SIMPLE PENDULUM

A simple pendulum isolated from nonconseiwative forces would oscillate forever. Complete isolation can never be achieved, and the pendulum will eventually stop because nonconsewative forces such as air resistance and surface friction always remove mechanical energy from a system. Unless there is a mechanism for putting the energy back, the mechan-... 

The consideration of the simple pendulum illustrates the basic problem behind devising a perpetual motion machine. The problem is the fact that energy exists in several forms and is transformed from one form to the other, especially when motion is involved. Even if friction is eliminated, there arc still the electromagnetic radiation and gravitational inter-... 

A vibration is a periodic motion or one that repeats itself after a certain interval of time. This time interval is referred to as the period of the vibration, T. A plot, or profile, of a vibration is shown in Figure 43.1, which shows the period, T, and the maximum displacement or amplitude, X - The inverse of the period, j, is called the frequency, f, of the vibration, which can be expressed in units of cycles per second (cps) or Hertz (Hz). A harmonic function is the simplest type of periodic motion and is shown in Figure 43.2, which is the harmonic function for the small oscillations of a simple pendulum. Such a relationship can be expressed by the equation ... 

Figure 43.2 Small oscillations of a simple pendulum, harmonic function...

The classical harmonic oscillator in one dimension was illustrated in Seetfon 5.2.2 by the simple pendulum. Hooke s law was employed in the fSfin / = —kx where / is the force acting on the mass and k is the force constant The force can also be expressed as the negative gradient of a scalar potential function, V(jc) = for the problem in one dimension [Eq. (4-88)]. Similarly, the three-dimensional harmonic oscillator in Cartesian coordinates can be represented by the potential function... 

The spherical pendulum, which consists of a mass attached by a massless rigid rod to a frictionless universal joint, exhibits complicated motion combining vertical oscillations similar to those of the simple pendulum, whose motion is constrained to a vertical plane, with rotation in a horizontal plane. Chaos in this system was first observed over 100 years ago by Webster [2] and the details of the motion discussed at length by Whittaker [3] and Pars [4]. All aspects of its possible motion are covered by the case, when the mass is projected with a horizontal speed V in a horizontal direction perpendicular to the vertical plane containing the initial position of the pendulum when it makes some acute angle with the downward vertical direction. In many respects, the motion is similar to that of the symmetric top with one point fixed, which has been studied ad nauseum by many of the early heroes of quantum mechanics [5]. 

Fig. 1. The geometry of the pendulum (the stick and balls). In the case of the simple pendulum, the system is displaced from its downward position released from rest with the initial value of its vertical coordinate z[0] measured from the support point, as shown. The most general motion involves giving the mass an initial velocity V in the plane formed by the two lines, which are perpendicular to the pendulum and thus tangent to the sphere on which the pendulum moves. It is sufficiently general to consider the initial velocity V to be horizontal (in the direction of the vector perpendicular to the vertical plane containing the pendulum).

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. 

Fig. 3. The first integral of the equation of motion for vertical coordinate z[t] of the pendulum, plotted for three values of the kinetic energy K. In the case of the simple pendulum K=Q, the curve cutting the horizontal axis at z = — 1), we see that the first integral is only non-negative if z[t] lies between — 1, its lowest possible value and W, the vertical coordinate of the point from which it is released. As K increases, the zero at z= — 1 moves to the right. For small values of K (0.2), the left zero of the first integral, representing the lower limit of the vertical coordinate, lies to the left of the initial value, so that the initial vertical motion is downwards. As K increases, this zero moves above the initial value, and the initial motion is upwards.

Fig. 4. The acceleration associated with the vertical coordinate for the simple pendulum (the lowest curve), the slow spherical pendulum (the middle curve) and the fast spherical pendulum (the upper curve).

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... 

Figure 6 shows the approximate frequency of the spherical pendulum relative to the ideal value for the simple pendulum, plotted as a function of the dimensionless kinetic energy. 

To make a suspension-spring system with a natural frequency of 1 Hz, the weight of the mass should stretched the spring by 25 cm. Notice that Eq. (10.23) is exactly the formula for the natural frequency of a simple pendulum with length AL. To isolate the horizontal vibration, a pendulum is the... 

Example. Calculate the uncertainty of the acceleration of gravity measured with a simple pendulum based on the length of a pendulum and the period of the pendulum. The acceleration of gravity g is related to... 

ASTM D105410 now specifies only the Goodyear-Healey pendulum (it previously also included the Schob). The Goodyear-Healey is a simple pendulum consisting of a rod mounted on ball races with an additional mechanism for measuring the depth of penetration of the indentor. A note... 

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].)... 

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... 

We may stow by a simple example how this can be done. Consider a simple pendulum (fig. 3) whose length can be altered, say by drawing the thread over a pulley. If we shorten the thread... 

Fig. 3.—Simple pendulum of variable length. If the length is reduced slowly enough, the ratio of energy to frequency is constant.

In order to make clear the conception of adiabatic invariance, we consider the example of a simple pendulum consisting of a hob of mass m on a thread whose length l is slowly decreased by drawing the thread up through the point of suspension. This shortening causes an alteration of the energy W and the frequency v of the pendulum we can show, however, that for small oscillations the magnitude W/v remains invariant. 

For example, mass (M), length (L), and time (T) are the three basic mechanical dimensions. If we wished to consider the case of a simple pendulum, the relevant variables and their dimensions are ... 

It may not be immediately clear why the sine and cosine function, which we probably first en-coimtered in trigonometry, have an ihing to do with waveforms or speech. In fact it turns out that the sinusoid function has important interpretations beyond trigonometry and is found in many places in the physical world where oscillation and periodicity are involved. For example, bolli llie movement of a simple pendulum and a bouncing spring are described by sinusoid functions. 

Foucault pondulum A simple pendulum in which a heavy bob attached to a long wire is firee to swing in any direction. As a result of the earth s rotation, the plane of the pendulum s swing slowly turns (at the poles of the earth it makes one complete revolution in 24 hours). It was devised by the French physi-... 

Kater s pendulum A complex pendulum designed by Henry Kater (1777-1835) to measure the acceleration of free fall. It consists of a metal bar with knife edges attached near the ends and two weights that can slide between the knife edges. The bar is pivoted from each knife edge in turn and the positions of the weights are adjusted so that the period of the pendulum is the same with both pivots. The period is then given by the formula for a simple pendulum, which enables gto be calculated. 

Pendulum pen-j3-lom [NL, fr. L, neuter of pendulus] (1660) n. For a simple pendulum of length /, for a small amplitude, the complete period... 

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