Flexible string

Figure 4.5 Mechanical model [94] constructed with a practice golf ball suspended by flexible strings in a wooden framework.

J. Hutchinson and R.J. Farris, Rev. Sci. Instr., in press.). These quantities are then related to the mass/length using the wave equations for a flexible string which yields ... 

V/ is the mass per unit length of a perfectly flexible string T is the tension in the string... 

Heyman [48] discusses in some detail a second report on the dome by Poleni. Poleni s method is one which would have been reproduced almost exactly by a modern analyst using the safe theorem of plasticity. He sliced the dome into 50 portions approximating half spherical lunes (orange slices) and worked on the premise that if each lune would stand, then so would the dome. The thrust line was determined experimentally by loading a flexible string and was found to lie within the thickness of the dome. He thus observed that the cracking was not critical but he agreed with the three mathematicians that further ties should be provided. 

FIGURE 2 Vibration of a flexible string (a) Initial vibration (b) Element dx. 

The chain OA can take up an enormous number of different conformations, each characterized by a value of r. By conformation is meant a particular shape of the chain. The chain can be thought of in a simple manner, as a flexible string. When its ends are held at fixed r it can take up a certain number of conformations. Each value of r will have a specific probability the greater the number of conformations for a particular r, the greater the probability of occurrence of that value of r. What is the probability that the chain displacement vector reaches fiom the origin to the point r and lies within the volume element dV = dx dy dz ... 

Figure 3.3 Morphology of thermoplastic polyurethanes (TPUs). The solid prisms illustrate the urethane linkages, which have been designated as the hard segments. The polyol soft segments are indicated as flexible strings. Note that a portion of hard...

As discussed in Chapter 2, xanthan has a structure that is not quite a rigid rod since it has some degree of flexibility. This type of structure was described by Porod and Kratky as the worm-like chain model (Richards, 1980, p. 88). Although this may be visualised intuitively to be rather like a semi-flexible string of plastic pop-in beads, it requires the definition of the persistence length, /p, in order to develop the idea in a more quantitative way. This quantity is defined for an infinite polymer chain as follows ... 

The Kratky-Porod wormlike chain model [20,21] is widely used for describing conformational characteristics of less flexible chains. The polymer is viewed as a semi-flexible string (or worm) of overall contour length L with a continuous curvature. The chain is subdivided into N segments of length AL, which are linked at a supplementary angle r. The persistence length q (Fig. 1) is defined as... 

We consider the classical equation of motion of a flexible string. There are important similarities between this equation and the Schrbdinger equation of quantum mechanics. The flexible string is a model system designed to provide an approximate representation of a real string such as those found in musical instruments. This flexible string is defined as follows ... 

Both of these equations are similar to the equation that we solved for the flexible string. Equation (12.109) can be rewritten... 

In order to illustrate the mathematics of classical waves, we now analyze the vibrations of a flexible string, which is a model system designed to resemble a real vibrating string. It is defined to have the following properties ... 

The displacement and velocity are fimetions of x as well as functions of t. The derivative dz/dt is a partial derivative, taken with a fixed value of x. The classical equation of motion of the flexible string is derived in Appendix E from Newton s second law. From... 

Figure 14.7 Standing Waves in a Flexible String, (a) The wave function for n = 1.

Newton s second law, F = ma, provides an equation of motion for a system that obeys classical mechanics. The solution of the classical equation of motion for the harmonic oscillator provides formulas for the position and velocity that correspond to uniform harmonic motion. The solution of the classical equation of motion for a flexible string prescribes the position and velocity of each point of the string as a function of time. These solutions are deterministic, which means that if the initial conditions are precisely specified, the motion is determined for all times. 

Figure E.1 The Position of a Flexible String and the Forces on a Segment of the String. This arbitrary conformation of the string as a function of length along the string is used to describe the forces on a small segment of the string.

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