# Conservation Laws

# CONSERVATION LAWS

## CONCEPT

The term "conservation laws" might sound at first like a body of legal statutes geared toward protecting the environment. In physics, however, the term refers to a set of principles describing certain aspects of the physical universe that are preserved throughout any number of reactions and interactions. Among the properties conserved are energy, linear momentum, angular momentum, and electrical charge. (Mass, too, is conserved, though only in situations well below the speed of light.) The conservation of these properties can be illustrated by examples as diverse as dropping a ball (energy); the motion of a skater spinning on ice (angular momentum); and the recoil of a rifle (linear momentum).

## HOW IT WORKS

The conservation laws describe physical properties that remain constant throughout the various processes that occur in the physical world. In physics, "to conserve" something means "to result in no net loss of" that particular component. For each such component, the input is the same as the output: if one puts a certain amount of energy into a physical system, the energy that results from that system will be the same as the energy put into it.

The energy may, however, change forms. In addition, the operations of the conservation laws are—on Earth, at least—usually affected by a number of other forces, such as gravity, friction, and air resistance. The effects of these forces, combined with the changes in form that take place within a given conserved property, sometimes make it difficult to perceive the working of the conservation laws. It was stated above that the resulting energy of a physical system will be the same as the energy that was introduced to it. Note, however, that the usable energy output of a system will not be equal to the energy input. This is simply impossible, due to the factors mentioned above—particularly friction.

When one puts gasoline into a motor, for instance, the energy that the motor puts out will never be as great as the energy contained in the gasoline, because part of the input energy is expended in the operation of the motor itself. Similarly, the angular momentum of a skater on ice will ultimately be dissipated by the resistant force of friction, just as that of a Frisbee thrown through the air is opposed both by gravity and air resistance—itself a specific form of friction.

In each of these cases, however, the property is still conserved, even if it does not seem so to the unaided senses of the observer. Because the motor has a usable energy output less than the input, it seems as though energy has been lost. In fact, however, the energy has only changed forms, and some of it has been diverted to areas other than the desired output. (Both the noise and the heat of the motor, for instance, represent uses of energy that are typically considered undesirable.) Thus, upon closer study of the motor—itself an example of a system—it becomes clear that the resulting energy, if not the desired usable output, is the same as the energy input.

As for the angular momentum examples in which friction, or air resistance, plays a part, here too (despite all apparent evidence to the contrary) the property is conserved. This is easier to understand if one imagines an object spinning in outer space, free from the opposing force of friction. Thanks to the conservation of angular momentum, an object set into rotation in space will continue to spin indefinitely. Thus, if an astronaut in the 1960s, on a spacewalk from his capsule, had set a screwdriver spinning in the emptiness of the exosphere, the screwdriver would still be spinning today!

### Energy and Mass

Among the most fundamental statements in all of science is the conservation of energy: a system isolated from all outside factors will maintain the same total amount of energy, even though energy transformations from one form or another take place.

Energy is manifested in many varieties, including thermal, electromagnetic, sound, chemical, and nuclear energy, but all these are merely reflections of three basic types of energy. There is potential energy, which an object possesses by virtue of its position; kinetic energy, which it possesses by virtue of its motion; and rest energy, which it possesses by virtue of its mass.

The last of these three will be discussed in the context of the relationship between energy and mass; at present the concern is with potential and kinetic energy. Every system possesses a certain quantity of both, and the sum of its potential and kinetic energy is known as mechanical energy. The mechanical energy within a system does not change, but the relative values of potential and kinetic energy may be altered.

#### A SIMPLE EXAMPLE OF MECHANICAL ENERGY.

If one held a baseball at the top of a tall building, it would have a certain amount of potential energy. Once it was dropped, it would immediately begin losing potential energy and gaining kinetic energy proportional to the potential energy it lost. The relationship between the two forms, in fact, is inverse: as the value of one variable decreases, that of the other increases in exact proportion.

The ball cannot keep falling forever, losing potential energy and gaining kinetic energy. In fact, it can never gain an amount of kinetic energy greater than the potential energy it possessed in the first place. At the moment before it hits the ground, the ball's kinetic energy is equal to the potential energy it possessed at the top of the building. Correspondingly, its potential energy is zero—the same amount of kinetic energy it possessed before it was dropped.

Then, as the ball hits the ground, the energy is dispersed. Most of it goes into the ground, and depending on the rigidity of the ball and the ground, this energy may cause the ball to bounce. Some of the energy may appear in the form of sound, produced as the ball hits bottom, and some will manifest as heat. The total energy, however, will not be lost: it will simply have changed form.

#### REST ENERGY.

The values for mechanical energy in the above illustration would most likely be very small; on the other hand, the rest or mass energy of the baseball would be staggering. Given the weight of 0.333 pounds for a regulation baseball, which on Earth converts to 0.15 kg in mass, it would possess enough energy by virtue of its mass to provide a year's worth of electrical power to more than 150,000 American homes. This leads to two obvious questions: how can a mere baseball possess all that energy? And if it does, how can the energy be extracted and put to use?

The answer to the second question is, "By accelerating it to something close to the speed of light"—which is more than 27,000 times faster than the fastest speed ever achieved by humans. (The astronauts on *Apollo 10* in May 1969 reached nearly 25,000 MPH (40,000 km/h), which is more than 33 times the speed of sound but still insignificant when compared to the speed of light.) The answer to the first question lies in the most well-known physics formula of all time: *E* = *mc* ^{2}

In 1905, Albert Einstein (1879-1955) published his Special Theory of Relativity, which he followed a decade later with his General Theory of Relativity. These works introduced the world to the above-mentioned formula, which holds that energy is equal to mass multiplied by the squared speed of light. This formula gained its widespread prominence due to the many implications of Einstein's Relativity, which quite literally changed humanity's perspective on the universe. Most concrete among those implications was the atom bomb, made possible by the understanding of mass and energy achieved by Einstein.

In fact, *E* = *mc* ^{2} is the formula for rest energy, sometimes called mass energy. Though rest energy is "outside" of kinetic and potential energy in the sense that it is not defined by the above-described interactions within the larger system of
mechanical energy, its relation to the other forms can be easily shown. All three are defined in terms of mass. Potential energy is equal to *mgh,* where *m* is mass, *g* is gravity, and *h* is height. Kinetic energy is equal to ½ *mv* ^{2}, where *v* is velocity. In fact—using a series of steps that will not be demonstrated here—it is possible to directly relate the kinetic and rest energy formulae.

The kinetic energy formula describes the behavior of objects at speeds well below the speed of light, which is 186,000 mi (297,600 km) per second. But at speeds close to that of the speed of light, ½ *mv* ^{2} does not accurately reflect the energy possessed by the object. For instance, if *v* were equal to 0.999 *c* (where *c* represents the speed of light), then the application of the formula ½ *mv* ^{2} would yield a value equal to less than 3% of the object's real energy. In order to calculate the true energy of an object at 0.999 *c,* it would be necessary to apply a different formula for total energy, one that takes into account the fact that, at such a speed, mass itself becomes energy.

#### CONSERVATION OF MASS.

Mass itself is relative at speeds approaching *c,* and, in fact, becomes greater and greater the closer an object comes to the speed of light. This may seem strange in light of the fact that there is, after all, a law stating that mass is conserved. But mass is only conserved at speeds well below c: as an object approaches 186,000 mi (297,600 km) per second, the rules are altered.

The conservation of mass states that total mass is constant, and is unaffected by factors such as position, velocity, or temperature, in any system that does not exchange any matter with its environment. Yet, at speeds close to *c,* the mass of an object increases dramatically.

In such a situation, the mass would be equal to the rest, or starting mass, of the object divided by √(1 − (*v* ^{2}/*c* ^{2}), where *v* is the object's speed of relative motion. The denominator of this equation will always be less than one, and the greater the value of *v,* the smaller the value of the denominator. This means that at a speed of *c,* the denominator is zero—in other words, the object's mass is infinite! Obviously, this is not possible, and indeed, what the formula actually shows is that no object can travel faster than the speed of light.

Of particular interest to the present discussion, however, is the fact, established by relativity theory, that mass can be converted into energy. Hence, as noted earlier, a baseball or indeed any object can be converted into energy—and since the formula for rest energy requires that the mass be multiplied by *c* ^{2}, clearly, even an object of virtually negligible mass can generate a staggering amount of energy. This conversion of mass to energy happens well below the speed of light, in a very small way, when a stick of dynamite explodes. A portion of that stick becomes energy, and the fact that this portion is equal to just 6 parts out of 100 billion indicates the vast proportions of energy available from converted mass.

