Angular momentum

Type of physical science:Momentum, Classical physics

Field of study: Mechanics

Angular momentum is the tendency of an object (or a system of objects rigidly held together in some way) to keep spinning or moving in a circle. Conservation of angular momentum is one of the most fundamental principles of physics, with a wide range of applications.

Overview

Sir Isaac Newton described the motion of objects in three basic laws. Newton's first law, the law of inertia, states that an object at rest will remain at rest, and an object in motion will continue to move in a straight line at a constant speed, unless acted upon by some external agent. The inertia of an object, or its resistance to change, is quantified by the mass of the object. An object's mass is simply a measure of the amount of matter or material in the object. (Mass is not to be confused with weight, which is the force that gravity exerts upon the mass.) The second of Newton's laws specifies the nature of the external agent. The push or pull that gets things moving (or slows them down and stops them) is called a "force." When a force is exerted on an object, the object changes its motion. This change in the object's motion, speeding up or slowing down, is the acceleration of the object (the change in the velocity of the object over time). The more massive the object, the harder one must push on it to make it move; in other words, more force must be exerted. Also, for a given object, the harder one pushes, the faster the object accelerates or decelerates. Thus, the force depends directly on both the mass and the acceleration. A closer look at Newton's second law suggests a more complete definition of the inertia of an object. Mass does not completely describe the motional aspect of inertia; a new quantity, momentum, is a more complete description. The momentum of an object is the product of its mass and its velocity. Momentum is a vector quantity (one requiring both a magnitude and a direction) and has a very important property: It is conserved. In terms of momentum, Newton's second law can be expressed thus: The force is equal to the change in the momentum over time. Thus, if there are no external forces acting on an object or a system of objects, the momentum will not change; it will be constant, or conserved, in both magnitude and direction. Conservation of momentum is one of the most fundamental of physical concepts.

These laws also apply to objects in rotational motion. Rotational motion includes both an object spinning about an axis and an object (or a group of objects held rigidly together) moving in a circle. The rotational inertia of an object (sometimes also referred to as the "moment of inertia") is a little more complicated to define than the inertia for linear motion. It involves both the mass of the object and how that mass is distributed. For a mass moving in a circle, the rotational inertia is defined as the mass times the square of the radius of the circle. For a solid object spinning about an axis, the mass is distributed along a whole series of radii. Thus, it is necessary to find an effective average radius for the mass. This radius will be a fraction of the original radius and will depend on the shape of the object. (This effective radius is often referred to as the "radius of gyration.") For an object spinning about an axis, therefore, the rotational inertia will be some fraction times the mass times the radius squared, where the fraction depends on the shape of the object.

The basic quantities of rotational motion are defined analogously to the linear quantities. Thus, angular velocity is defined as the change in the angular position of an object per unit time. Angular velocity depends on the angle the object sweeps out or rotates through in some amount of time. The unit for angular velocity that is encountered in everyday usage is revolutions per minute (abbreviated rpm), although a more fundamental unit is radians per second; a radian is a unit of angular measurement such that one revolution (360 degrees) is equal to 2p radians. Similarly, angular acceleration is defined as the change in angular velocity per unit time.

Restating Newton's first two laws in terms of rotations thus yields the statement that an object at rest will remain at rest, while a rotating object will continue to rotate at a constant angular speed in the same direction (clockwise or counterclockwise), unless acted upon by an external agent. In rotations, this external agent is called a "torque." In order to create this twisting force, or torque, one must apply a force to the object. However, this force must be applied in a specific manner. First, the force must be applied some radial distance from the center of rotation. The farther out the force is applied, the greater the torque. Second, the force must be applied perpendicular to the radius. Pushing along the radius merely makes the object slide linearly. In order to get a rotation (in this case, clockwise), the force F must be applied perpendicular to the radius (or lever arm) R as illustrated below.

The torque thus depends directly on both the radius and the perpendicular force. The result of the torque will be that the object begins to rotate (or to change its speed of rotation if it is already rotating). The amount of angular acceleration (or deceleration) the object experiences will depend on the amount of torque and the rotational inertia of the object. Thus, the amount of torque placed on an object will be directly related to its rotational inertia and its angular acceleration.

The rotational analog to linear momentum is angular momentum. It is defined in the same way as linear momentum. Angular momentum is the product of the rotational inertia times the angular velocity. Like linear momentum, angular momentum is a vector quantity. How can a rotation be described with a vector? Rotations are either clockwise or counterclockwise. In order to give the direction of rotation, and thus the angular momentum a vector, a right-hand rule is applied. If one curls the fingers of one's right hand in the direction of rotation, one's thumb will point in the direction of the vector. Thus the vector's direction will always lie perpendicular to the plane of the rotation, as shown on the following page.

Finally, as with linear momentum, it is possible to rewrite Newton's second law for a rotation. Applying a torque to an object will result in a change in angular momentum over time. The implications of this are the same as with linear momentum and force. If there are no external torques acting on an object, the angular momentum will not change; it will remain constant, or conserved, in both magnitude and direction. It is this last aspect, conservation of angular momentum, that has many applications.

