM frame amounts to the requirement that "p" = 0 both before and after collision. In the Newtonian analysis, conservation of mass dictates that . In the simplified, one-dimensional scenarios that we have been considering, only one additional constraint is necessary before the outgoing momenta of the particles can be determined—an energy condition. In the one-dimensional case of a

neg: George Hadley the direction taken by an air mass moving relative to the Earth, an element that had been missing in Halley's proposal. Later, in the second half of the 19th century, Hadley's theory was itself shown to be deficient in that it was based on an assumption that when air mass travels from one latitude to another its linear momentum is conserved. However, since the air mass is at all times in a state of circumnavigating the Earth axis, it is in fact the angular momentum that is conserved, causing an effect known as the Coriolis effect. When using the correct

neg: Conserved sequence As with proteins, nucleic acids that are important for the structure and function of non-coding RNA (ncRNA) can also be conserved. However, sequence conservation in ncRNAs is generally poor compared to protein-coding sequences, and base pairs that contribute to structure or function are often conserved instead. Conserved sequences are typically identified by bioinformatics approaches based on sequence alignment. Advances in high-throughput DNA sequencing and protein mass spectrometry has substantially increased the availability of protein sequences and whole genomes for comparison since the early 2000s. Conserved sequences may be identified by homology search, using tools such as BLAST, HMMER and Infernal.

neg: Bohr–Einstein debates "S", and since it is only the interaction with this screen that can cause a deflection from the original direction of propagation, by the law of conservation of impulse which implies that the sum of the impulses of two systems which interact is conserved, if the incident particle is deviated toward the top, the screen will recoil toward the bottom and vice versa. In realistic conditions the mass of the screen is so large that it will remain stationary, but, in principle, it is possible to measure even an infinitesimal recoil. If we imagine taking the measurement of the impulse

neg: Conservation of energy the heat energy may be written where formula_21 is the temperature and formula_22 is a small change in the entropy of the system. Temperature and entropy are variables of state of a system. If an open system (in which mass may be exchanged with the environment) has several walls such that the mass transfer is through rigid walls separate from the heat and work transfers, then the first law may be written: where formula_24 is the added mass and formula_25 is the internal energy per unit mass of the added mass, measured in the surroundings before the process. The conservation

neg: Isolated system An isolated system obeys the conservation law that its total energy–mass stays constant. Most often, in thermodynamics, matter and energy are treated as separately conserved. Because of the requirement of enclosure, and the near ubiquity of gravity, strictly and ideally isolated systems do not actually occur in experiments or in nature. Though very useful, they are strictly hypothetical. Classical thermodynamics is usually presented as postulating the existence of isolated systems. It is also usually presented as the fruit of experience. Obviously, no experience has been reported of an ideally isolated system. It is, however, the fruit of experience that some

neg: Lorentz ether theory concluded in 1906 that the inertia of (electromagnetic) energy is a necessary condition for the center of mass theorem to hold in systems, in which electromagnetic fields and matter are acting on each other. Based on the mass–energy equivalence he showed that emission and absorption of em-radiation and therefore the transport of inertia solves all problems. On that occasion, Einstein referred to Poincaré's 1900-paper and wrote: Also Poincaré's rejection of the reaction principle due to the violation of the mass conservation law can be avoided through Einstein's formula_28, because mass conservation appears as a special case of the energy conservation

neg: Mass in general relativity in question. However, when the symmetry is an infinite parameter continuous group, the existence of a conserved energy is not guaranteed. In a similar manner, Noether's theorem associates conserved momenta with space-translations, when the symmetry group of the translations is finite-dimensional. Because General Relativity is a diffeomorphism invariant theory, it has an infinite continuous group of symmetries rather than a finite-parameter group of symmetries, and hence has the wrong group structure to guarantee a conserved energy. Noether's theorem has been extremely influential in inspiring and unifying various ideas of mass, system energy, and system momentum in General Relativity. As an