### Other Conservation Laws

In addition to the conservation of energy, as well as the limited conservation of mass, there are laws governing the conservation of momentum, both for an object in linear (straight-line) motion, and for one in angular (rotational) motion. Momentum is a property that a moving body possesses by virtue of its mass and velocity, which determines the amount of force and time required to stop it. Linear momentum is equal to mass multiplied by velocity, and the conservation of linear momentum law states that when the sum of the external force vectors acting on a physical system is equal to zero, the total linear momentum of the system remains unchanged, or conserved.

Angular momentum, or the momentum of an object in rotational motion, is equal to *mr* ^{2}ω, where *m* is mass, *r* is the radius of rotation, and ω (the Greek letter omega) stands for angular velocity. According to the conservation of angular momentum law, when the sum of the external torques acting on a physical system is equal to zero, the total angular momentum of the system remains unchanged. Torque is a force applied around an axis of rotation. When playing the old game of "spin the bottle," for instance, one is applying torque to the bottle and causing it to rotate.

#### ELECTRIC CHARGE.

The conservation of both linear and angular momentum are best explained in the context of real-life examples, provided below. Before going on to those examples, however, it is appropriate here to discuss a conservation law that is outside the realm of everyday experience: the conservation of electric charge, which holds that for an isolated system, the net electric charge is constant.

This law is "outside the realm of everyday experience" such that one cannot experience it through the senses, but at every moment, it is happening everywhere. Every atom has positively charged protons, negatively charged electrons, and uncharged neutrons. Most atoms are neutral, possessing equal numbers of protons and electrons; but, as a result of some disruption, an atom may have more protons than electrons, and thus, become positively charged. Conversely, it may end up with a net negative charge due to a greater number of electrons. But the protons or electrons it released or gained did not simply appear or disappear: they moved from one part of the system to another—that is, from one atom to another atom, or to several other atoms.

Throughout these changes, the charge of each proton and electron remains the same, and the net charge of the system is always the sum of its positive and negative charges. Thus, it is impossible for any electrical charge in the universe to be smaller than that of a proton or electron. Likewise, throughout the universe, there is always the same number of negative and positive electrical charges: just as energy changes form, the charges simply change position.

There are also conservation laws describing the behavior of subatomic particles, such as the positron and the neutrino. However, the most significant of the conservation laws are those involving energy (and mass, though with the limitations discussed above), linear momentum, angular momentum, and electrical charge.

## REAL-LIFE APPLICATIONS

### Conservation of Linear Momentum: Rifles and Rockets

#### FIRING A RIFLE.

The conservation of linear momentum is reflected in operations as simple as the recoil of a rifle when it is fired, and in those as complex as the propulsion of a rocket through space. In accordance with the conservation of momentum, the momentum of a system must be the same after it undergoes an operation as it was before the process began. Before firing, the momentum of a rifle and bullet is zero, and therefore, the rifle-bullet system must return to that same zero-level of momentum after it is fired. Thus, the momentum of the bullet must be matched—and "cancelled" within the system under study—by a corresponding backward momentum.

When a person shooting a gun pulls the trigger, it releases the bullet, which flies out of the barrel toward the target. The bullet has mass and velocity, and it clearly has momentum; but this is only half of the story. At the same time it is fired, the rifle produces a "kick," or sharp jolt, against the shoulder of the person who fired it. This backward kick, with a velocity in the opposite direction of the bullet's trajectory, has a momentum exactly the same as that of the bullet itself: hence, momentum is conserved.

But how can the rearward kick have the same momentum as that of the bullet? After all, the bullet can kill a person, whereas, if one holds the rifle correctly, the kick will not even cause any injury. The answer lies in several properties of linear momentum. First of all, as noted earlier, momentum is equal to mass multiplied by velocity; the actual proportions of mass and velocity, however, are not important as long as the backward momentum is the same as the forward momentum. The bullet is an object of relatively small mass and high velocity, whereas the rifle is much larger in mass, and hence, its rearward velocity is correspondingly small.

In addition, there is the element of impulse, or change in momentum. Impulse is the product of force multiplied by change or interval in time. Again, the proportions of force and time interval do not matter, as long as they are equal to the momentum change—that is, the difference in momentum that occurs when the rifle is fired. To avoid injury to one's shoulder, clearly force must be minimized, and for this to happen, time interval must be extended.

If one were to fire the rifle with the stock (the rear end of the rifle) held at some distance from one's shoulder, it would kick back and could very well produce a serious injury. This is because the force was delivered over a very short time interval—in other words, force was maximized and time interval minimized. However, if one holds the rifle stock firmly against one's shoulder, this slows down the delivery of the kick, thus maximizing time interval and minimizing force.

#### ROCKETING THROUGH SPACE.

Contrary to popular belief, rockets do not move by pushing against a surface such as a launchpad. If that were the case, then a rocket would have nothing to propel it once it had been launched, and certainly there would be no way for a rocket to move through the vacuum of outer space. Instead, what propels a rocket is the conservation of momentum.

Upon ignition, the rocket sends exhaust gases shooting downward at a high rate of velocity. The gases themselves have mass, and thus, they have momentum. To balance this downward momentum, the rocket moves upward—though, because its mass is greater than that of the gases it expels, it will not move at a velocity as high as that of the gases. Once again, the upward or forward momentum is exactly the same as the downward or backward momentum, and linear momentum is conserved.

Rather than needing something to push against, a rocket in fact performs best in outer space, where there is nothing—neither launch-pad nor even air—against which to push. Not only is "pushing" irrelevant to the operation of the rocket, but the rocket moves much more efficiently without the presence of air resistance. In the same way, on the relatively frictionless surface of an ice-skating rink, conservation of linear momentum (and hence, the process that makes possible the flight of a rocket through space) is easy to demonstrate.

If, while standing on the ice, one throws an object in one direction, one will be pushed in the opposite direction with a corresponding level of momentum. However, since a person's mass is presumably greater than that of the object thrown, the rearward velocity (and, therefore, distance) will be smaller.

Friction, as noted earlier, is not the only force that counters conservation of linear momentum on Earth: so too does gravity, and thus, once again, a rocket operates much better in space than it does when under the influence of Earth's gravitational field. If a bullet is fired at a bottle thrown into the air, the linear momentum of the spent bullet and the shattered pieces of glass in the infinitesimal moment just after the collision will be the same as that of the bullet and the bottle a moment before impact. An instant later, however, gravity will accelerate the bullet and the pieces downward, thus leading to a change in total momentum.

### Conservation of Angular Momentum: Skaters and Other Spinners

As noted earlier, angular momentum is equal to *mr* ^{2}ω, where *m* is mass, *r* is the radius of rotation, and ω stands for angular velocity. In fact, the first two quantities, *mr* ^{2}, are together known as moment of inertia. For an object in rotation, moment of inertia is the property whereby objects further from the axis of rotation move faster, and thus, contribute a greater share to the overall kinetic energy of the body.

One of the most oft-cited examples of angular momentum—and of its conservation—involves a skater or ballet dancer executing a spin. As the skater begins the spin, she has one leg planted on the ice, with the other stretched behind her. Likewise, her arms are outstretched, thus creating a large moment of inertia. But when she goes into the spin, she draws in her arms and leg, reducing the moment of inertia. In accordance with conservation of angular momentum, *mr* ^{2}ω will remain constant, and therefore, her angular velocity will increase, meaning that she will spin much faster.

#### CONSTANT ORIENTATION.

The motion of a spinning top and a Frisbee in flight also illustrate the conservation of angular momentum. Particularly interesting is the tendency of such an object to maintain a constant orientation. Thus, a top remains perfectly vertical while it spins, and only loses its orientation once friction from the floor dissipates its velocity and brings it to a stop. On a frictionless surface, however, it would remain spinning—and therefore upright—forever.

A Frisbee thrown without spin does not provide much entertainment; it will simply fall to the ground like any other object. But if it is tossed with the proper spin, delivered from the wrist, conservation of angular momentum will keep it in a horizontal position as it flies through the air. Once again, the Frisbee will eventually be brought to ground by the forces of air resistance and gravity, but a Frisbee hurled through empty space would keep spinning for eternity.

### WHERE TO LEARN MORE

Beiser, Arthur. *Physics,* 5th ed. Reading, MA: Addison-Wesley, 1991.

