Hooke’s equation

As an example let s consider the hydrogen bond between an N—H and a C=0 group. The presence of hydrogen bond here lowers the force constant of both these bonds to stretching. In the Hooke s equation, force constant occurs in the numerator. It is obvious that the frequency of these stretching vibrations will be reduced. 

Write the rotational analog of Hooke s law for the torque x driving the oseillation in Problem 3. Write the rotational analog of Newton s second law. Combine the two laws to obtain the rotational analog of the Newton-Hooke equation, Eq. (4-1). 

This equation shows that at small deformations individual chains obey Hooke s law with the force constant kj = 3kT/nlo. This result may be derived directly from random flight statistics without considering a network. 

Just as the electrical behaviour of a real diatomic molecule is not accurately harmonic, neither is its mechanical behaviour. The potential function, vibrational energy levels and wave functions shown in Figure f.i3 were derived by assuming that vibrational motion obeys Hooke s law, as expressed by Equation (1.63), but this assumption is reasonable only... 

However, Hooke s law, Equation (2.1), can be derived from Equation (2.5) ... 

Here E is Young modulus. Comparison with Equation (3.95) clearly shows that the parameter k, usually called spring stiffness, is inversely proportional to its length. Sometimes k is also called the elastic constant but it may easily cause confusion because of its dependence on length. By definition, Hooke s law is valid when there is a linear relationship between the stress and the strain. Equation (3.97). For instance, if /q = 0.1 m then an extension (/ — /q) cannot usually exceed 1 mm. After this introduction let us write down the condition when all elements of the system mass-spring are at the rest (equilibrium) ... 

Here ko is the stiffness of the spring with the restoring force and Equation (3.127) can be treated as Hooke s law for such spring. Suppose that the mass performs vibrations near a point of equilibrium. Then the equation of motion changes slightly and we have ... 

Equation (9.19) is a chemical version of Hooke s law, and only applies where the Morse curve is parabolic, i.e. near the bottom of the curve where molecular vibrations are of low energy. 

In short, near-infrared spectra arise from the same source as mid-range (or normal ) infrared spectroscopy vibrations, stretches, and rotations of atoms about a chemical bond. In a classical model of the vibrations between two atoms, Hooke s Law was used to provide a basis for the math. This equation gave the lowest or base energies that arise from a harmonic (diatomic) oscillator, namely ... 

Solving this equation gives compbcated values for the ground and excited states, as well. Using a simpbfied version of the equation, more usable levels may be discerned (here, the echoes of Hooke s Law are seen)... 

If we work at small strains so that we are in the linear (Hooke s law) region of the rheogram, then Equation (2.2) reduces to... 

For a simple diatomic molecule X-Y the sole vibration which may take place in a periodic stretching along the X-Y band. Thus, the stretching vibrations may be visualized as the oscillations of two entities connected by a spring and the same mathematical treatment, known as Hooke s Law, holds good to a first approximation. Hence, for stretching of the band X-Y, the vibrational frequency (cm-1) may be expressed by the following equation ... 

Let us consider a diatomic molecule and assume that it behaves as a harmonic oscillator with two masses, nii and m2, connected by an ideal (constant-force) spring. At equilibrium, the two masses are at a distance Xq by extending or compressing the distance by an amount X, a force F will be generated between the two masses, described by Hooke s law (cf equation 1.14) ... 

MICHAELIS-MENTEN EQUATION FIRST-ORDER REACTION ZERO POINT ENERGY HOOKE S LAW SPRING KINETIC ISOTOPE EFFECTS Zeroth law of thermodynamics, THERMODYNAMICS, LAWS OF ZETA... 

The elastic component is dominant in solids, hence their mechanical properties may be described by Hooke s law (Equation 14.1), which states that the applied stress (.s) is proportional to the resultant strain (y) but is independent of the rate of this strain (dy/dt). 

Equation (5.5) is known as Hooke s Law and simply states that in the elastic region, the stress and strain are related through a proportionality constant, E. Note the similarity in form to Newton s Law of Viscosity [Eq. (4.3)], where the shear stress, r, is proportional to the strain rate, y. The primary differences are that we are now describing a solid, not a fluid, the response is to a tensile force, not a shear force, and we do not (yet) consider time dependency in our tensile stress or strain. 

In general, Hooke s law is the basic constitutive equation giving the relationship between stress and strain. Generalized Hooke s law is often expressed in the following form [20,108] ... 

All crystals are anisotropic many other structures also have elastic anisotropy. The propagation of elastic waves in anisotropic media is described by the Christoffel equation. This still depends on Newton s law and Hooke s law, but it is expressed in tensor form so that elastic anisotropy may be included. The tensor description of elastic stiffness was summarized in 6.2, especially eqns (6.23)—(6.29). The Christoffel equation is... 

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