neg: Mass balance and exclusion of detectably erroneous measurements. Since all real world measured values contain inherent error, the reconciled measurements provide a better basis than the measured values do for financial reporting, optimization, and regulatory reporting. Software packages exist to make this commercially feasible on a daily basis. Mass balance A mass balance, also called a material balance, is an application of conservation of mass to the analysis of physical systems. By accounting for material entering and leaving a system, mass flows can be identified which might have been unknown, or difficult to measure without this technique. The exact conservation law used

neg: Chemical equation specific types of energy or radiation. The law of conservation of mass dictates that the quantity of each element does not change in a chemical reaction. Thus, each side of the chemical equation must represent the same quantity of any particular element. Likewise, the charge is conserved in a chemical reaction. Therefore, the same charge must be present on both sides of the balanced equation. One balances a chemical equation by changing the scalar number for each chemical formula. Simple chemical equations can be balanced by inspection, that is, by trial and error. Another technique involves solving a system of

neg: Measure (mathematics) and a number of other sources. For more details, see the article on Radon measures. Some important measures are listed here. Other 'named' measures used in various theories include: Borel measure, Jordan measure, ergodic measure, Euler measure, Gaussian measure, Baire measure, Radon measure, Young measure, and Loeb measure. In physics an example of a measure is spatial distribution of mass (see e.g., gravity potential), or another non-negative extensive property, conserved (see conservation law for a list of these) or not. Negative values lead to signed measures, see "generalizations" below. Let be a measure. If and are measurable sets with then

neg: Mass–energy equivalence by calculating the invariant mass of pairs of photons in a particle detector, pairs can be identified that were probably produced by pion disintegration. A similar calculation illustrates that the invariant mass of systems is conserved, even when massive particles (particles with rest mass) within the system are converted to massless particles (such as photons). In such cases, the photons contribute invariant mass to the system, even though they individually have no invariant mass or rest mass. Thus, an electron and positron (each of which has rest mass) may undergo annihilation with each other to produce two photons, each of

neg: Spacetime two bodies stick together and continue moving as a single particle. This second case is the case of completely inelastic collision. For both cases (1) and (2), momentum, mass, and total energy are conserved. However, kinetic energy is not conserved in cases of inelastic collision. A certain fraction of the initial kinetic energy is converted to heat. In case (2), two masses with momentums and collide to produce a single particle of conserved mass traveling at the center of mass velocity of the original system, . The total momentum is conserved. Fig. 3‑10 illustrates the inelastic collision of two particles

neg: Mass–energy equivalence the surroundings, which absorb the heat (so that total mass is conserved). Such a change in mass may only happen when the system is open, and the energy and mass escapes. Thus, 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 would still be equal to the original mass of the chamber and dynamite. If sitting on a scale, the weight and mass would not change. This would in theory also happen even with a nuclear bomb, if it could be kept in an

neg: Quadrupole mass-energy in a system, which is conserved—thus it gives off no radiation. Similarly, the mass dipole corresponds to the center of mass of a system and its first derivative represents momentum which is also a conserved quantity so the mass dipole also emits no radiation. The mass quadrupole, however, can change in time, and is the lowest-order contribution to gravitational radiation. The simplest and most important example of a radiating system is a pair of mass points with equal masses orbiting each other on a circular orbit, an approximation to e.g. special case of binary black holes. Since the dipole

neg: Conservation of mass law in chemical reactions, even though they were carried out with other intentions. His research indicated that in certain reactions the loss or gain could not have been more than from 2 to 4 parts in 100,000. The difference in the accuracy aimed at and attained by Lavoisier on the one hand, and by Morley and Stas on the other, is enormous. The law of conservation of mass was challenged with the advent of special relativity. In one of the Annus Mirabilis papers of Albert Einstein in 1905, he suggested an equivalence between mass and energy. This theory implied several

neg: Viscosity mass diffusivity and thermal conductivity. The fact that mass, momentum, and energy (heat) transport are among the most relevant processes in continuum mechanics is not a coincidence: these are among the few physical quantities that are conserved at the "microscopic level" in interparticle collisions. Thus, rather than being dictated by the fast and complex microscopic interaction timescale, their dynamics occurs on macroscopic timescales, as described by the various equations of transport theory and hydrodynamics. Newton's law of viscosity is a constitutive equation (like Hooke's law, Fick's law, and Ohm's law): it is not a fundamental law of nature but an