*"Conservation Laws: An Online Physics Textbook"* (Web site). <http://www.lightandmatter.com/area1book2.html> (March 12, 2001).

*"Conservation Laws: The Most Powerful Laws of Physics"* (Web site). <http://webug.physics.uiuc.edu/courses/phys150/fall 99/slides/lect07/> (March 12, 2001).

*"Conservation of Energy." NASA* (Web site). <http://www.grc.nasa.gov/WWW/K-12/airplane/thermo1f.html> (March 12, 2001).

Elkana, Yehuda. *The Discovery of the Conservation of Energy.* With a foreword by I. Bernard Cohen. Cambridge, MA: Harvard University Press, 1974.

*"Momentum and Its Conservation"* (Web site). <http://www.glenbrook.k12.il.us/gbssci/phys/Class/momentum/momtoc.html> (March 12, 2001).

Rutherford, F. James; Gerald Holton; and Fletcher G. Watson. *Project Physics.* New York: Holt, Rinehart, and Winston, 1981.

Suplee, Curt. *Everyday Science Explained.* Washington, D.C.: National Geographic Society, 1996.

## KEY TERMS

### CONSERVATION LAWS:

A set of principles describing physical properties that remain constant—that is, are conserved—throughout the various processes that occur in the physical world. The most significant of these laws concerns the conservation of energy (as well as, with qualifications, the conservation of mass); conservation of linear momentum; conservation of angular momentum; and conservation of electrical charge.

### CONSERVATION OF ANGULAR MOMENTUM:

A physical law stating that when the sum of the external torques acting on a physical system is equal to zero, the total angular momentum of the system remains unchanged. Angular momentum is the momentum of an object in rotational motion, and torque is a force applied around an axis of rotation.

### CONSERVATION OF ELECTRICALCHARGE:

A physical law which holds that for an isolated system, the net electrical charge is constant.

### CONSERVATION OF ENERGY:

A law of physics stating that within a system isolated from all other outside factors, the total amount of energy remains the same, though transformations of energy from one form to another take place.

### CONSERVATION OF LINEAR MOMENTUM:

A physical law stating that when the sum of the external force vectorsacting on a physical system is equal to zero, the total linear momentum of the system remains unchanged—or is conserved.

### CONSERVATION OF MASS:

A physical principle stating that total mass is constant, and is unaffected by factors such asposition, velocity, or temperature, in any system that does not exchange any matter with its environment. Unlike the other conservation laws, however, conservation of mass is not universally applicable, but applies only at speeds significant lower than that of light—186,000 mi (297,600 km) per second. Close to the speed of light, mass begins converting to energy.

### CONSERVE:

In physics, "to conserve" something means "to result in no net loss of" that particular component. It is possible that within a given system, the component may change form or position, but as long as the net value of the component remains the same, it has been conserved.

### FRICTION:

The force that resists motion when the surface of one object comes into contact with the surface of another.

### MOMENTUM:

A property that a moving body possesses by virtue of its mass and velocity, which determines the amount of force and time required to stop it.

### SYSTEM:

In physics, the term "system" usually refers to any set of physical interactions isolated from the rest of the universe. Anything outside of the system, including all factors and forces irrelevant to a discussion of that system, is known as the environment.

# Conservation Laws

# CONSERVATION LAWS

A conservation law is a statement of constancy in nature. One quantity (the conserved quantity) remains constant while other quantities may change. For example, in a collision of elementary particles, the total momentum is conserved (it is the same after the collision as before) while other quantities, such as the speeds and directions of the particles, and even the number of particles and their masses, may change.

The world of particles, like the larger-scale world around us, is a world of incessant change. Probing the small-scale world, one might not be surprised to find that *everything* changes, that nothing is constant. Yet scientists have, in fact, identified a limited number of conserved quantities. These quantities have special significance because constancy is an idea of such power and simplicity and because conserved quantities are related to symmetry principles in nature.

Conservation laws are not just keystones of theory, they are practical tools of analysis. They can be applied to processes whose complex details are beyond any capability of measurement or calculation. In a particle collision, conserved quantities such as energy, momentum, angular momentum, and electric charge are the same after the collision as before, even though incredible complexity, with countless interactions, may attend the process.

Conservation laws are tested not so much by measuring some quantity before and after an interaction to see if it is the same as by looking for, and *not* finding, evidence of a process, which, if it occurred, would violate the conservation law. For example, the decay of an electron into lighter neutral particles has never been observed. A process consistent with all known conservation laws except charge conservation is the decay of an electron into a neutrino of the electron type (called an electron neutrino) and a photon (or gamma ray), indicated by Energy, momentum, and angular momentum could all be conserved in this process, but not charge. The slash through the arrow means that the process does *not* occur—or, more accurately stated, has never been seen. We can say that the electron is stabilized by charge conservation. Since the electron is the lightest charged particle, its decay would necessarily violate the law of charge conservation. Experiment puts the lifetime of the electron at more than 4 × 10^{24} years. This means, roughly, that in the lifetime of the universe, no more than one out of a million billion electrons could have decayed. It is on this basis (together with a theoretical underpinning) that we call charge conservation an absolute conservation law.

Conservation laws apply to isolated systems, those for which external influences are absent or too small to be significant. Particle processes are almost always isolated in this sense. Gravity, electric fields, magnetic fields, and neighboring atoms have no appreciable effect during the brief moment of a particle collision. (There are certain examples of nuclear gamma decay within an atom in a crystal where neighboring atoms do have an effect.)

## Absolute and Partial Conservation Laws

Conservation laws may be absolute or partial. An absolute conservation law is one for which no confirmed violation has ever been seen and which is believed to be valid under all circumstances. By this definition, momentum, energy, angular momentum, and charge are absolutely conserved, as is the color charge of the strong interaction. (Although color charge is an attribute of strongly interacting particles only, its conservation can be considered universal because leptons and their associated bosons have zero color—they are "colorless.") Another absolutely conserved quantity is the combined symmetry called TCP, standing for time reversal, charge conjugation (or particle-antiparticle inversion), and parity (or left-right inversion). (The individual symmetries T, C, and P are only partially conserved.)

Baryon conservation occupies a special and ambiguous place. Baryons are a class of heavy particles (those made up of three quarks) that include the proton and the neutron. The law of baryon conservation, which states that the number of baryons minus the number of antibaryons never changes, is valid experimentally. Its most stringent test is the absence of proton decay. Since the proton is the lightest baryon, its decay would imply a change of baryon number, a violation of the law of baryon conservation. Experiment puts the lifetime of the proton at greater than 10^{33} years. This leaves room for only an incredibly tiny probability of decay (much less even than the limit on electron decay probability), yet theorists are reluctant to call baryon conservation an absolute law. The law has no known theoretical basis, and indeed there is theoretical reason to expect a tiny but nonzero probability of proton decay. Searches for it continue.

Like baryon conservation, lepton conservation is absolute so far as experiment is concerned. The law of lepton conservation states that the total number of leptons of all types (electron type, muon type, and tau type) minus the number of antileptons never changes. No violation of this law has ever been reported. Yet there is no known theoretical reason for it, and physicists expect that in the end it will prove to be partial, not absolute. For many years, the numbers of leptons of the three individual types appeared to be separately conserved (from which it followed, of course, that the total of all lepton types is conserved). Because of recent evidence for the "oscillation" of one type of neutrino into another type, the conservation laws of individual lepton types are now recognized to be partial, not absolute (although the transformation of one type of charged lepton into another has yet to be seen).

What is a partial conservation law? At first, it sounds like a contradiction in terms, like partial pregnancy. A conservation law is called partial if the quantity it governs is conserved when certain interactions are at work but not conserved for all interactions. Stated differently, a partial conservation law is one obeyed by one or more kinds of interaction and violated by at least one kind of interaction. For example, conservation of the number of quarks of a particular "flavor" (up, down, strange, charm, top, or bottom—in each case counting particles minus anti-particles) is obeyed by the strong and electromagnetic interactions but not by the weak interaction. The weak interaction can cause one type of quark to turn into another type.

The weak interaction is recognized as a "symmetry-breaker" that prevents several conservation laws from being absolute. Another symmetry-breaker is the "Higgs interaction," an interaction between every particle and the Higgs boson. The Higgs boson (named after one of its inventors, Scotland's Peter Higgs) is the quantum of a still-hypothetical field believed to permeate all space and to account, through its interactions, for particle masses. The Higgs interaction, along with the strong and electroweak interactions, is incorporated into the so-called Standard Model of particles. Whether gravity, which lies outside the Standard Model, is an additional symmetry-breaker is unknown. If it is, its effects in the particle world will surely be tiny and hard to detect. The present status of conservation laws is shown in Table 1.

