Energy of a Spring

Here x — Xo is the extension of the spring apart from it equilibrium position and k is the force constant. This means that when the spring extends to its equilibrium position, its energy U equals zero. We consider this spring as an ideal spring. We can see easily, if we form the derivative with respect to the position, Hooke s law will emerge  

If we have more than one identical spring with equal force constants k and equal equilibrium position Xo, we insert an index 

The total length of the springs when switched in series is x = xi + X2 +. ... In dimensionless variables we find 

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The Bond Stretching Term The increase in the energy of a spring (remember that we are modelling the molecule as a collection of balls held together by springs) when it is stretched (Fig. 3.2) is approximately proportional to the square of the extension ... 

In the electronic Hamiltonian t +i, is the transfer integral, i.e. the re-electron wavefunction overlap between nearest neighbour sites in the polymer chain, and is equivalent to the parameter /3 in Equation (4.20), and c 1+,s. and clhS are creation and annihilation operators that create an electron of spin s ( 1/2) on the carbon atom at site n-f 1 and destroy an electron of spin, s at the carbon atom on site n, i.e. in effect transfer an electron between adjacent carbon atoms in the polymer chain. The elastic term is just the energy of a spring of force constant k extended by an amount ( +1— u ), where the u are the displacements along the chain axis of the carbon atoms from their positions in the equal bond length structure, as indicated in Fig. 9.8(b). The extent of the overlap of 7i-electron wavefunction will depend on the separation of nearest neighbour carbon atoms and is approximated by ... 

The argument, at its simplest, is as follows. The primary function of a spring is that of storing elastic energy and - when required - releasing it again. The elastic energy stored per unit volume in a block of material stressed uniformly to a stress a is ... 

The molecular mechanics method, often likened to a ball and spring model of the molecule, represents the total energy of a system of molecules with a set of simple analytical functions representing different interactions between bonded and non-bonded atoms, as shown schematically in Figure 1. 

The static tests considered in Chapter 8 treat the rubber as being essentially an elastic, or rather high elastic, material whereas it is in fact viscoelastic and, hence, its response to dynamic stressing is a combination of an elastic response and a viscous response and energy is lost in each cycle. This behaviour can be conveniently envisaged by a simple empirical model of a spring and dashpot in parallel (Voigt-Kelvin model). 

Absorption of infrared radiation causes transitions between vibrational energy states of a molecule. A simple diatomic molecule, such as H—Cl, has only one vibrational mode available to it, a stretching vibration somewhat like balls on the ends of a spring ... 

Molecular Mechanics. Molecular mechanics (MM), or empirical force field methods (EFF), are so called because they are a model based on equations from Newtonian mechanics. This model assumes that atoms are hard spheres attached by networks of springs, with discrete force constants. The force constants in the equations are adjusted empirically to repro duce experimental observations. The net result is a model which relates the "mechanical" forces within a structure to its properties. Force fields are made up of sets of equations each of which represents an element of the decomposition of the total energy of a system (not a quantum mechanical eneigy, but a classical mechanical one). The sum of the components is called the force field eneigy, or steric energy, which also routinely includes the electrostatic eneigy components. Typically, the steric energy is expressed as... 

We have shown the molecular orbital theory origin of structure - function relationships for electronic hyperpolarizability. Yet, much of the common language of nonlinear optics is phrased in terms of anharmonic oscillators. How are the molecular orbital and oscillator models reconciled with one another The potential energy function of a spring maps the distortion energy as a function of its displacement. A connection can indeed be drawn between the molecular orbitals of a molecule and its corresponding effective oscillator . 

The potential energy of a mass connected to an ideal spring is given by ... 

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