Conservation Laws

Conservation Laws

Conservation of linear momentum

Conservation of angular momentum

Conservation of energy and mass

Conservation of electric charge

Other conservation laws

Resources

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