**TABLE 1**

Status of Conservation Laws and Invariance Principles | |

Conserved or Invariant Quantity | Comment |

Energy | Believed to be absolutely conserved |

Momentum | Believed to be absolutely conserved |

Angular momentum (orbital spin) | Believed to be absolutely conserved |

Electric charge | Believed to be absolutely conserved |

Time inversion, or time reversal (T) | T and the combination CP are violated, presumably by the Higgs interaction |

Particle-antiparticle inversion, or charge conjugation (C) | Violated by the weak interaction |

Space inversion, or mirror inversion, called parity (P) | Violated by the weak interaction |

Combined inversions, TCP | Believed to be absolutely conserved Violated by the electromagnetic and Higgs interactions (masses depend on charge) |

Isospin (charge independence of interactions) | |

Color | Believed to be absolutely conserved. (Quarks and gluons have color. Leptons, photons, and W and Z bosons are colorless.) |

Individual quark flavors, i.e., upness, downness, charm, strangeness, topness, bottomness | Violated by the electroweak interaction and ultimately the Higgs interaction |

Baryon conservation (combined number of all quark types) | Experiment consistent with absolute conservation, but theorists predict a very weak violation that would be evidenced by proton decay |

Electron-family number, Muon-family number, Tau-family number | Observed violation for neutrinos and predicted minute violation for charged leptons via the Higgs mechanism |

Combined lepton number (e-family + mu-family + tau-family) | Experiment consistent with absolute conservation, but violation by the Higgs interaction is predicted |

Note: Except for the lepton-number rules, which are not relevant to this generalization, all of the conservation laws and invariance principles enumerated above are believed to be valid principles for the strong interactions. Violations occur through the electroweak interaction and/or the Higgs interaction. Gravitys role, if any, is unknown. | |

credit: Courtesy of Kenneth W. Ford. |

## Conservation Laws and Feynman Diagrams

Quantum theory replaces smooth change with explosive change. Every interaction is believed to be driven ultimately by the creation and annihilation of particles. Simple space-time diagrams, or Feynman diagrams (Figure 1), illustrate this idea and show how conservation laws work at the most fundamental level.

For all interaction types, Feynman diagrams have the same "three-prong" structure. At a space-time "vertex," one fermion world line ends, another fermion world line begins, and a boson world line begins or ends. No particle survives an interaction. (If an electron enters an interaction event and an electron leaves it, theory treats them as different particles, one being annihilated and one created.) What do survive an interaction are conserved quantities— whichever ones are preserved by the interaction in question (with a subtlety involving energy and momentum, to be discussed below). So conservation

**FIGURE 1**

laws apply not just to the before and after of a process in the laboratory. They are believed to apply to the before and after of every separate interaction. From that base, they reach out to all that happens, at whatever scale of size.

It turns out that each of the simple Feynman diagrams shown in Figure 1 represents only a "virtual," not a real, process because energy and momentum cannot be simultaneously conserved across the vertex. So real processes that conserve all relevant quantities involve a "daisy chain" of at least two such interaction vertices (and perhaps millions of them). Illustrating this point is a diagram for a real process, the decay of a negative muon, shown in Figure 2. The time sequence is from the bottom to the top of the diagram. In this process, muon family number, electron family number, and electric charge are conserved.

## Examples of Absolute Conservation Laws

Momentum is a vector (directed) quantity. Following a particle collision, the vector sum of the momenta has the same direction and the same magnitude as before the collision. Nonrelativistically, the momentum of an object is its mass times its velocity (**p** = *m***v** ). Relativistically, the definition is different but the conservation law remains valid. Momentum conservation is related to the homogeneity of space (that the laws of physics are the same at every point).

**FIGURE 2**

Energy is a scalar (numerical) quantity that takes many forms. Its relevant forms in the particle world are kinetic energy (energy of motion) and mass. The energy locked up in mass is *E = mc*^{2}, where *c* is the speed of light. Initial kinetic energy is required if new mass is to be created. If mass decreases, as in a decay process, kinetic energy increases. A simple implication of energy conservation is that all spontaneous decay processes must be "downhill" in mass—that is, the total mass of the products must be less than the mass of the initial particle. Energy and momentum conservation taken together prevent one particle from decaying into another single particle. If the initial particle is at rest, the final particle would have to be at rest, too, to preserve zero momentum, but then energy would not be conserved. And if the final particle moves in such a way as to conserve energy, momentum is not conserved. So every decay process results in at least two particles. Energy conservation is related to the homogeneity of time (nature's laws are the same at one time as another).

Angular momentum, like momentum, is a vector quantity. It measures the strength of rotational motion and is directed along the axis of rotation. Particles may have *spin* angular momentum and *orbital* angular momentum, roughly analogous to the spin of the Earth about its axis and the orbiting of the Earth around the Sun. Remarkably, a particle may possess spin even if it has no spatial extent. In units of Planck's constant *h* divided by 2π, written ħ, fermions have half-odd-integer spin (1/2, 3/2, etc.) and bosons have integer spin (0, 1, 2, etc.). The spins of the fundamental particles are as follows: quarks and leptons, ½ gluons, photon, and the *W* and *Z* bosons, 1; the Higgs boson, 0. Orbital angular momentum is always integral (0, 1, 2, etc.). One consequence of angular momentum conservation is that if the number of fermions before a reaction is even, the number afterward must also be even, and if the number before is odd, the number after must be odd (Figure 2, for example, shows a process where one fermion decays into three fermions.) Angular momentum is related to the isotropy of space (nature's laws don't depend on direction).

The law of charge conservation goes back to Benjamin Franklin in the eighteenth century. It has withstood the test of time. It is related now to the masslessness of the photon through a principle called gauge invariance. In units of the proton charge *e,* leptons have charges 0 and -1 (antileptons 0 and +1); quarks have charges +2/3;; and -1/3 (anti quarks -2/3 and +1/3); gluons, the photon, the *Z* boson, and the Higgs boson have charge 0; and the *W* boson has charge -1 (its antiparticle +1). As noted above, the most salutary effect of charge conservation is to prevent the decay of the electron.

The conservation of "color" (or "color charge") is much like the conservation of electric charge. Both color charge and electric charge are quantized properties of particles that are preserved in every interaction, even when no particle survives the interaction. (Think of runners in a relay race as particles and the baton they carry as color charge or electric charge. When one runner stops and another starts, the baton continues on.) Quarks may have any one of three "colors," arbitrarily called red, green, and blue, so there are really eighteen different quarks, not six. There are no "colorless" quarks. Anti-quarks are said to have anticolor. Gluons carry a color-anticolor mixture such as red-antiblue, of which there are eight independent combinations. Figure 1a shows an example of a color-conserving quark-gluon interaction.

Lepton conservation means that the number of leptons minus the number of antileptons is preserved in every interaction. Figure 2 illustrates the principle with a decay process in which one lepton turns into two leptons and an antilepton. Another example is the decay of the neutron into a proton, an electron, and an antineutrino, The lepton number is zero before and after. This process illustrates several other conservation laws as well: charge conservation (0 to 0), baryon conservation (1 to 1), energy conservation ("downhill" in mass), and angular momentum conservation (odd number of fermions to an odd number of fermions). This process and the one in Figure 2 also show conservation of the individual lepton families (electron family and muon family), for which there are no known exceptions involving charged leptons.

## Examples of Partial Conservation Laws

The concept of isospin was introduced in the 1930s to describe the similarities of the proton and neutron—they have nearly the same mass and appeared to have the same strong interaction. The proton and neutron came to be regarded as two states of a single underlying particle, the nucleon, to which the mathematics of a spin-½ particle with its two orientations of spin could be applied (thus the name isospin, which otherwise has nothing to do with spin). Later other particle "multiplets" were found, such as the pion triplet and the xi-particle doublet (and some singlets such as the lambda particle). The law of isospin conservation states that the strong interaction is identically the same for the members of each multiplet. Isospin conservation is clearly violated by the electromagnetic interaction because particles within a given multiplet don't have the same charge (and also differ slightly in mass).

