Simple harmonic motion

Harmonic Motion. Simple harmonic motion is one in which an object repeatedly cycles through the same set of positions. Examples (ignoring friction) include the vibration of a string, the motion of a... 

The hydrogen atom attached to an alkane molecule vibrates along the bond axis at a frequency of about 3000 cm. What wavelength of electromagnetic radiation is resonant with this vibration What is the frequency in hertz What is the force constant of the C II bond if the alkane is taken to be a stationary mass because of its size and the H atom is assumed to execute simple harmonic motion ... 

Vibrational energy, which is associated with the alternate extension and compression of die chemical bonds. For small displacements from the low-temperature equilibrium distance, the vibrational properties are those of simple harmonic motion, but at higher levels of vibrational energy, an anharmonic effect appears which plays an important role in the way in which atoms separate from tire molecule. The vibrational energy of a molecule is described in tire quantum theory by the equation... 

Treating tire atomic vibration as simple harmonic motion yields the expression... 

Discuss how to compute vibrational frequencies using a simple harmonic oscillator model of nuclear motion. 

There are many different solutions for X1 and X2 to this pair of coupled equations, but it proves possible to find two particularly simple ones called normal modes of vibration. These have the property that both particles execute simple harmonic motion at the same angular frequency. Not only that, every possible vibrational motion of the two particles can be described in terms of the normal modes, so they are obviously very important. 

There are thus two frequencies at which the two particles will show simple harmonic motion at the same frequency. 

First of all, we have to take account of every bond-stretching motion. We could write a simple harmonic potential for each bond, as discussed above. For a bond A-B, we would therefore write... 

The above treatment has made some assumptions, such as harmonic frequencies and sufficiently small energy spacing between the rotational levels. If a more elaborate treatment is required, the summation for the partition functions must be carried out explicitly. Many molecules also have internal rotations with quite small barriers, hi the above they are assumed to be described by simple harmonic vibrations, which may be a poor approximation. Calculating the energy levels for a hindered rotor is somewhat complicated, and is rarely done. If the barrier is very low, the motion may be treated as a free rotor, in which case it contributes a constant factor of RT to the enthalpy and R/2 to the entropy. 

The solution of this equation describes simple harmonic motion, which is given below ... 

We recall that in this terminology the center is the singular point (the state of rest) for simple harmonic motion represented in the phase plane by a circle (or by an ellipse). The trajectories in this case axe closed curves not having any tendency to approach the singular point (the center). 

A single-acting reciprocating pump has a cylinder diameter of 115 mm and a stroke of 230 mm. The suction line is 6 m long and 50 mm in diameter, and the level of the water in the suction tank is 3 m below the cylinder of the pump. What is the maximum speed at which the pump can run without an air vessel if separation is not to occur in the suction line The piston undergoes approximately simple harmonic motion. Atmospheric pressure is equivalent to a head of 10.4 m of water and separation occurs at a pressure corresponding to a head of 1,22 in of water. 

In order to elucidate the physical origin of second-order Doppler shift, sod, we consider the Mossbauer nucleus Fe with mass M executing simple harmonic motion [1] (see Sect. 2.3). The equation of motion under isotropic and harmonic approximations can be written as... 

The vibrational and rotational motions of the chemically bound constituents of matter have frequencies in the IR region. Industrial IR spectroscopy is concerned primarily with molecular vibrations, as transitions between individual rotational states can be measured only in IR spectra of small molecules in the gas phase. Rotational - vibrational transitions are analysed by quantum mechanics. To a first approximation, the vibrational frequency of a bond in the mid-IR can be treated as a simple harmonic oscillator by the following equation ... 

For small displacements molecular vibrations obey Hooke s law for simple harmonic motion of a system that vibrates about an equilibrium configuration. In this case the restoring force on a particle of mass m is proportional to the displacement x of the particle from its equilibrium position, and acts in the opposite direction. In terms of Newton s second law ... 

An important example of one-dimensional motion is provided by a simple harmonic oscillator. The equation of motion is... 

Mathematically, the movement of vibrating atoms at either end of a bond can be approximated to simple-harmonic motion (SHM), like two balls separated by a spring. From classical mechanics, the force necessary to shift an atom or group away from its equilibrium position is given by... 

Imagine a Maxwell liquid placed between two parallel plates and sheared by moving the upper plate in its own plane. However, instead of moving the plate at a constant velocity as discussed in Chapter 1, let the displacement of the plate vary sinusoidally with time, ie the plate undergoes simple harmonic motion. If the maximum displacement of the upper plate is X and the distance between the plates is h, then the amplitude A of the shear strain in the liquid is given by... 

Figure 2.4 Simple harmonic motion of the mass m caused by an initial displacement of a0 from its equilibrium position...

