Mechanics

Type of physical science: Classical physics

Field of study: Mechanics

The laws and methods of mechanics provide a quantitative description of the motion of ordinary particles and systems of particles, in relation to the forces responsible for the motion. Mechanics explains numerous physical properties of materials and serves as a basis and a limiting case of classical and modern physics.

Overview

Classical mechanics involves the motion or static equilibrium of massive particles or systems of particles. The basic divisions of mechanics are kinematics, dynamics, and statics.

Kinematics provides a mathematical framework on which rests most of mechanics.

Basic concepts of kinematics include trajectory, displacement, velocity, and acceleration. The velocity of a moving particle or system is the time rate of change of its displacement vector. The magnitude of the velocity is the speed. The speed is measured in meters per second. The time rate of change of a quantity is understood best by noting that it is the amount by which the quantity changes from any second to the next, in the cases where the rate is constant. The acceleration of a motion is the time rate of change of the velocity. Like the displacement vector, the velocity and the acceleration are specified by providing not only their magnitudes but also their directions. Ten meters per second specifies a speed, a scalar quantity for which no direction is needed. In contrast, going from north to south at 10 meters per second specifies a velocity, a vector quantity for which a direction (north to south) is needed in addition to its magnitude of ten. Forces, like displacements, velocities, and accelerations, are vector quantities. Energy, time, and speed are scalar quantities completely specified by providing their magnitudes and the associated units. In general, the time rate of change of displacement is not equal to displacement divided by time. It is equal to displacement over time only when the displacement changes by the same amount during any one second of the motion. In this latter case, the motion is said to be linear and uniform and the velocity is constant in magnitude and in direction.

Sir Isaac Newton's laws of motion are the foundations of dynamics and classical mechanics. Newton's first law of motion states that when the sum of the external forces acting on a system is zero, then the system remains at rest if it were at rest; otherwise, its motion is linear and uniform. The concept of linear momentum is central to the statement of Newton's second law. The linear momentum of a particle is its mass multiplied by its velocity. For a system of particles, the linear momentum is the total mass of the system multiplied by the velocity of its center of mass. For a uniform and symmetric object, the center of mass is the geometric center.

The center of mass of a uniform sphere is the center of the sphere. Newton's second law of motion states that the sum of the external forces acting on a system (or a particle) is equal to the time rate of change of its linear momentum. In this relation, the unit of force is the newton, where the units for mass, velocity, and time are kilograms, meters per second, and seconds, respectively. A commonly used form of Newton's second law, for systems with constant mass, is that the sum of the external forces is equal to the mass of the system multiplied by the acceleration of the center of mass of the system (F = ma). In this relation, letters in bold denote vector quantities: F and a represent the net force and the acceleration, respectively. For circular motions, Newton's second law states that the time rate of change of the angular momentum of a system is equal to the sum of the external torques acting on the system. For rigid bodies, such as a symmetric top, the law is reduced. The sum of the external torques is equal to the moment of inertia times the angular acceleration. The angular momentum is a vector whose magnitude is that of a linear momentum multiplied by length.

Similarly, angular accelerations and torques are vector quantities whose magnitudes are those of an acceleration and of a force, respectively, multiplied by length. The moment of inertia is a scalar quantity of the nature of mass multiplied by the square of length. The magnitude of the angular momentum is the moment of inertia times the angular speed.

Newton's third law of motion states that mechanical action and reaction forces are equal in magnitude and opposite in direction. An object set atop a horizontal surface exerts a downward force on that surface equal to the weight of the object. This weight is the object mass multiplied by the acceleration of gravity (w = mg). Both w and g denote vector quantities. In accordance with Newton's third law, the surface exerts an upward force on the object, whose magnitude is equal to that of the weight of the object.

Weight is the gravitational force exerted by Earth on an object. Any two objects (for example, Earth and a ball) exert on each other attractive gravitational forces of equal magnitude and in opposite directions, whose magnitude is proportional to the product of their masses and inversely proportional to the square of the distance between their centers of mass. Newton also developed the law of gravitation.