Isospin conservation is now recognized to be a consequence of "flavor invariance." The six quark flavors are called up, down, charm, strange, top, and bottom; flavor invariance means that the strong interactions do not distinguish among flavors. Thus replacing an up quark by a down quark (which could mean changing a proton to a neutron), or even replacing a strange quark by a top quark, does not change any property of the strong interaction. The electroweak interaction does, however, depend on flavor, or quark type—that is, it violates the law of flavor invariance. This is attributed, ultimately, to effects of the Higgs boson.

The strong interaction goes further and conserves each quark flavor separately. If the conservation of quark flavor were an absolute law, there would be no transformations among different quark types. An observed decay process that violates the law of flavor conservation is the decay of a lambda particle into a proton and a negative pion, Using *u* for up, *d* for down, and *s* for strange, this process is represented in terms of its quark constituents by Balancing the books here requires that a strange quark be transformed into a down quark. This is a process forbidden by the strong interaction but allowed by the weak interaction. The mean life of the lambda is many orders of magnitude greater than if the process occurred through the strong interaction, confirming that the flavor violation is provided by the weak interaction.

According to the reductionist view that dominates modern science, conservation laws in the large-scale world result from the action of such laws in the particle world. There is extensive evidence that this is the case. But how fascinating it will be if future discovery reveals large-scale regularity not attributable to small-scale laws.

*See also:*Energy; Momentum; Noether, Emmy; Symmetry Principles

## Bibliography

Bernstein, J. *The Tenth Dimension* (McGraw-Hill, New York, 1989).

Bernstein, J.; Fishbane, P.; and Gasiorowicz, S. *Modern Physics* (Prentice Hall, Upper Saddle River, NJ, 2000).

Feynman, R. *The Character of Physical Law* (MIT Press, Cambridge, MA, 1967).

Pagels, H. *The Cosmic Code: Quantum Physics as the Language of Nature* (Simon & Schuster, New York, 1982).

*Kenneth W. Ford*

# Conservation Laws

# Conservation Laws

Conservation of linear momentum

Conservation of angular momentum

Conservation of energy and mass

Conservation of electric charge

Conservation laws refer to those laws of physics which describe quantities that remain constant in nature. If these physical quantities are carefully measured, and if all known sources are taken into account, they will always remain unchanged. The validity of conservation laws is tested through experiments. The conservation laws include the conservation of linear momentum, the conservation of angular momentum, the conservation of energy and mass, and the conservation of electric charge. In addition, there are many conservation laws that deal with subatomic particles, that is, particles smaller than the atom.

## Conservation of linear momentum

A rocket ship taking off, the recoil of a rifle, and a bank-shot in a pool game are examples that demonstrate the conservation of linear momentum. Linear momentum is defined as the product of an object’s mass and its velocity. For example, the linear momentum of a 220 lb (100 kg) football-linebacker traveling at a speed of 10 mph(16 km/h) is exactly the same as the momentum of a 110 lb (50 kg) sprinter traveling at 20 mph (32 km/h). Since the velocity is both the speed and direction of an object, the linear momentum is also specified by a certain direction.

The linear momentum of one or more objects is conserved when there are no external forces acting on those objects. For example, consider a rocket ship in deep outer space, where the force of gravity is negligible. Linear momentum will be conserved, since the external force of gravity is absent. If the rocket ship is initially at rest, its momentum is zero, since its speed is zero (Figure 1a). If the rocket engines are suddenly fired, the rocket ship will be propelled forward (Figure 1b). For linear momentum to be conserved, the final momentum must be equal to the initial momentum, which is zero. Linear momentum is conserved if one takes into account the momentum both of the rocket and of the gasses ejected out the back. The positive momentum of the rocket ship going forward is equal to the negative momentum of the fuel going backward. (Note that the direction of motion is used to define positive and negative.) Adding these two quantities yields zero. It is important to realize that the rocket’s propulsion is not achieved by the fuel pushing on anything. In outer space there is nothing to push on. Propulsion is achieved by the conservation of linear momentum. An easy way to demonstrate this type of propulsion is by propelling yourself on a frozen pond. Since there is little friction between your ice skates and

the ice, linear momentum is conserved. Throwing an object in one direction will cause you to travel in the opposite direction.

Even in cases where the external forces are significant, the concept of conservation of linear momentum can be applied to a limited extent. An instance would be the momentum of objects that are affected by the external force of gravity. For example, a bullet is fired at a clay pigeon that has been launched into the air. The linear momentum of the bullet and clay pigeon at the instant just before impact is equal to the linear momentum of the bullet and hundreds of shattered clay pieces at the instant just after impact. Linear momentum is conserved just before, during, and just after the collision (Figure 2). This is true because the external force of gravity does not significantly affect the momentum of the objects within this narrow time period. Many seconds later, however, gravity will have had a significant influence, and the total momentum of the objects will not be the same as just before the collision.

There are many illustrations of the conservation of linear momentum. Begin to walk forward in a rowboat and you will notice that the boat begins to travel backward relative to the water: the momentum your legs impart to your body is equal and opposite to the momentum they impart to the boat. When a rifle is fired, the recoil one feels against one’s shoulder is due to the momentum of the rifle, which is equal but in the opposite direction to the momentum of the bullet. Again, since the rifle is so much heavier than the bullet, its velocity is correspondingly less. Conservation of

linear (and angular) momentum is used to give space probes an extra boost when they pass planets. The momentum of the planet as it circles the sun in its orbit is given to the passing space probe, increasing its velocity on its way to the next planet. In all of the experiments ever attempted, there has been never been a violation of the law of conservation of linear momentum.

## Conservation of angular momentum

Just as there is the conservation of motion for objects traveling in straight lines, there is also a conservation of motion for objects traveling along curved paths. This conservation of rotational motion is known as the conservation of angular momentum. An object that is traveling at a constant speed in a circle (compare this to a race car on a circular track) is shown in Figure 3. The angular momentum for this object is defined as the product of the object’s mass, its velocity, and the radius of the circle. For example, a 2, 200-lb (1, 000 kg) car traveling at 30 mph (50 km/h) on a 2–mi-radius (3 km) track, a 4, 400-lb (2, 000-kg) truck traveling at 30 mph on a 1–mi-radius (1.6 km) track, and a 2, 200-lb car traveling at 60 mph (97 km/h) on a 1–mi-radius track will all have the same value of angular momentum. In addition, objects that are spinning, such as a top or an ice skater, have angular momentum that is defined by their mass, their shape, and the velocity at which they spin.

In the absence of external forces that tend to change an object’s rotation, the angular momentum will be conserved. Close to Earth, gravity is uniform and will not tend to alter an object’s rotation. Consequently, many instances of angular momentum conservation can be seen every day. When an ice skater goes from a slow spin, with her arms stretched out, into a fast spin, with her arms at her sides, we are witnessing the conservation of angular momentum. With arms stretched, the radius of the rotation circle is large and the rotation speed is small. With arms at her side, the radius of the rotation circle is now small and the speed must increase to keep the angular momentum constant.

An additional consequence of the conservation of angular momentum is that the rotation axis of a spinning object will tend to keep a constant orientation. For example, a spinning Frisbee thrown horizontally will tend to keep its horizontal orientation even if tapped from below. To test this, try throwing a Frisbee without spin and see how unstable it is. A spinning top remains vertical as long as it keeps spinning fast enough. Earth itself maintains a constant orientation of its spin axis due to the conservation of angular momentum.

As is the case for linear momentum, there has never been a violation of the law of conservation of angular momentum. This applies to all objects, large and small. In accordance with the Bohr model of subatomic particles, the electrons that surround the nucleus of the atom are found to possess angular momentum of only certain discrete values. Intermediate values are not found. Even with these constraints, the angular momentum is always conserved.

## Conservation of energy and mass

Energy is a state function that can be described in many forms. One form of energy is kinetic energy, which is the energy of motion. A moving object has

kinetic energy (as well as other forms of energy) because it is moving. However, many non-moving objects contain energy in the form of potential or stored energy. A boulder on the top of a cliff has potential energy. This implies that the boulder could convert this potential energy into kinetic energy if it were to fall off the cliff. A stretched bow and arrow have potential energy also. This implies that the stored energy in the bow could be converted into the kinetic energy of the arrow after it is released. Stored energy may be more complicated than these mechanical examples, however, as in the stored electrical energy in a car battery. We know that the battery has stored energy because this energy can be converted into the kinetic energy of a cranking engine. There is stored chemical energy in many substances, for example gasoline. Again we know this because the energy of the gasoline can be converted into the kinetic energy of a car moving down the road. This stored chemical energy could alternately be converted into thermal energy by burning the gasoline and using the heat to increase the temperature of water in a bath, for instance. In all these instances, energy can be converted from one form to another, but the total energy remains constant.