neg: Mass in general relativity example of the application of Noether's theorem is the example of stationary space-times and their associated Komar mass.(Komar 1959). While general space-times lack a finite-parameter time-translation symmetry, stationary space-times have such a symmetry, known as a Killing vector. Noether's theorem proves that such stationary space-times must have an associated conserved energy. This conserved energy defines a conserved mass, the Komar mass. ADM mass was introduced (Arnowitt et al., 1960) from an initial-value formulation of general relativity. It was later reformulated in terms of the group of asymptotic symmetries at spatial infinity, the SPI group, by various authors. (Held, 1980). This

neg: Conservation law electric current into or out of the volume that carries the difference in charge. Since it only involves continuous "local" changes, this stronger type of conservation law is Lorentz invariant; a quantity conserved in one reference frame is conserved in all moving reference frames. This is called a "local conservation" law. Local conservation also implies global conservation; that the total amount of the conserved quantity in the Universe remains constant. All of the conservation laws listed above are local conservation laws. A local conservation law is expressed mathematically by a "continuity equation", which states that the change in the quantity

neg: Negative mass required condition for the mathematical consistency of the theory. Ever since Newton first formulated his theory of gravity, there have been at least three conceptually distinct quantities called mass: Einstein’s equivalence principle postulates that inertial mass must equal passive gravitational mass. The law of conservation of momentum requires that active and passive gravitational mass be identical. All experimental evidence to date has found these are, indeed, always the same. In considering negative mass, it is important to consider which of these concepts of mass are negative. In most analyses of negative mass, it is assumed that the equivalence principle and

neg: Numerical modeling (geology) means no empty space can be found in the object. This equation is commonly used in numerical modeling in geology. One example is the continuity equation of mass of fluid. Based on the law of "conservation of mass", for a fluid with density formula_1 at position formula_2 in a fixed volume formula_3 of fluid, the rate of change of mass is equal to the outward fluid flow across the boundary formula_4: formula_5 where formula_6 is the volume element and formula_7 is the velocity at formula_2. In Lagrangian form: formula_9 In Eulerian form: formula_10 This equation is useful when the model

neg: Pierre Louis Maupertuis such bodies in collisions. Cartesian and Newtonian physicists argued that in their collisions, point masses conserved both momentum and relative velocity. Leibnizians, on the other hand, argued that they also conserved what was called "live force" or "vis viva". This was unacceptable for their opponents for two reasons: the first that live force conservation did not apply to so-called ‘hard’ bodies, bodies that were totally incompressible, whereas the other two conservation principles did; the second was that live force was defined by the product of mass and square of velocity. Why did the velocity appear twice in this quantity, as

neg: Mass–energy equivalence mass conservation appears as a special case of the energy conservation law. During the nineteenth century there were several speculative attempts to show that mass and energy were proportional in various ether theories. In 1873 Nikolay Umov pointed out a relation between mass and energy for ether in the form of , where . The writings of Samuel Tolver Preston, and a 1903 paper by Olinto De Pretto, presented a mass–energy relation. Bartocci (1999) observed that there were only three degrees of separation linking De Pretto to Einstein, concluding that Einstein was probably aware of De Pretto's work. Preston and

neg: Equivalent weight zinc oxide, implying that the zinc has reacted with 12.24 g of oxygen (from the Law of conservation of mass): the equivalent weight of zinc is the mass which will react with eight grams of oxygen, hence 50 g × 8 g/12.24 g = 32.7 g. When choosing primary standards in analytical chemistry, compounds with higher equivalent weights are generally more desirable because weighing errors are reduced. An example is the volumetric standardisation of a solution of sodium hydroxide which has been prepared to approximately 0.1 . It is necessary to calculate the mass of a solid acid which will