The laws of statics are derived from the two expressions of Newton's second law for linear and circular motions. Specifically, a system is in static equilibrium if the sum of the external forces and that of the external torques acting on it are both zero. For example, a book atop a horizontal table is in equilibrium because the two external forces acting on it--its weight and the reaction of the table--cancel each other, and the torques of both forces with respect to the center of mass of the book are zero.

Classical mechanics problems often can be solved by using the energy approach. The work-energy theorem is a very powerful tool; it states that when a system moves from one configuration (position) to another, then the change in its total mechanical energy is equal to the work done on the system by the nonconservative external forces acting on it. A typical nonconservative force emanates from a frictional origin. Such forces include ordinary frictional forces between plane surfaces in contact, as well as forces resulting from the resistance of fluids (air and water, for example) to the motion of an object.

Conservation laws, derived from Newton's second law of motion and from the work-energy theorem, play a key role in mechanics. The law of conservation of linear momentum applies to a system when, according to Newton's second law, the net external force acting on the system is zero. When the net force is zero, then the time rate of change of the linear momentum is zero, and the momentum is therefore a constant in time. Similarly, the angular momentum of a system is conserved when the sum of the external torques acting on the system is zero. From the work-energy theorem, it follows that the total mechanical energy of a system is constant in the absence of nonconservative external forces. In this latter case, any gain in kinetic energy is equal to the loss in potential energy and vice versa. An object of mass m kilograms, held at a height of h meters above ground level on a planet where the acceleration of gravity has a magnitude of g meters per second, possesses a gravitational potential energy of mgh joules. For h not greater than 1 kilometer, g is assumed to be equal at its value at ground level. The kinetic energy of the object in this fixed position is zero and the total energy is therefore equal to mgh. When the object falls or is released from rest, at a height h' between zero and h, the loss of potential energy (mgh - mgh') is exactly equal to the kinetic energy gained by the object, assuming no nonconservative forces, such as air resistance, are acting on the object.

The above laws constitute the key building blocks of classical mechanics. In addition to kinematics, dynamics, and statics, mechanics also encompasses several special subfields, which are generally characterized by the systems considered. These topical subfields of mechanics include fluid dynamics (hydrodynamics and aerodynamics), elasticity, and acoustics. Mechanics explains the elastic and other mechanical properties of materials. At the core of these explanations is the oscillatory motion of a particle in an extended object or large structure.

Oscillatory or vibrational motions are caused by restoring forces, that is, forces that always act to restore equilibrium. The back-and-forth motion of a mass hanging on a spring is a typical oscillatory or vibrational motion. Acoustics studies the propagation and properties of sound in various media. Sound is a classical wave. When sound propagates through a medium, particles in the medium undergo back-and-forth motions, with a zero net displacement. Energy, however, is transported in this process. A similar phenomenon occurs when waves propagate along a string.

The mathematical complication of mechanics problems often stems from the complication of the applied forces which, as in the case of fluid dynamics, depends partly on the system. The extensions of mechanics and the basic laws of mechanics can be cast in elaborate mathematical formalisms, including variational principles.

Applications

Mechanics is applied pervasively. These applications may be classified in three general categories: explanations of natural laws and phenomena; the use of these laws in the design and manufacture of man-made systems, products, and processes; and the use of the concepts, laws, and methods of mechanics to other fields of study.

Mechanics explains the flow of fluids, from rivers and rainwater to urban water supplies. This flow goes from higher points to lower-lying areas, partly converting potential energy into kinetic energy. Daily life motions of various kinds and the motions of planets around the sun are explained by mechanics. The gravitational force between a planet and the sun is the force causing it to rotate around the sun, which is much more massive. In short, mechanics explains most phenomena entailing linear, circular, oscillatory motions, or combinations thereof, of ordinary massive systems and their components. The longitudinal sound wave propagation through a medium and the transverse wave propagation at the surface of water are typical examples where energy is transported via vibrations of particles with a zero net displacement.

The stability of natural structures, from caverns and trees to mountains, is caused by the net zero force and net zero torque acting on these structures.