In certain instances, even mass can be converted into energy. For example, in a nuclear reactor, the nucleus of the uranium atom is split into fragments. The sum of the masses of the fragments is always less than the original uranium nucleus. What happened to this original mass? This mass has been converted into thermal energy, which heats the water to drive steam turbines, which ultimately produce electrical energy. As first discovered by Albert Einstein (1879–1955), there is a precise relationship defining the amount of energy that is equivalent to a certain amount of mass. In instances in which mass is converted into energy, or visa versa, this relationship must be taken into account.

In general, therefore, there is a universal law of conservation of energy and mass that applies to all of nature. The sum of all the forms of energy and mass in the universe is a certain amount, which remains constant. As is the case for angular momentum, the energies of the electrons that surround the nucleus of the atom can possess only certain discrete values. And again, even with these constraints, the conservation of energy and mass is always obeyed.

## Conservation of electric charge

Electric charge is the property of matter that makes you experience a spark when you touch a metal doorknob after shuffling your feet across a rug. It is also the property that produces lightning and is the basis of all electrical machines and all electrical phenomena in nature, including the behaviors of molecules. Electric charge comes in two varieties, positive and negative. Like charges repel, that is, they tend to push one another apart, and unlike charges attract, that is, they tend to pull one another together. Therefore, two negative charges repel one another and, likewise, two positive charges repel one another. On the other hand, a positive charge will attract a negative charge. The net electric charge on an object is found by adding all the negative charge to all the positive charge residing on the object. Therefore, the net electric charge on an object with an equal amount of positive and negative charge is exactly zero. The more net electric charge an object has, the greater will be the force of attraction or repulsion for another object containing a net electric charge.

Electric charge is a property of the particles that make up an atom. The electrons that surround the nucleus of the atom have a negative electric charge. The protons, which partly make up the nucleus, have a positive electric charge. The neutrons, which also make up the nucleus, have no electric charge. The negative charge of the electron is exactly equal and opposite to the positive charge of the proton. For example, two electrons separated by a certain distance will repel one another with the same force as two protons separated by the same distance and, likewise, a proton and electron separated by this same distance will attract one another with the same force.

Electric charge is only available in discrete units. These discrete units are exactly equal to the amount of electric charge that is found on the electron or the proton (which have equal and opposite charges). It is impossible to find a naturally occurring amount of electric charge that is smaller than what is found on the proton or the electron. All objects contain an amount of electric charge, which is made up of a combination of these discrete units. An analogy can be made to the winnings and losses in a penny ante game of poker. If you are ahead, you have a greater amount of winnings (positive charges) than losses (negative charges); if you are in the hole, you have a greater amount of losses than winnings. Note that the amount that you are ahead or in the hole can only be an exact amount of pennies or cents, as in 49 cents up or 78 cents down. You cannot be ahead by 32 and 1/4 cents. This is the analogy to electric charge. You can only be positive or negative by a discrete amount of charge.

If one were to add all the positive and negative electric units of charge in the universe together, one would arrive at a number that never changes. This would be analogous to remaining always with the same amount of money in poker. If you go down by five cents in a given hand, you have to simultaneously go up by five cents in the same hand. This is the statement of the law of conservation of electric charge. If a positive charge turns up in one place, a negative charge must turn up in the same place so that the net electric charge of the universe never changes. There are many other subatomic particles besides protons and electrons that have discrete units of electric charge. Even in interactions involving these particles, the law of conservation of electric charge is always obeyed.

## Other conservation laws

In addition to the conservation laws already described, there are conservation laws that describe reactions between subatomic particles. Several hundred subatomic particles have been discovered since the discovery of the proton, electron, and the neutron. By observing which processes and reactions occur between these particles, physicists can determine new conservation laws governing these processes. For example, there exists a subatomic particle called the positron, which is very much like the electron except that it carries a positive electric charge. The law of conservation of charge would allow a process whereby a proton could change into a positron. However, the fact that this process does not occur leads physicists to define a new conservation law restricting the allowable transformations between different types of subatomic particles.

Occasionally, a conservation law can be used to predict the existence of new particles. In the 1920s, it was discovered that a neutron could change into a proton and an electron. However, the energy and mass before the reaction was not equal to the energy and mass after the reaction. Although seemingly a violation of energy and mass conservation, it was instead proposed that the missing energy was carried by a new particle, unheard of at the time. In 1956, this new particle, named the neutrino, was discovered. As new subatomic particles are discovered and more processes are studied, the conservation laws will be an important asset to our understanding of the universe.

## Resources

### BOOKS

Feynman, Richard. *The Character of Physical Law.* Cambridge, MA: MIT Press, 1965.

Petrova, Liudmila. *Evolutionary Differential Forms: Conservation Laws and Causality.* New York: Springer, 2006.

Villard, Ray, ed. *Changes Within Physical Systems and/or Conservation of Energy and Momentum: An Anthology of Current Thought.* New York: Rosen, 2005.

### OTHER

Crowell, Benjamin. “Conservation Laws.” *LightandMatter.com.* <http://www.lightandmatter.com/area1book2.html> (accessed October 23, 2006).

Kurt Vandervoort

# Conservation Laws

# Conservation laws

Conservation laws refer to physical quantities that remain constant throughout the multitude of processes which occur in nature. If these physical quantities are carefully measured, and if all known sources are taken into account, they will always yield the same result. The validity of the conservation laws is tested through experiments. However, many of the conservation laws are suggested from theoretical considerations. The conservation laws include: the **conservation** of linear **momentum** , the conservation of angular momentum, the conservation
of **energy** and **mass** , and the conservation of **electric charge** . In addition, there are many conservation laws that deal with **subatomic particles** , that is, particles that are smaller than the atom.

## Conservation of linear momentum

A rocket ship taking off, the recoil of a rifle, and a bank-shot in a pool are examples which demonstrate the conservation of linear momentum. Linear momentum is defined as the product of an object's mass and its **velocity** . For example, the linear momentum of a 220 lb (100 kg) football-linebacker traveling at a speed of 10 MPH (16 km/h) is exactly the same as the momentum of a 110 lb (50 kg) sprinter traveling at 20 MPH (32 km/h). Since the velocity is both the speed and direction of an object, the linear momentum is also specified by a certain direction.

The linear momentum of one or more objects is conserved when there are no external forces acting on those objects. For example, consider a rocket-ship in deep outer **space** where the **force** of gravity is negligible. Linear momentum will be conserved since the external force of gravity is absent. If the rocket-ship is initially at rest, its momentum is **zero** since its speed is zero (Figure 1a). If the rocket engines are suddenly fired, the rocket-ship will be propelled forward (Figure 1b). For linear momentum to be conserved, the final momentum must be equal to the initial momentum, which is zero. Linear momentum is conserved if one takes into account the burnt fuel that is ejected out the back of the rocket. The positive momentum of the rocket-ship going forward is equal to the **negative** momentum of the fuel going backward (note that the direction of **motion** is used to define positive and negative). Adding these two quantities yields
zero. It is important to realize that the rocket's propulsion is not achieved by the fuel pushing on anything. In outer space there is nothing to push on! Propulsion is achieved by the conservation of linear momentum. An easy way to demonstrate this type of propulsion is by propelling yourself on a frozen pond. Since there is little **friction** between your ice skates and the **ice** , linear momentum is conserved. Throwing an object in one direction will cause you to travel in the opposite direction.

Even in cases where the external forces are significant, the concept of conservation of linear momentum can be applied to a limited extent. An instance would be the momentum of objects that are affected by the external force of gravity. For example, a bullet is fired at a clay pigeon that has been launched into the air. The linear momentum of the bullet and clay pigeon at the instant just before impact is equal to the linear momentum of the bullet and hundreds of shattered clay pieces at the instant just after impact. Linear momentum is conserved just before, during, and just after the collision (Figure 2). This is true is because the external force of gravity does not significantly affect the momentum of the objects within this narrow **time** period. Many seconds later, however, gravity will have had a significant influence and the total momentum of the objects will not be the same as just before the collision.