neg: Chemical revolution practiced chemistry. Lavoisier's work was characterized by his systematic determination of weights and his strong emphasis on precision and accuracy. The law of the conservation of mass, that the weight of matter would be conserved through any reaction, was discovered by Antoine Lavoisier in the late 18th century and represented his careful conduction of research. Lavoisier also contributed to chemistry a method of understanding combustion and respiration and proof of the composition of water by decomposition into its constituent parts. He explained the theory of combustion, and challenged the phlogiston theory with his views on caloric. The "Traité" incorporates notions

neg: Probability mass function integrated over an interval to yield a probability. The value of the random variable having the largest probability mass is called the mode. Suppose that "X": "S" → "A" (A formula_1 R) is a discrete random variable defined on a sample space "S". Then the probability mass function "f": "A" → [0, 1] for "X" is defined as Thinking of probability as mass helps to avoid mistakes since the physical mass is conserved as is the total probability for all hypothetical outcomes "x": When there is a natural order among the potential outcomes "x", it may be convenient to assign

neg: Mass in special relativity that, even though it is conserved for any observer during a reaction, its absolute "value" will change with the frame of the observer, and for different observers in different frames. By contrast, the rest mass and invariant masses of systems and particles are "both" conserved "and" also invariant. For example: A closed container of gas (closed to energy as well) has a system "rest mass" in the sense that it can be weighed on a resting scale, even while it contains moving components. This mass is the invariant mass, which is equal to the total relativistic energy of the container

neg: Conservation of energy as much to the inertia (and to any weight) of the system as did the rest mass of the electron and positron before their demise. Likewise, non-material forms of energy can perish into matter, which has rest mass. Thus, conservation of energy ("total", including material or "rest" energy), and conservation of mass ("total", not just "rest"), each still holds as an (equivalent) law. In the 18th century these had appeared as two seemingly-distinct laws. The discovery in 1911 that electrons emitted in beta decay have a continuous rather than a discrete spectrum appeared to contradict conservation of energy, under the

neg: Fluid flow through porous media filtration through sand beds. For transient processes in which the flux varies from point to-point, the following differential form of Darcy’s law is used. Darcy's law is valid for situation where the porous material is already saturated with the fluid. For the calculation of capillary imbibition speed of a liquid to an initially dry medium, Washburn's or Bosanquet's equations are used. Mass conservation of fluid across the porous medium involves the basic principle that mass flux in minus mass flux out equals the increase in amount stored by a medium. This means that total mass of the fluid is always

neg: Spacetime conserved independently, because it has been subsumed into the total relativistic energy. This makes the relativistic conservation of energy a simpler concept than in nonrelativistic mechanics, because the total energy is conserved without any qualifications. Kinetic energy converted into heat or internal potential energy shows up as an increase in mass. The topics in this section are of significantly greater technical difficulty than those in the preceding sections and are not essential for understanding "Introduction to curved spacetime." Lorentz transformations relate coordinates of events in one reference frame to those of another frame. Relativistic composition of velocities is used to

neg: Mass in general relativity reformulation did much to clarify the theory, including explaining why ADM momentum and ADM energy transforms as a 4-vector (Held, 1980). Note that the SPI group is actually infinite-dimensional. The existence of conserved quantities is because the SPI group of "super-translations" has a preferred 4-parameter subgroup of "pure" translations, which, by Noether's theorem, generates a conserved 4-parameter energy–momentum. The norm of this 4-parameter energy–momentum is the ADM mass. The Bondi mass was introduced (Bondi, 1962) in a paper that studied the loss of mass of physical systems via gravitational radiation. The Bondi mass is also associated with a group of

neg: Periodic boundary conditions that this approach is not consistent. It fails to explain the absence of conservation of angular momentum of a single particle moving in a periodic cell. Lagrangian of the particle is constant and therefore rotationally invariant, while angular momentum of the particle is not conserved. This contradiction is caused by the fact that Noether's theorem is usually formulated for closed systems. The periodic cell exchanges mass momentum, angular momentum, and energy with the neighboring cells. When applied to the microcanonical ensemble (constant particle number, volume, and energy, abbreviated NVE), using PBC rather than reflecting walls slightly alters the sampling of