When an object is subjected to external forces whose sum is constant in time, then the motion of the object is uniformly varied. In particular, the speed of the object increases or decreases by an amount equal to the magnitude of the constant acceleration from any second of the motion to the next. For example, when the brakes of a moving car are fully applied, the car is acted on by friction forces (between the tires and the ground) whose sum is constant on the average. Hence, the speed of the car will decrease until it stops completely.

When a standing ice skater spins, he or she uses the conservation of angular momentum. The closer most of the mass of the skater is to a vertical axis passing through the skater's center of mass, the faster the skater turns. By extending a leg or the arms away from the axis, the moment of inertia increases, while the speed decreases to keep their product--the angular momentum--constant. Also, the recoil phenomenon of a gun upon firing is a common illustration of the conservation of the total linear momentum of the gun and bullet system. The linear momentum of the fired bullet is equal and opposite to that of the gun, thus maintaining their sum at zero and conserving the total linear momentum of the system.

The design, fabrication, and operation of most macroscopic structures, systems, and products rely on the laws of mechanics. Elaborate applications of statics have made possible the construction of such structures as houses, skyscrapers, huge football and baseball fields with or without domes, bridges, and numerous other fixed or mobile structures or systems. The design and operation of cars, airplanes, and space rockets heavily utilize the laws of mechanics. The principle of rocket propulsion is based on the recoil example cited in connection with the gun-bullet system. For a rocket, the exhaust gas plays a role similar to that of the bullet. This process can be demonstrated easily with an inflated balloon. By letting the gas escape from the balloon, the latter is propelled in the direction opposite to that in which the escaping gas is going.

When objects move in fluids, such as air or water, they experience a reaction force, part of which is often directed upward. This force increases with the speed of the moving system, and it depends on its shape. An airplane takes off when it moves fast enough to have the upward part of this force become greater than its weight.

One characteristic of mechanical systems (structures of all kinds) stems from the fact that they possess natural or resonance frequencies. The frequency of an oscillatory or vibrational motion is the number of back-and-forth displacement cycles in one second. The design of mechanical structures utilizes the fact that their resonance frequencies must be out of the range of the frequencies of all vibrations to which these structures may be subjected. The reason for this stems from the knowledge of resonance phenomena. A macroscopic structure is in resonance when its natural or resonance frequency is equal to that of the stimulating vibrations; the structure, in this condition, continuously takes energy from the external vibrations until it tears apart as a result of the large size of the displacement of its parts from their equilibrium positions.

In earthquake-prone areas, construction regulations take into account the possible frequencies of the tremors of the earth's crust. The design of airplanes, space vehicles, and space stations pay particular attention to the vibrational stability issue for safety and other reasons. Also, the magnification of the back-and-forth displacement during a resonance is used in some sensitive scientific instruments.

Computers, though they are not direct products of mechanics, have opened new horizons for its applications. Indeed, computer simulations and modeling are used increasingly in the design of complex systems. They are used equally to understand large-scale mechanical systems and processes, including planetary crust, surface fluid and atmospheric dynamics, and circulations with related disturbances. Progress is being made in the understanding of global oceanic and atmospheric circulations for Earth.

Context

The work of Nicolaus Copernicus (1473-1543), Johannes Kepler (1571-1630), and Galileo (1564-1642) partly provided the background on which Sir Isaac Newton (1642-1927) built classical mechanics. While Kepler provided a correct description of planetary motion, he did not understand the force responsible for the motion. The inventions of calculus of Newton and Gottfried Wilhelm Leibniz (1646-1716), and its utilization in the formulation and applications of the basic laws of mechanics, constituted radical departures from pre-Newtonian understanding of motions.

To appreciate the central place of mechanics among scientific and technological fields, it should be noted that Newton's second law of motion is universal in the sense that it does not depend on the chemical constitution of the mass of the systems and that it holds whether the external forces are of mechanical, electric, or magnetic nature. Further, the kinetic and potential energies, the momenta, and other key quantities of classical mechanics remain such in most classical physics fields. These fields include statistical mechanics, thermodynamics, and electricity and magnetism (electromagnetism). Essential mathematical entities used in statistical mechanics--the distribution functions--are obtained by direct use of the potential and kinetic energies of classical mechanics. Several key concepts and laws of thermodynamics, including temperature and pressure, are, in turn, directly derivable from the ones for statistical mechanics.