There are many examples that illustrate the conservation of linear momentum. When we walk down the road, our momentum traveling forward is equal to the momentum of the **earth** traveling backward. Of course, the mass of Earth is so large compared to us that its velocity will be negligible. (A simple calculation using the 220 lb [100 kg] linebacker shows that as he travels forward at 10 mph [16 kph]. Earth travels backward at a speed of 9 trillionths of an inch per century!) A better illustration is to walk forward in a row-boat and you will notice that the boat travels backward relative to the **water** . When a rifle is fired, the recoil you feel against your shoulder is due to the momentum of the rifle which is equal but in the opposite direction to the momentum of the bullet. Again, since the rifle is so much heavier than the bullet, its velocity will be correspondingly less than the bullet's. Conservation of linear momentum is the chief reason that heavier cars are safer than lighter cars. In a head-on collision with two cars traveling at the same speed, the motion of the two cars after the collision will be along the original direction of the larger car due to its larger momentum. Conservation of linear momentum is used to give space probes an extra boost when they pass planets. The momentum of the **planet** as it circles the **sun** in its **orbit** is given to the passing **space probe** , increasing its velocity on its way to the next planet. In all of the experiments ever attempted, there has been never been a violation of the law of conservation of linear momentum. This applies to all objects ranging in size from galaxies to subatomic particles.

## Conservation of angular momentum

Just as there is the conservation of motion for objects traveling in straight lines, there is also a conservation of motion for objects traveling along curved paths. This conservation of rotational motion is known as the conservation of angular momentum. An object which is traveling at a constant speed in a **circle** (compare this to a race car on a circular track) is shown in Figure 3. The angular momentum for this object is defined as the product of the object's mass, its velocity, and the radius of the circle. For example, a 2,200 lb (1000 kg) car traveling at 30 MPH (50 km/h) on a 2–mi-radius (3-km) track, a 4,400 lb (2000 kg) truck traveling at 30 MPH (50 km/h) on a 1–mi-radius (1.6 km) track, and a 2,22200 lb (1000 kg) car traveling at 60 MPH (97 km/h) on a 1–mi-radius (1.6-km) track will all have the same value of angular momentum. In addition, objects which are spinning, such as a top or an ice skater, have angular momentum which is defined by their mass, their shape, and the velocity at which they spin.

In the absence of external forces that tend to change an object's **rotation** , the angular momentum will be conserved. Close to Earth, gravity is uniform and will not tend to alter an object's rotation. Consequently, many instances of angular momentum conservation can be seen every day. When an ice skater goes from a slow spin with her arms stretched into a fast spin with her arms at her sides, we are witnessing the conservation of angular momentum. With arms stretched, the radius of the rotation circle is large and the rotation speed is small. With arms at her side, the radius of the rotation circle is now small and the speed must increase to keep the angular momentum constant.

An additional consequence of the conservation of angular momentum is that the rotation axis of a spinning object will tend to keep a constant orientation. For example, a spinning Frisbee thrown horizontally will tend to keep its horizontal orientation even if tapped from below. To test this, try throwing a Frisbee without spin and see how unstable it is. A spinning top remains vertical as long as it keeps spinning fast enough. Earth itself maintains a constant orientation of its spin axis due to the conservation of angular momentum.

As is the case for linear momentum, there has never been a violation of the law of conservation of angular momentum. This applies to all objects, large and small. In accordance with the **Bohr model** of subatomic particles, the electrons that surround the nucleus of the atom are found to possess angular momentum of only certain discrete values. Intermediate values are not found. Even with these constraints, the angular momentum is always conserved.

## Conservation of energy and mass

Energy is a state function that can be described in many forms. The most basic form of energy is kinetic energy, which is the energy of motion. A moving object has energy solely due to the fact that it is moving. However, many non-moving objects contain energy in the form of potential or stored energy. A boulder on the top of a cliff has potential energy. This implies that the boulder could convert this potential energy into kinetic energy if it were to fall off the cliff. A stretched bow and arrow have potential energy also. This implies that the stored energy in the bow could be converted into the kinetic energy of the arrow, after it is released. Stored energy may be more
complicated than these mechanical examples, however, as in the stored electrical energy in a car **battery** . We know that the battery has stored energy because this energy can be converted into the kinetic energy of a cranking engine. There is stored chemical energy in many substances, for example gasoline. Again we know this because the energy of the gasoline can be converted into the kinetic energy of a car moving down the road. This stored chemical energy could alternately be converted into thermal energy by burning the gasoline and using the **heat** to increase the **temperature** of a bath of water. In all these instances, energy can be converted from one form to another, but it is always found that the total energy remains constant.

In certain instances, even mass can be converted into energy. For example, in a **nuclear reactor** the nucleus of the **uranium** atom is split into fragments. The sum of the masses of the fragments is always less than the original uranium nucleus. What happened to this original mass? This mass has been converted into thermal energy which heats the water to drive steam turbines which ultimately produces electrical energy. As first discovered by Albert Einstein (1879-1955), there is a precise relationship defining the amount of energy that is equivalent to a certain amount of mass. In instances where mass is converted into energy, or visa versa, this relationship must be taken into account.

In general, therefore, there is a universal law of conservation of energy and mass that applies to all of nature. The sum of all the forms of energy and mass in the universe is a certain amount which remains constant. As is the case for angular momentum, the energies of the electrons that surround the nucleus of the atom can possess only certain discrete values. And again, even with these constraints, the conservation of energy and mass is always obeyed.

## Conservation of electric charge

Electric charge is the property that makes you experience a spark when you touch a **metal** door knob after shuffling your feet across a rug. It is also the property that produces **lightning** . Electric charge comes in two varieties, positive and negative. Like charges repel, that is, they tend to push one another apart, and unlike charges attract, that is, they tend to pull one another together. Therefore, two negative charges repel one another and, likewise, two positive charges repel one another. On the other hand, a positive charge will attract a negative charge. The net electric charge on an object is found by adding all the negative charge to all the positive charge residing on the object. Therefore, the net electric charge on an object with an equal amount of positive and negative charge is exactly zero. The more net electric charge an object has, the greater will be the force of attraction or repulsion for another object containing a net electric charge.

Electric charge is a property of the particles that make up an atom. The electrons that surround the nucleus of the atom have a negative electric charge. The protons which partly make up the nucleus have a positive electric charge. The neutrons which also make up the nucleus have no electric charge. The negative charge of the **electron** is exactly equal and opposite to the positive charge of the **proton** . For example, two electrons separated by a certain **distance** will repel one another with the same force as two protons separated by the same distance and, likewise, a proton and electron separated by this same distance will attract one another with the same force.

The amount of electric charge is only available in discrete units. These discrete units are exactly equal to the amount of electric charge that is found on the electron or the proton. It is impossible to find a naturally occurring amount of electric charge that is smaller than what is found on the proton or the electron. All objects contain an amount of electric charge which is made up of a combination of these discrete units. An analogy can be made to the winnings and losses in a penny ante game of poker. If you are ahead, you have a greater amount of winnings (positive charges) than losses (negative charges), and if you are in the hole you have a greater amount of losses than winnings. Note that the amount that you are ahead or in the hole can only be an exact amount of pennies or cents, as in 49 cents up or 78 cents down. You cannot be ahead by 32 and 1/4 cents. This is the analogy to electric charge. You can only be positive or negative by a discrete amount of charge.

If one were to add all the positive and negative electric units of charge in the universe together, one would arrive at a number that never changes. This would be analogous to remaining always with the same amount of money in poker. If you go down by five cents in a given hand, you have to simultaneously go up by five cents in the same hand. This is the statement of the law of conservation of electric charge. If a positive charge turns up in one place, a negative charge must turn up in the same place so that the net electric charge of the universe never changes. There are many other subatomic particles besides protons and electrons which have discrete units of electric charge. Even in interactions involving these particles, the law of conservation of electric charge is always obeyed.

## Other conservation laws

In addition to the conservation laws already described, there are conservation laws that describe reactions between subatomic particles. Several hundred sub-atomic particles have been discovered since the discovery of the proton, electron, and the **neutron** . By observing which processes and reactions occur between these particles, physicists can determine new conservation laws governing these processes. For example, there exists a subatomic particle called the positron which is very much like the electron except that it carries a positive electric charge. The law of conservation of charge would allow a process whereby a proton could change into a positron. However, the fact that this process does not occur leads physicists to define a new conservation law restricting the allowable transformations between different types of subatomic particles.