The concept of conservation of angular momentum is one of the fundamental principles of much of modern atomic, nuclear, and elementary particle physics. Experimental evidence has led to a model of the atom with a large, positively charged nuclear core orbited by electrons. The angular momentum of the electrons must be conserved, since they are going in a circle. An early objection to this model was that in order for any object to go in a circle, it must experience a force toward the center of the circle and thus will have an acceleration. Accelerated charges radiate; the electron would continuously lose energy and spiral into the nucleus. Niels Bohr hypothesized that the electrons can only be found at certain radii from the nucleus. Since the angular momentum of the electron as it moves in a circle depends on the radius of the circle, Bohr expressed his hypothesis by saying that the angular momentum of the electrons must be "quantized"; not only is it conserved, but it can only take on certain discrete values. This quantization condition on the angular momentum was fundamental to working out the energies of the allowed orbitals and yielded the explanation of atomic spectra. When an electron moves from one orbital to another, energy must be absorbed or emitted, usually in the form of light. The light emitted is a characteristic of the atom.

In expanding this model to multielectron atoms, it was found that these electron orbitals could take on several possible shapes at a given radius; thus, the angular momentum of these different shapes would also be slightly different. This further complicated the atomic spectra, because only certain transitions were allowed in order to conserve angular momentum. On further inspection, it was found that the spectral lines were sometimes actually composed of two or more lines very close together. The explanation for this fine structure seems to lie in the fact that the electrons have an intrinsic angular momentum. The only classical situation that corresponds to this is an object spinning about its axis, and so this intrinsic angular momentum is called "spin." The effect, however, is a purely quantum-mechanical one and does not really imply that the electrons are truly spinning. Very specific rules regarding the addition of these orbital and spin angular momenta in an atom govern the order in which the electrons fill the available energy levels. The order in which the orbitals are filled in turn determines the chemical properties of the atom and helps to explain the relationships seen in the periodic table of the elements. Conservation of these different types of angular momenta also plays a large role in the construction of properties of molecules.

It was subsequently discovered that several other fundamental particles, such as neutrons and protons, also exhibit this intrinsic angular momentum, or spin, to one degree or another. Further investigations of the implications of this intrinsic angular momentum in these elementary particles has led to an understanding of nuclear structure, electrical conduction in metals, the interactions of elementary particles, and the unusual properties of superfluidity and superconductivity.

Applications

The most familiar application of conservation of angular momentum is seen in ice skating. At the end of a routine, an ice skater usually performs a complicated maneuver, spinning with arms extended and then bringing the arms in to the body. As the arms are brought inward, rotational speed increases. This increase is the result of the decrease in rotational inertia caused by redistributing the body's mass over a much smaller radius. In order for angular momentum to be conserved, when the rotational inertia decreases, the angular velocity must increase. At the end of the spin, the skater's arms are thrown out wide in a dramatic gesture. The purpose of this movement is actually more physical than dramatic. By opening up the arms, the skater increases rotational inertia, thus decreasing angular velocity and making it easier to stop. Ballet dancers also use this technique, although they tend to be more subtle about it.

Conservation of angular momentum also includes conservation of the direction of angular momentum. The most obvious application of this is seen in a top. Trying to balance a top on its tip without spinning it borders on the impossible, but when the top is spinning, even a child can balance it. When the top is spinning, it has an angular momentum that points straight upward along the axis of the rotation. The direction of this angular momentum wants to be conserved; thus, when the top is pushed slightly to the side, it tends to right itself, so that its angular momentum returns to its original direction. The same principle is seen in riding a bicycle and in throwing a football. The spin imparted to a football helps it to travel straighter in order to conserve its angular momentum. The spin of the wheels on a bicycle acts as virtual training wheels, keeping the bicycle upright as long as it is moving. Tipping the bicycle to the side changes the direction of the angular-momentum vector of the wheels. In order to compensate for this, the bicycle needs to create another angular momentum to add to that of the wheels so that the total is the same direction as the original angular momentum. In order to do this, the bicycle goes in a circle. This is how it is possible to turn a bicycle without using one's hands.

One very useful application of the conservation of angular momentum is seen in a gyroscope. A gyroscope consists of a rotor, usually a solid, flat disk (called a "flywheel") on a shaft, which is mounted on frictionless bearings in a supporting frame. Once the rotor is spinning at a fairly high, constant rate, it will want to maintain its spin axis in order to conserve the direction of the angular momentum. If the bearings are truly frictionless and there are no other torques acting on the rotor, then the rotor will always stay aligned in the same direction, regardless of how the frame holding it is moved. This makes the gyroscope an excellent compass. Once it is set spinning in a preset direction, it will always return to that chosen direction. Gyroscopes and gyrocompasses are used in ship navigation and to guide the controls of airplane autopilots. Most important, they are used to maintain a spacecraft in a fixed orientation in space and can be used to help in automatic course corrections where a magnetic compass would be useless.

