Physics
Dissipative Force
Dissipative force is a type of force that opposes motion and converts kinetic energy into heat or other forms of energy. It is a non-conservative force that causes a decrease in mechanical energy of a system. Examples of dissipative forces include friction, air resistance, and viscosity.
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3 Key excerpts on "Dissipative Force"
eBook - PDFAdvanced Quantum Mechanics
A Practical Guide
- Yuli V. Nazarov, Jeroen Danon(Authors)
- 2013(Publication Date)
- Cambridge University Press(Publisher)
PART IV DISSIPATIVE QUANTUM MECHANICS 11 Dissipative quantum mechanics Everyday experience tells us that any process, motion, or oscillation will stop after some time if we do not keep it going, by providing energy from some external source. The reason for this is the fact that energy is dissipated: no system is perfectly isolated from its environment, and any energy accumulated in the system is eventually dissipated into the environment. Table 11.1 summarizes the content of this chapter. 11.1 Classical damped oscillator Let us first see how energy dissipation is treated in classical mechanics, by considering a generic model of a classical damped oscillator (Fig. 11.1). We take a single degree of freedom, x, and associate it with the coordinate of a particle. This particle has a mass M, is confined in a parabolic potential U(x) = 1 2 ax 2 , and may be subjected to an external force F ext . Dissipation of energy in the oscillator is due to friction, which is described by an extra force F f , a friction force acting on the particle. The equations of motion for the particle then read ˙ p = −ax + F f + F ext and ˙ x = p M , (11.1) with p being the momentum of the particle. The simplest assumption one can make about the friction force F f is that it is proportional to the velocity of the particle, F f = −γ ˙ x, (11.2) where γ is the damping coefficient. This simplest expression implies an instant relation between velocity and force. However, generally one expects some retardation: the force at the moment t depends not only on the velocity at the same time ˙ x(t), but also on the velocity of the particle in the past. In the time domain, the more general relation thus reads F f (t) = − ∞ 0 dτ γ (τ ) ˙ x(t − τ ). (11.3) This relation is in fact simpler to express in terms of Fourier components of the force and coordinate. Using the definition of the Fourier transform of a function f (t) of time, f (ω) = ∞ −∞ dt f (t)e iωt , (11.4) 269
- Mohsen Razavy(Author)
- 2017(Publication Date)
- World Scientific(Publisher)
(1.71) In Sec. 1.1 we discussed the motion of an object in the air where we assumed that the retarding force is proportional to -˙ x | ˙ x | n -1 , with 1 ≤ n ≤ 2. Therefore this model gives us the maximum resisting force, and for the motion of an object in the air it is not realistic. The reason being the one-dimensionality of the model. Since all collisions are “head-on” collisions, the body compresses the air in front of it as it moves. In two or three dimensions because of the glancing collisions between the body and molecules, there is a reduction of the forward momentum loss rate compared to the case that we solved, thus reducing the quadratic dependence of the resistive force [22]. In Chapter 7 we will consider other models where a particle M interacts with a group of particles and the resulting resistive force has a complicated dependence on the velocity of M . 1.5 Non-Newtonian and Nonlocal Dissipative Forces Dissipative Forces which explicitly depend on acceleration, i.e. F = F ( r , ˙ r , ¨ r , t ) are called non-Newtonian [23]. These forces violate some of the basic principles of the Newtonian dynamics. For instance, in the presence of these forces, the total acceleration is not given by the vector sum of accelerations produced by each individual force. As we will see later, Eq. (1.101) describing the relativistic motion of an electron provides an example of a non-Newtonian force law. When a particle interacts with a system of particles, or when an extended object moves in a resistive medium, then the effective force on the particle or on the extended object is nonlocal. This nonlocality can be spatial or temporal. Phenomenological Equations of Motion for Dissipative Systems 17 Examples of the latter type of nonlocality are given in Chapter 7, where the effective force felt by the particle at the time t is given by F ( x, t ) = Z t 0 K ( t -t 0 ) x ( t 0 ) dt 0 . (1.72) Here the motion is assumed to be one-dimensional.
eBook - PDF- P. C. Deshmukh(Author)
- 2019(Publication Date)
- Cambridge University Press(Publisher)
There is no easy way of including these effects at a fundamental level in the equation of motion. One may, however, include the effect of dissipation in the equation of motion, even if only approximately, using empirical knowledge. Friction arises primarily due to the movement (r . ≠ 0 ) of the oscillator through the medium, (including at the point of support of a pendulum). This movement is a common aspect of dissipation in oscillators, including the electrical LC oscillator where the resistance to the electrical charges in motion causes obstruction to the flow of the electric current and causes energy dissipation. It is thus natural to expect that the damping effect can be represented, at least approximately, by an additional force in the equation of motion that would (a) oppose motion and (b) be proportional to the oscillator’s instantaneous speed. We are of course free to suspect that the effective damping term may be proportional to the square of the instantaneous speed, or perhaps to some other polynomial function of the speed. It is, however, most common to consider the effective damping force to be linearly proportional to the instantaneous speed of the oscillator. It is also natural to expect that this term would be independent of the direction of motion of the oscillator, whether toward the equilibrium or away from it. Hence only the absolute magnitude of the instantaneous speed of the oscillator would matter. For the 1-dimensional oscillator along the X-axis, we may therefore write the effective force due to the multitude of the unspecified degrees of freedom as: F friction = –bv = –bx , (5.1) where b > 0, and is known as the damping coefficient. The negative sign in Eq. 5.1 implies that the frictional force opposes movement of the oscillator. The unspecified degrees of freedom would take away energy from the oscillator, and hence slow it down.
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