Time-dependent (electromagnetism) and time-independent (electrostatics and magnetostatics) theories of electricity and magnetism rely heavily on classical mechanics to explain the motion or state of equilibrium of charged particles.

The laws and methods of classical mechanics contributed to the development of two physics fields that are radically different. One of these fields is relativistic mechanics, established by Albert Einstein (1879-1955) from 1905 to 1915. Although relativistic mechanics is very different from classical mechanics, a form of it practically reduces to classical mechanics when the speeds of the objects considered are much lower than that of light in a vacuum. In relativity, time becomes a fourth dimension of space-time, whereas in classical mechanics, it is basically a parameter; also, mass can be converted into energy. The second of these physics fields is quantum mechanics. The mathematical framework of quantum mechanics is also radically different from that of classical mechanics. Classical mechanics provided many concepts and methods at the core of quantum mechanics. Quantum mechanics describes the motion of subatomic particles.

Several other scientific and technological fields widely utilize classical mechanics.

Classical civil and mechanical engineering fields are mostly applications of classical mechanics, particularly in the studies of mechanical properties of materials, in hydro-geological applications, and when addressing the issue of vibrational and other stabilities of structures and systems.

Chemistry and biology, particularly classical biophysics, heavily use concepts and laws of classical mechanics.

Principal terms

DISPLACEMENT or POSITION VECTOR: the distance between the reference point and the position of the system with a direction going from the reference point to the system

DYNAMICS: the study of motions, taking their causes into account

FORCE: the causes of motion or change of a system

KINEMATICS: the study of motions without considering their causes

KINETIC ENERGY: the energy of a system that stems solely from its motion

POTENTIAL ENERGY: the work that will be done if the system is released

STATICS: the study of the state of rest of a system, that is, static equilibrium

TOTAL MECHANICAL ENERGY: the sum of the kinetic and potential energies

TRAJECTORY: the path followed by a moving particle

Bibliography

Feynman, Richard P., Robert B. Leighton, and Matthew Sands. THE FEYNMAN LECTURES ON PHYSICS: COMMEMORATIVE ISSUE. Vol. 1. Reading, Mass.: Addison-Wesley, 1990. This classic text, like other volumes in the series, is intended for college students. The clarity of the presentation of principles, laws, and systems of mechanics is an invitation to most readers. Contains easy-to-understand graphical presentations.

Frautschi, S. C., R. P. Olenick, T. M. Apostol, and D. L. Goodstein. THE MECHANICAL UNIVERSE: MECHANICS AND HEAT. New York: Cambridge University Press, 1986. This college freshman textbook handles the mathematics used in mechanics with extreme care so as to keep in mind the underlying physical principles and laws. The material is the subject of a television series by Goodstein.

Goldstein, Herbert. CLASSICAL MECHANICS. Reading, Mass.: Addison-Wesley, 1950. This classical graduate textbook discusses the mathematical intricacies of classical mechanics. The relationship between the principles and methods of classical mechanics and those of relativistic and quantum theories, as well as electromagnetism, is elucidated exhaustively.

Greider, Ken. INVITATION TO PHYSICS. New York: Harcourt Brace Jovanovich, 1973. This book presents physicists, from Aristotle to Murray Gell-Mann, and their contributions to physics. Principles and laws of physics are discussed insightfully without the use of calculus. Informative biographical sketches add flavor to the delightful reading of this inspiring work.

Halliday, David, and Robert Resnick. FUNDAMENTALS OF PHYSICS. New York: John Wiley & Sons, 1988. This superb college-level textbook contains eighteen essays mostly devoted to the applications of physics. These essays cover topics ranging from physics and sports to the use of ultrasound in medicine.

Hawking, Stephen. A BRIEF HISTORY OF TIME. New York: Bantam, 1988. This masterpiece is intended for general audiences. It retraces the evolution of physics and the changes of the concept of time from antiquity to the present. Pre-Newtonian physics and Newtonian mechanics are discussed and are placed in the context of the search for a unifying theory of physics.

Centrifugal/Centripetal and Coriolis Accelerations

Tops and Gyroscopes