neg: Charge conservation important, well established property of the electromagnetic field and has many testable consequences. The theoretical justification for charge conservation is greatly strengthened by being linked to this symmetry. For example, gauge invariance also requires that the photon be massless, so the good experimental evidence that the photon has zero mass is also strong evidence that charge is conserved. Even if gauge symmetry is exact, however, there might be apparent electric charge non-conservation if charge could leak from our normal 3-dimensional space into hidden extra dimensions. Simple arguments rule out some types of charge nonconservation. For example, the magnitude of the

neg: Nuclear reaction to take place (endothermic reaction). This can be calculated by reference to a table of very accurate particle rest masses, as follows: according to the reference tables, the nucleus has a standard atomic weight of 6.015 atomic mass units (abbreviated u), the deuterium has 2.014 u, and the helium-4 nucleus has 4.0026 u. Thus: In a nuclear reaction, the total (relativistic) energy is conserved. The "missing" rest mass must therefore reappear as kinetic energy released in the reaction; its source is the nuclear binding energy. Using Einstein's mass-energy equivalence formula "E" = "mc"², the amount of energy released can be

neg: Non-topological soliton Here formula_3 is a constant inside the ball except for a thin surface coat where it sharply drops to the global U(1) symmetrical minimum of formula_4. The value formula_3 is adjusted so that it minimises the energy of the configuration Since the "U"(1) symmetry gives the conserved current formula_7 the ball possesses the conserved charge The minimization of the energy (1) with R gives The charge conservation allows the decay of the ball into Q particles exactly. This decay is energetically unprofitable if the sum mass Qm exceed the energy (2). Therefore, for the NTS existence it is necessary to

neg: Magnetohydrodynamics system. In a closed system such as the cloud of gas and dust from which the Sun was formed, mass and angular momentum are both conserved. That conservation would imply that as the mass concentrated in the center of the cloud to form the Sun, it would spin faster, much like a skater pulling their arms in. The high speed of rotation predicted by early theories would have flung the proto-Sun apart before it could have formed. However, magnetohydrodynamic effects transfer the Sun's angular momentum into the outer solar system, slowing its rotation. Breakdown of ideal MHD (in the form

neg: Atomic theory had become accepted by portions of the scientific community. Near the end of the 18th century, two laws about chemical reactions emerged without referring to the notion of an atomic theory. The first was the law of conservation of mass, closely associated with the work of Antoine Lavoisier, which states that the total mass in a chemical reaction remains constant (that is, the reactants have the same mass as the products). The second was the law of definite proportions. First established by the French chemist Joseph Louis Proust in 1799, this law states that if a compound is broken down

neg: Groundwater flow equation for a given control volume, aside from sources or sinks, mass cannot be created or destroyed. The conservation of mass states that, for a given increment of time ("Δt"), the difference between the mass flowing in across the boundaries, the mass flowing out across the boundaries, and the sources within the volume, is the change in storage. Mass can be represented as density times volume, and under most conditions, water can be considered incompressible (density does not depend on pressure). The mass fluxes across the boundaries then become volume fluxes (as are found in Darcy's law). Using Taylor series to

neg: Derivation of the Navier–Stokes equations equation), three important concepts may be concisely written: conservation of mass, conservation of momentum, and conservation of energy. Validity is retained if formula_5 is a vector, in which case the vector-vector product in the second term will be a dyad. A general momentum equation is obtained when the conservation relation is applied to momentum. If the intensive property formula_5; considered is the mass flux (also "momentum density"), i.e. the product of mass density and flow velocity formula_17, by substitution in the general continuum equation: where formula_19 is a dyad, a special case of tensor product, which results in a second

neg: Henri Poincaré 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. However, 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. Poincaré performed a Lorentz boost (to order "v"/"c") to the frame of the moving source. He noted that energy conservation holds in both frames, but that the law of conservation

