# Dissipative Systems

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## Dissipative Systems

mechanical systems whose total mechanical energy (the sum of kinetic and potential energy) decreases upon motion, changing into other forms of energy, such as heat. This process is called the process of dissipation of mechanical energy; it arises because of the presence of various forces of resistance (friction), which are also called dissipative forces. Examples of dissipative systems include a solid body moving along the surface of another solid in the presence of friction and a liquid or gas, among whose particles forces of viscosity (viscous friction) act upon motion.

The motion of dissipative systems may be retarded (damped) or accelerated. For example, the oscillations of a weight suspended from a spring will decay because of resistance from the surroundings and because of internal (viscous) resistance arising in the material of the spring itself upon deformation. On the other hand, the motion of a load along a rough inclined plane, which occurs when the sliding force is greater than the force of friction, will be accelerated. In this case the velocity v of the load, and consequently its kinetic energy T = mv2/2, where m is the mass of the load, are constantly increasing, but this increase proceeds more slowly than the decrease in potential energy Π = mgh (g is the acceleration of gravity and h is the height of the weight). As a result, the total mechanical energy of the weight, T + Π, is constantly decreasing.

The concept of dissipative systems is also applied in physics to nonmechanical systems in all cases when the energy of an orderly process is transformed into energy of a disorderly process—in the final analysis, into thermal energy. Thus, a system of circuits in which oscillations of electric current occur that die out because of the presence of resistance is also a dissipative system; in this case the electrical energy becomes joule heat.

Under terrestrial conditions, because of the inevitable presence of forces of resistance, virtually all systems in which no energy input from without takes place are dissipative systems. They may be regarded as conservative systems—that is, systems in which the conservation of mechanical energy takes place—only approximately, if forces of resistance are not taken into account. However, a nonconservative system may also not be a dissipative system, if the dissipation of energy in it is compensated for by the intake of energy from without. For example, the balance wheel of a clock, taken separately, is a dissipative system because of the presence of frictional resistance, and its oscillations (just as those of the weight) will decay. However, with a periodic input of energy from without by means of a spring or descending weight, the dissipation of energy is compensated, and the balance wheel performs self-oscillations.

S. M. TARG

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