Occasionally, a conservation law can be used to predict the existence of new particles. In the 1920s, it was discovered that a neutron could change into a proton and an electron. However, the energy and mass before the reaction was not equal to the energy and mass after the reaction. Although seemingly a violation of energy and mass conservation, it was instead proposed that the missing energy was carried by a new particle, unheard of at the time. In 1956, this new particle named the **neutrino** was discovered. As new subatomic particles are discovered and more processes are studied, the conservation laws will be an important asset to our understanding of the Universe.

Feynman, Richard. *The Character of Physical Law.* Cambridge, MA: MIT Press, 1965.

Feynman, Richard. *Six Easy Pieces.* Reading, MA: Addison-Wesley, 1995.

Giancoli, Douglas. *Physics.* Englewood Cliffs, NJ: Prentice Hall, 1995.

Schwarz, Cindy. *A Tour of the Subatomic Zoo.* New York: American Institute of Physics, 1992.

Young, Hugh. *University Physics.* Reading, MA: Addison-Wesley, 1992.

Kurt Vandervoort

# Conservation Laws

# Conservation laws

Conservation laws are scientific statements that describe the amount of some quantity before and after a physical or chemical change.

## Conservation of mass and energy

One of the first conservation laws to be discovered was the conservation of mass (or matter). Suppose that you combine a very accurately weighed amount of iron and sulfur with each other. The product of that reaction is a compound known as iron(II) sulfide. If you also weigh very accurately the amount of iron(II) sulfide formed in that reaction, you will discover a simple relationship: The weight of the beginning materials (iron plus sulfur) is exactly equal to the weight of the product or products of the reaction (iron(II) sulfide). This statement is one way to express the law of conservation of mass. A more formal definition of the law is that mass (or matter) cannot be created or destroyed in a chemical reaction.

A similar law exists for energy. When you turn on an electric heater, electrical energy is converted to heat energy. If you measure the amount of electricity supplied to the heater and the amount of heat produced by the heater, you will find the amounts are equal. In other words, energy is conserved in the heater. It may take various forms, such as electrical energy, heat, magnetism, or kinetic energy (the energy of an object due to its motion), but the relationship is always the same: The amount of energy used to initiate a change is the same as the amount of energy detected at the end of the change. In other words, energy cannot be created or destroyed in a physical or chemical change. This statement summarizes the law of conservation of energy.

## Words to Know

**Angular momentum:** For objects in rotational (or spinning) motion, the product of the object's mass, its speed, and its distance from the axis of rotation.

**Conserved quantities:** Physical quantities, the amounts of which remain constant before, during, and after some physical or chemical process.

**Linear momentum:** The product of an object's mass and its velocity.

**Mass:** A measure of the quantity of matter.

**Subatomic particle:** A particle smaller than an atom, such as a proton, neutron, or electron.

**Velocity:** The rate at which the position of an object changes with time, including both the speed and the direction.

At one time, scientists thought that the law of conservation of mass and the law of conservation of energy were two distinct laws. In the early part of the twentieth century, however, German-born American physicist Albert Einstein (1879–1955) demonstrated that matter and energy are two forms of the same thing. He showed that matter can change into energy and that energy can change into matter. Einstein's discovery required a restatement of the laws of conservation of mass and energy. In some instances, a tiny bit of matter can be created or destroyed in a change. The quantity is too small to be measured by ordinary balances, but it still amounts to something. Similarly, a small amount of energy can be created or destroyed in a change. But, the *total* amount of matter PLUS energy before and after a change still remains constant. This statement is now accepted as the law of conservation of mass and energy.

Examples of the law of conservation of mass and energy are common in everyday life. The manufacturer of an electric heater can tell consumers how much heat will be produced by a given model of heater. The amount of heat produced is determined by the amount of electrical current that goes into the heater. Similarly, the amount of gasoline that can be formed in the breakdown of petroleum can be calculated by the amount of petroleum used in the process. And the amount of nuclear energy produced by a nuclear power plant can be calculated by the amount of uranium-235 used in the plant.

Calculations such as these are never quite as simple as they sound. We think of an electric lightbulb, for example, as a way of changing electrical energy into light. Yet, more than 90 percent of that electricity is actually converted to heat. (Baby chicks are kept warm by the heat of lightbulbs.) Still, the conservation law holds true. The total amount of energy produced in a lightbulb (heat plus light) is equal to the total amount of energy put into the bulb in the form of electricity.

## Other conservation laws

**Conservation of electric charge.** Most physical properties abide by conservation laws. Electric charge is another example. Electric charge is the property that makes you experience a shock or spark when you touch a metal doorknob after shuffling your feet across a rug. It is also the property that produces lightning. Electric charge comes in two varieties: positive and negative.

The law of conservation of electric charge states that the total electric charge in a system is the same before and after any kind of change. Imagine a large cloud of gas with 1,000 positive (+) charges and 950 negative (−) charges. The total electrical charge on the gas would be 1,000+ + 950−= 50+. Next, imagine that the gas is pushed together into a much smaller volume. Whatever else you may find out about this change, you can know one fact for certain: the total electric charge on the gas will continue to be 50+.

**Conservation of momentum.** Two of the most useful conservation laws apply to the property known as momentum. Linear momentum is defined as the product of an object's mass and its velocity. (Velocity is the rate at which the position of an object changes with time, including both the speed and the direction.) A 200-pound football player moving with a speed of 10 miles per hour has a linear momentum of 200 pounds × 10 miles per hour, or 2,000 pound-miles per hour. In comparison, a 100-pound sprinter running at a speed of 20 miles per hour has exactly the same liner momentum: 100 pounds × 20 miles per hour, or 2,000 pound-miles per hour.

Linear momentum is consumed in any change. For example, imagine a rocket ship about to be fired into space (Figure 1a). If the rocket ship is initially at rest, its speed is 0, so its momentum must be 0. No matter what its mass is, the linear momentum of the rocket is mass × 0 miles per hour = 0. The important fact that the conservation of linear momentum tells us is that, whatever else happens to the rocket ship, its final momentum will also be 0.

What happens when the rocket is fired, then, as in Figure 1b? Hot gases escape from the rear of the rocket ship. The momentum of those gases is equal to their total mass (call that m_{g}) times their velocity (v_{g}), or m_{g}v_{g}. We'll give this number a negative sign (−m_{g}v_{g}) to indicate that the gases are escaping backward, or to the left.

The law of conservation of linear momentum says, then, that the rocket has to move in the opposite direction, to the right or the + direction, with a momentum of m_{g}v_{g}. That must be true because then −m_{g}v_{g}(from the gases) plus +m_{g}v_{g} (from the rocket) = 0. If you know the mass of the rocket, you can find the speed with which it will travel to the right.

A second kind of momentum is angular momentum. The most familiar example of angular momentum is probably a figure skater spinning on the ice. The skater's angular momentum depends on three properties: her mass (or weight), the speed with which she is spinning, and the radius of her body.

At the beginning of a spin, the skater's arms may be extended outward, producing a large radius (the distance from the center of her body to the outermost part of his body). As she spins, she may pull her arms inward, bringing them to her side. What happens to the skater's angular momentum during the spin?

We can neglect the skater's mass, since she won't gain or lose any weight during the spin. The only factors to consider are the speed of her spin and her body radius. The law of conservation of angular momentum says that the product of these two quantities at the beginning of the spin (v_{1}r_{1}) must be the same as the product of the two quantities at the end of the spin (m_{2}r_{2}). So m_{1}r_{1} = m_{2}r_{2} must be true. But if the skater makes the radius of her body smaller, this equality can be true only if her velocity increases. This fact explains what you actually see on the ice. As a spinning skater pulls her arms in (and the body radius gets smaller), her spinning speed increases (and her velocity gets larger).

**Conservation of parity.** Conservation laws are now widely regarded as some of the most fundamental laws in all of nature. It was a great shock, therefore, when two American physicists, Val Lodgson Fitch (1923– ) and James Watson Cronin (1931– ), discovered in the mid-1960s that certain subatomic particles known as K-mesons appear to violate a conservation law. That law is known as the conservation of parity, which defines the basic symmetry of nature: that an object and its mirror image will behave the same way. Scientists have not yet fully explained this unexpected experimental result.

# conservation, laws of

**conservation, laws of** Physical laws stating that some property of a closed system is unaltered by change in the system; it is conserved. The most important are the laws of conservation of matter and energy. Mass and energy are interconvertible according to the equation *E = mc*^{2}; what is conserved is the total mass and its equivalent in energy.

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**conservation laws**