The mathematical pendulum

At small pendulum s angle deflection 1), sina a and the returning force F = 

These formulas are valid only for small displacements (if /), under which approximation sin a = a is also valid. This approximate equality will be executed if angle a 1. So, for instance, at a = 5 (a 0.1 rad) replacing sin a by a brings about an inaccuracy of the order 0.2%. On reducing angle a this inaccuracy quickly decreases at a = 1°, it reaches an 

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We have studied small oscillations of the mathematical pendulum. Next, we solve the same problem for larger values of displacements, and with this purpose in mind consider both equations of the set (3.22). Multiplying them by unit vectors i and k, respectively, and adding we obtain one equation of motion... 

Next we consider the influence of the earth rotation on the motion of the mathematical pendulum. We will proceed from Equation (3.72). As is seen from Fig. 3.5b, the components of the tension S of the string are... 

We saw that the equations of motion can be formulated as an index-3 system by using constraints on position level. By using constraints on velocity or A level the equations are obtained in index-2 or index-1 form, respectively. The scheme (5.2.1) is formally applicable to each of these formulations though there are significant differences in performance and accuracy. First, we will demonstrate this by integrating the mathematical pendulum formulated in Cartesian coordinates as DAE. 

Another example of harmonic oscillations is that of a mathematical pendulum. An MP suspended on a weightless, nonstretched and ideally flexible thread, is referred to as a mathematical pendulum. Consider small displacements of a pendulum from the equilibrium position, i.e., I, where I is the length of the mathematical pendulum. At a certain instant of time, let the pendulum occupy the position depicted in Figure 2.9. Using the second Newtonian law, equation of motion can be written as... 

Recent mathematical work suggests that—especially for nonlinear phenomena—certain geometric properties can be as important as accuracy and (linear) stability. It has long been known that the flows of Hamiltonian systems posess invariants and symmetries which describe the behavior of groups of nearby trajectories. Consider, for example, a two-dimensional Hamiltonian system such as the planar pendulum H = — cos(g)) or the... 

A difference between these two concepts can be illustrated in many ways. Consider, for example, a mathematical pendulum in this case the old concept of trajectories around a center holds. On the other hand, in the case of a wound clock at standstill, clearly it is immaterial whether the starting impulse is small or large (as long as it is sufficient for starting, the ultimate motion will be exactly the same). Electron tube circuits and other self-excited devices exhibit similar features their ultimate motion depends on the differential equation itself and not on the initial conditions. 

Until now we have considered the theory of the mathematical model of the real pendulum. Next, suppose that a solid body of finite dimensions swings in the plane XZ around a horizontal axis, and that the motion takes place with the angular velocity o(t), Fig. 3.4. 

There is a considerable literature [10-13] devoted to finding approximate formulas for the frequency of the simple pendulum for non-zero amplitudes, usually based on mathematical arguments designed to approximate elliptic functions. 

There is actually a considerable literature on the approximate amplitude dependence of the simple pendulum [9-11], although this is the only one we know of which is based on approximating the physics rather than the mathematics. The formula is remarkably accurate even for initial angular displacements of 90° from the downward vertical. The corresponding equations for the spherical pendulum in generalised coordinates are altogether more complicated, very... 

Pn is sometimes said to represent the coupling of q with term resonance integral has similar roots (Coulson C. A., Valence, Oxford University Press, Oxford, 2nd edn, p. 79). 

In the special case of mathematical pendulums in this last case, we have... 

In order to understand the mathematical importance of the chemostat, one must look at the broader picture of the subject of nonlinear differential equations. Linear differential equations have been studied for more than two hundred years their solutions have a rich structure that has been well worked out and exploited in physics, chemistry, and biology. Avast and challenging new world opens up when one turns to nonlinear differential equations. There is an almost incomprehensible variety of non-linearities to be studied, and there is little common structure among them. Models of the physical and biological world provide classes of nonlinearities that are worthy of study. Some of the classic and most studied nonlinear differential equations are those associated with the simple pendulum. Other famous equations include those associated with the names of... 

Due to the exploratory nature of these investigations at this stage, rather than pretending to study realistic systems, we will limit ourselves to the consideration of simple models, which we believe contain an indication of methods to be further pursued and a useful phenomenology in nuce. Accordingly, in this Introductory Section, we will show that the semiclassical approach naturally leads to a search for a quasiseparable variable, and to adiabatic and diabatic representations (see the Appendix). A general analysis of transition between modes will then illustrate the role of local breakdown of adiabaticity and a semiclassical study of the pendulum motion will provide an introduction to the mathematical techniques involved. 

