conservation laws

(physics) A law which states that some physical quantity associated with an isolated system is constant.

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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 = mc²

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 = mc² 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 mv², 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 mv² 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 mv² 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 − (v²/c²), 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 c², 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 mr²ω, 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 mr²ω, where m is mass, r is the radius of rotation, and ω stands for angular velocity. In fact, the first two quantities, mr², 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, mr²ω 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.

Principles which state that the total values of specified quantities remain constant in time for an isolated system. Conservation laws occupy enormously important positions both at the foundations of physics and in its applications.

Realization in classical mechanics

There are three great conservation laws of mechanics: the conservation of linear momentum, often referred to simply as the conservation of momentum; the conservation of angular momentum; and the conservation of energy.

The linear momentum, or simply momentum, of a particle is equal to the product of its mass and velocity. It is a vector quantity. The total momentum of a system of particles is simply the sum of the momenta of each particle considered separately. The law of conservation of momentum states that this total momentum does not change in time. See also Conservation of momentum; Momentum.

The angular momentum of a particle is more complicated. It is defined by the vector product of the position and momentum vectors. The law of conservation of angular momentum states that the total angular momentum of an isolated system is constant in time. See also Angular momentum.

The conservation of energy is perhaps the most important law of all. Energy is a scalar quantity, and takes two forms: kinetic and potential. The kinetic energy of a particle is defined to be one-half the product of its mass and the square of its velocity. The potential energy is loosely defined as the ability to do work. The total energy is the sum of the kinetic and potential energies, and according to the conservation law it remains constant in time for an isolated system.

The essential difficulty in applying the conservation of energy law can be appreciated by considering the problem of two colliding bodies. In general, the bodies emerge from the collision moving more slowly than when they entered. This phenomenon seems to violate the conservation of energy, until it is recognized that the bodies involved may consist of smaller particles. Their random small-scale motions will require kinetic energy, which robs kinetic energy from the overall coherent large-scale motion of the bodies that are observed directly. One of the greatest achievements of nineteenth-century physics was the recognition that small-scale motion within macroscopic bodies could be identified with the perceived property of heat. See also Conservation of energy; Energy; Kinetic theory of matter.

Position in modern physics

As physics has evolved, the great conservation laws have likewise evolved in both form and content, but have never ceased to be important guiding principles.

In order to account for the phenomena of electromagnetism, it was necessary to go beyond the notion of point particles, to postulate the existence of continuous electric and magnetic fields filling all space. To obtain valid conservation laws, energy, momentum, and angular momentum must be ascribed to the electromagnetic fields. See also Electromagnetic radiation; Maxwell's equations; Poynting's vector.

In the special theory of relativity, energy and momentum are not independent concepts. Einstein discovered perhaps the most important consequence of special relativity, that is, the equivalence of mass and energy, as a consequence of the conservation laws. The “law” of conservation of mass is understood as an approximate consequence of the conservation of energy. See also Conservation of mass; Relativity.

A remarkable, beautiful, and very fruitful connection has been established between symmetries and conservation laws. Thus the law of conservation of linear momentum is understood as a consequence of the homogeneity of space, the conservation of angular momentum as a consequence of the isotropy of space, and the conservation of energy as a consequence of the homogeneity of time. See also Symmetry laws (physics).

The development of general relativity, the modern theory of gravitation, necessitates attention to a fundamental question for the conservation laws: The laws refer to an “isolated system,” but it is not clear that any system is truly isolated. This is a particularly acute problem for gravitational forces, which are long range and add up over cosmological distances. It turns out that the symmetry of physical laws is actually a more fundamental property than the conservation laws themselves, for the symmetries remain valid while the conservation laws, strictly speaking, fail.

In quantum theory, the great conservation laws remain valid in a very strong sense. Generally, the formalism of quantum mechanics does not allow prediction of the outcome of individual experiments, but only the relative probability of different possible outcomes. One might therefore entertain the possibility that the conservation laws were valid only on the average. However, momentum, angular momentum, and energy are conserved in every experiment. See also Quantum mechanics; Quantum theory of measurement.

Conservation laws of particle type

There is another important class of conservation laws, associated not with the motion of particles but with their type. Perhaps the most practically important of these laws is the conservation of chemical elements. From a modern viewpoint, this principle results from the fact that the small amount of energy involved in chemical transformations is inadequate to disrupt the nuclei deep within atoms. It is not an absolute law, because some nuclei decay spontaneously, and at sufficiently high energies it is grossly violated. See also Radioactivity.

Several conservation laws in particle physics are of the same character: They are useful even though they are not exact because, while known processes violate them, such processes are either unusually slow or require extremely high energy. See also Elementary particle.

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