Consider the situation shown in Figure 2.4 where a mass m is caused to oscillate by an initial displacement up to an amount oq at t = 0. The amplitude a would have to be smaller than shown for simple harmonic motion as a real spring would only obey Hooke s law over a limited strain amplitude. However the assumption is that Hooke s law is obeyed and the restoring force from both spring displacements is — IJcoq where k is the force constant or elastic modulus of the spring. So we may write the force at any position as... 

For elements that have three or more isotopes, isotopic fractionations may be defined using two or more isotopic ratios. Assuming that isotopic fractionation occurs through a mass-dependent process, the extent of fractionation will be a function of the relative mass differences of the two isotope ratios. For example, assuming a simple harmonic oscillator for molecular motion, the isotopic fractionation of may be related to as ... 

To separate the effects of static and dynamic disorder, and to obtain an assessment of the height of the potential barrier that is involved in a particular mean-square displacement (here abbreviated (x )), it is necessary to find a parameter whose variation is sensitive to these quantities. Temperature is the obvious choice. A static disorder will be temperature independent, whereas a dynamic disorder will have a temperature dependence related to the shape of the potential well in which the atom moves, and to the height of any barriers it must cross (Frauenfelder et ai, 1979). Simple harmonic thermal vibration decreases linearly with temperature until the Debye temperature Td below To the mean-square displacement due to vibration is temperature independent and has a value characteristic of the zero-point vibrational (x ). The high-temperature portion of a curve of (x ) vs T will therefore extrapolate smoothly to 0 at T = 0 K if the sole or dominant contribution to the measured (x ) is simple harmonic vibration ((x )y). In such a plot the low-temperature limb is expected to have values of (x ) equal to about 0.01 A (Willis and Pryor, 1975). Departures from this behavior indicate more complex motion or static disorder. 

K for myoglobin (Parak et al., 1981). Thus, measurements of (x ) at temperatures below this value should show a much less steep temperature dependence than measurements above, if nonharmonic or collective motions (whose mean-square displacement is denoted (x )c) are a significant component of the total (x ). Figure 21 illustrates the expected behavior of (x )v, x, and their sum for a simple model system in which a small number of substates are separated by relatively large barriers. In practice, the relative contributions of simple harmonic vibrations and coUective modes will vary from residue to residue within a given protein. 

In any cavitation field most of the visible bubbles will be oscillating in a stable manner and it is perhaps pertinent that we concentrate our discussions first on the fate of such bubbles in the acoustic field. If we assume that we have a bubble with an equilibrium radius, R, existing in a liquid at atmospheric pressure Pjj, then the oscillation of the bubble and in particular the motion of the bubble wall, under the influence of the applied sinusoidal acoustic pressure (P ) is a simple dynamical problem, akin to simple harmonic motion for a spring. 

For negative values of , a molecule thus experiences a restoring force towards the axis, and the molecule can execute simple harmonic motion about the axis. Newton s equations predict that molecules entering the field with no radial component of velocity will be focused to a point on the axis when the voltage is ... 

The first term again represents drag in steady motion at the instantaneous velocity, with Cd an empirical function of Re as in Chapter 5. The other terms represent contributions from added mass and history, with empirical coefficients, Aa and Ah, to account for differences from creeping flow. From measurements of the drag on a sphere executing simple harmonic motion in a liquid, Aa and Ah appeared to depend only on the acceleration modulus according to ... 

The simple harmonic motion of a diatomic molecule was treated in Chapter 1, and will not be repeated here. Instead, emphasis is placed on polyatomic molecules whose electronic energy s dependence on the 3N Cartesian coordinates of its N atoms can be written (approximately) in terms of a Taylor series expansion about a stable local minimum. We therefore assume that the molecule of interest exists in an electronic state for which the geometry being considered is stable (i.e., not subject to spontaneous geometrical distortion). 

This can be seen from elementary considerations. The distortion of the surrounding medium that leads to the intermediate state with energy WH can be described by simple harmonic motion, with a parameter x for the displacement, a potential energy px2 and a wave function of the form r=const xexp(—ax2). The probability P 2 of a configuration with potential energy WH is thus... 

The attenuation of die dipole of the repeat unit owing to thermal oscillations was modeled by treating the dipole moment as a simple harmonic oscillator tied to the motion of the repeat unit and characterized by the excitation of a single lattice mode, the mode, which describes the in-phase rotation of the repeat unit as a whole about the chain axis. This mode was shown to capture accurately the oscillatory dynamics of the net dipole moment itself, by comparison with short molecular dynamics simulations. The average amplitude is determined from the frequency of this single mode, which comes directly out of the CLD calculation ... 

M for a simple harmonic oscillator where v, the vibrational quantum i AUmber has values 0, 1.2, 3, etc. The potential function F(r) for simple < harmonic motion as derived from Hooke s law is given by... 

The representation as a two-dimensional potential energy diagram is simple for diatomic molecules. But for polyatomic molecules, vibrational motion is more complex. If the vibrations are assumed to be simple harmonic, the net vibrational motion of TV-atomic molecule can be resolved into 3TV-6 components termed normal modes of ibrations (3TV-5 for... 

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