Conservation of angular momentum is seen in the solar system in a variety of ways. Earth orbits the Sun in an elliptical path. When it is closest to the Sun, at a point called "perihelion," it moves faster than when it is farthest away (a point called "aphelion"). This effect was observed by Johannes Kepler and is known as "Kepler's second law." The reason for this change in speed is simple conservation of angular momentum. When Earth is closest to the Sun, its radius is smaller, and thus its rotational inertia is smallest. Conservation of angular momentum requires that as the rotational inertia decreases, the angular speed must increase. Space probes use this effect to change their direction and speed as they go out to explore the far reaches of the solar system; they slingshot themselves around the inner planets.

Conservation of angular momentum also played an important role in the formation of the Sun and the solar system. The current theory of the origin of the solar system postulates that the solar system began as a giant, swirling cloud of interstellar gas and dust called a "solar nebula." As a result of gravitational attraction between the particles of this nebula and some external influence, such as a collision with another nebula or the concussion from a nearby supernova, the nebula began collapsing. As it got smaller, conservation of angular momentum required it to spin faster. This increased angular velocity caused the shape of the nebula to change and flatten out, with the heaviest material near the center and lighter material moving outward toward the edge. As this solar nebula collapsed, it also began to get hotter, with the hottest region near the center, where the pressure from the outlying material was the greatest. This hot central core eventually formed a protosun, which continued to heat up until fusion began at its core and the Sun was born. The rest of the disk began to cool; small clumps of matter began to condense around the dust particles, and this accretion process eventually led to formation of the planets.

Principal terms

angular acceleration: a change in angular velocity with time

angular velocity: equivalent to linear velocity, this is the change in the angular position of an object over time, measured in revolutions per unit time

inertia: the property of an object to resist change; an object at rest will remain at rest, while an object in motion will continue to move at a constant speed in the same direction, unless acted upon by some outside agent

mass: the amount of inertia of an object; a measure of the amount of matter or material in an object

quantized: any quantity that is said to be "quantized" can take on only certain values and no others; the quantity cannot vary continuously, but must discretely step from one accepted value to another

rotational inertia: the rotational equivalent of mass in linear motion, it depends not only on the mass but also on how that mass is distributed; rotational inertia is the product of a constant, the mass of the object, and the radius of the object squared

scalar: a quantity that can be described by a magnitude (amount) only; mass and time are both examples of scalar quantities

spectra: a plot of the intensity of light emitted from an atom or molecule as a function of wavelength (or energy)

torque: a twisting force that causes an object at rest to begin rotating or a rotating object to slow down and stop; it consists of a force applied perpendicular to some radius

vector: a quantity that requires both a magnitude (amount) and a direction to specify it; velocity and force are both examples of vector quantities

Bibliography

Brancazio, Peter J. Sport Science: Physical Laws and Optimum Performance. New York: Simon & Schuster, 1984. A fun book that offers some wonderful insights into both physics and various sports. The author is an astrophysicist by vocation but an athlete by avocation. His love of both sports and physics shows through. The physics is described with very little math; what math is used is set aside in "math boxes" and is not fundamental to understanding the concepts.

Chaisson, Eric, and Steve McMillan. Astronomy Today. Englewood Cliffs, N.J.: Prentice-Hall, 1993. A basic introductory text in astronomy. It is written with very little math and covers the basic methods of astronomy, the solar system, stars and stellar evolution, galaxies, and more.

Feynman, Richard P., R. B. Leighton, and M. Sands. The Feynman Lectures on Physics. Vol. 1. Reading, Mass.: Addison-Wesley, 1963. The classic set of lectures that Feynman presented to his freshman physics classes at the California Institute of Technology in the early 1960's. Feynman's ability to describe even the most complicated concepts clearly and concisely, cutting right to the meaning of things, shines through every page. His wit and dynamism are evident.

Griffith, W. Thomas. The Physics of Everyday Phenomena: A Conceptual Introduction to Physics. Dubuque, Iowa: William C. Brown, 1992. A standard college physics text for the nonscience student. Presents the basic concepts of physics in a straightforward, understandable way. The text is full of ample illustrations of the concepts, all of which come from everyday experiences.

Hewitt, Paul G. Conceptual Physics. New York: HarperCollins, 1993. As the title indicates, this book is a basic introduction to physics, presented nonmathematically. Very readable, with numerous illustrations and the author's own cartoons. The author is known for his humorous approach to physics, making this an excellent starting point for anyone wanting to know more about physics.

Morrison, Michael A. Understanding Quantum Physics: A User's Manual. Englewood Cliffs, N.J.: Prentice-Hall, 1990. It seems a little incongruous to say, but this is a truly funny quantum-mechanics text. Although it is highly technical, it is still very readable, even for those who do not have the math background to follow the derivations. The explanations of the material are the best anywhere.

Tipler, Paul A. Modern Physics. New York: Worth, 1978. A standard text in "modern" physics, which includes the study of atomic, molecular, and nuclear structure, wave packets, special and general relativity, quantum mechanics, statistical physics, solid-state physics, elementary particles, and cosmology.

Weaver, Jefferson Howe. The World of Physics: A Small Library of the Literature of Physics from Antiquity to the Present. New York: Simon & Schuster, 1987. This three-volume set is an excellent introduction to the historical development of physics in the words of the actual scientists who did the work. The books consist of nontechnical papers written for a general audience.