The nonlinear study of bifurcations of the elastic equilibrium of a straight bar involves, in a way, a change of the physical point of view, mostly due to the mathematical difficulties related to the direct approach of the Bernoulli-Euler (B.-E.) equation. This aspect gave rise to various models describing the same phenomenon, such as Kirchhoff s pendulum analogy [1], as well as to different methods of calculus, such as Thompson s potential energy method [2], [3]. In this paper, we use the linear equivalence method (LEM) to a B.-E. type model, thus deducing an approach for the critical and postcritical behavior of the cantilever bar. 

Figure 8 shows the geometry of a rotational slider block. The dynamic system of a landslide is approximated by a mathematical pendulum with the length R and the mass M concentrated to a point. The position of the mass is defined by the angle a times R. Before the landslide happens at a = aO, the driving gravitational force M g sin (—a) is in equilibrium with the frictional force pO M g cos(a), which always acts in opposition to the direction of movement (Eq. 4). A decrease of the friction coefficient from pO to p initiates the landslide. The excess driving force is... 

The density of ice is dj and that of seawater d,. The characteristic time scale of toppling can be estimated firom the period of an iceberg interpreted as a mathematical pendulum ... 

The length of such a mathematical pendulum, which is equal to the physical pendulum s oscillation period, is called the reduced length of a physical pendulum. An expression for the reduced length of a physical pendulum can be found by comparing eqs. (2.4.9) and (2.4.14) ... 

A mathematical pendulum of / = 40 cm in length and a physical pendulum in the form of a thin straight rod of length /2 = 60 cm oscillate around a common horizontal axis. Find the distance a between the rod CM and the oscillation axis. 

A mechanical system, typified by a pendulum, can oscillate around a position of final equilibrium. Chemical systems cannot do so, because of the fundamental law of thermodynamics that at all times AG > 0 when the system is not at equilibrium. There is nonetheless the occasional chemical system in which intermediates oscillate in concentration during the course of the reaction. Products, too, are formed at oscillating rates. This striking phenomenon of oscillatory behavior can be shown to occur when there are dual sets of solutions to the steady-state equations. The full mathematical treatment of this phenomenon and of instability will not be given, but a simplified version will be presented. With two sets of steady-state concentrations for the intermediates, no sooner is one set established than the consequent other changes cause the system to pass quickly to the other set, and vice versa. In effect, this establishes a chemical feedback loop. 

The convenience and value of the concept of resonance in discussing the problems of chemistry are so great as to make the disadvantage of the element of arbitrariness of little significance. This element occutb in the classical resonance phenomenon also—it is arbitrary to discuss the behavior of a system of pendulums with connecting springs in terms of the motion of independent pendulums, since the motion can be described in a way that is mathematically simpler by use of the normal coordinates of the system—but the convenience and usefulness of the concept have nevertheless caused it to be widely applied. 

A few words should be said about the difference between resonance and molecular vibrations. Although vibrations take place, they are oscillations about an equilibrium position determined by the structure of the resonance hybrid, and they should not be confused with the resonance among the contributing forms. The molecule does not resonate or vibrate" from one canonical structure to another. In this sense the term resonance is unfortunate because it has caused unnecessary confusion by invoking a picture of vibration. The term arises from a mathematical analogy between the molecule and the classical phenomenon of resonance between coupled pendulums, or other mechanical systems. 

How can the result of unique steady state be consistent with the observed oscillation in Figure 5.9 The answer is that the steady state, which mathematically exists, is physically impossible since it is unstable. By unstable, we mean that no matter how close the system comes to the unstable steady state, the dynamics leads the system away from the steady state rather than to it. This is analogous to the situation of a simple pendulum, which has an unstable steady state when the weight is suspended at exactly at 180° from its resting position. (Stability analysis, which is an important topic in model analysis and in differential equations in general, is discussed in detail in a number of texts, including [146].)... 

To the extent that science seeks to explain the mechanism of physical phenomena with mathematically expressible laws, it reduces the data of concrete observation in particular events to the status of pure abstractions. The abstractions existed antecedently to the physical phenomena they were found to describe. The complex of ideas surrounding the periodic functions had to be worked out, as pure mathematical theory, before their relations to such physical phenomena as the motion of a pendulum, the movements of the planets, and the physical properties of a vibrating string could be discerned. The point is that as mathematics became more abstract, it acquired an ever-increasing practical application to diverse concrete phenomena. Thus, abstraction, characterized by numerical operations, became the dominant conceptual mode used to describe concrete facts. 

Several performance characteristics of rubber such as abrasion resistance, pendulum rebound, Mooney viscosity, modulus, Taber die swell, and rheological properties can be modeled by Eq 7.34. " A complex mathematical model, called links-nodes-blobs was also developed and experimentally tested to express the properties of a filled rubber network system. Blobs are the filler aggregates, nodes are crosslinks and links are interconnecting chains. The model not only allows for... 

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