neg: Continuity equation include electric charge conservation where is the electric 4-current; and energy-momentum conservation where is the stress-energy tensor. In general relativity, where spacetime is curved, the continuity equation (in differential form) for energy, charge, or other conserved quantities involves the "covariant" divergence instead of the ordinary divergence. For example, the stress–energy tensor is a second-order tensor field containing energy–momentum densities, energy–momentum fluxes, and shear stresses, of a mass-energy distribution. The differential form of energy-momentum conservation in general relativity states that the "covariant" divergence of the stress-energy tensor is zero: This is an important constraint on the form the Einstein field equations

neg: Maupertuis's principle that light does not follow the same laws as material objects. A few months later, well before Maupertuis's work appeared in print, Leonhard Euler independently defined action in its modern abbreviated form formula_42 and applied it to the motion of a particle, but not to light (see ). Euler also recognized that the principle only held when the speed was a function only of position, i.e., when the total energy was conserved. (The mass factor in the action and the requirement for energy conservation were not relevant to Maupertuis, who was concerned only with light.) Euler used this principle to

neg: Antoine Lavoisier crucial step in the advancement of chemistry. In 1774, he showed that, although matter can change its state in a chemical reaction, the total mass of matter is the same at the end as at the beginning of every chemical change. Thus, for instance, if a piece of wood is burned to ashes, the total mass remains unchanged if gaseous reactants and products are included. Lavoisier's experiments supported the law of conservation of mass. In France it is taught as Lavoisier's Law and is paraphrased from a statement in his "Traité Élémentaire de Chimie" to ""Rien ne se perd, rien

neg: Thermodynamics of micellization the law of mass action and the pseudo-phase separation model. The law of mass action postulates that the micelle formation can be modeled as a chemical equilibrium process between the micelles formula_7 and its constituents, the surfactant momomers, formula_8: formula_9, where formula_6 is the average number of surfactant monomers in solution that associate into a micelle, commonly denoted the aggregation number. The equilibrium is characterized by an equilibrium constant defined by formula_11, where formula_12 and formula_13 are the concentrations of micelles and free surfactant monomers, respectively. In combination with the law of conservation of mass, the system is fully specified

neg: Onsager reciprocal relations is the mass flux vector. The formulation of energy conservation is generally not in the form of a continuity equation because it includes contributions both from the macroscopic mechanical energy of the fluid flow and of the microscopic internal energy. However, if we assume that the macroscopic velocity of the fluid is negligible, we obtain energy conservation in the following form: where formula_6 is the internal energy density and formula_15 is the internal energy flux. Since we are interested in a general imperfect fluid, entropy is locally not conserved and its local evolution can be given in the form of

neg: Conserved current Conserved current In physics a conserved current is a current, formula_1, that satisfies the continuity equation formula_2. The continuity equation represents a conservation law, hence the name. Indeed, integrating the continuity equation over a volume formula_3, large enough to have no net currents through its surface, leads to the conservation law where formula_5 is the conserved quantity. In gauge theories the gauge fields couple to conserved currents. For example, the electromagnetic field couples to the conserved electric current. Conserved current is the flow of the canonical conjugate of a quantity possessing a continuous translational symmetry. The continuity equation for the

neg: Physics of firearms a combined mass and the bullet has a mass . When the gun is fired, the two systems move away from one another with new velocities and respectively. But the law of conservation of momentum states that the magnitudes of their momenta must be equal: Since force equals the rate of change in momentum and the initial momenta are zero, the force on the bullet must therefore be the same as the force on the gun/shooter. Gunshot victims frequently fall or collapse when shot; this is less a result of the momentum of the bullet pushing them over, but is

neg: Potential vorticity turn spins more slowly. When the air converges horizontally, the air speed increases to maintain potential vorticity, and the vertical extent increases to conserve mass. On the other hand, divergence causes the vortex to spread, slowing down the rate of spin. Hans Ertel generalized Rossby's work via an independent paper published in 1942. By identifying a conserved quantity following the motion of an air parcel, it can be proved that a certain quantity called the Ertel potential vorticity is also conserved for an idealized continuous fluid. We look at the momentum equation and the mass continuity equation of an idealized