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Concept# Simple harmonic motion

Summary

In mechanics and physics, simple harmonic motion (sometimes abbreviated ) is a special type of periodic motion an object experiences due to a restoring force whose magnitude is directly proportional to the distance of the object from an equilibrium position and acts towards the equilibrium position. It results in an oscillation that is described by a sinusoid which continues indefinitely (if uninhibited by friction or any other dissipation of energy).
Simple harmonic motion can serve as a mathematical model for a variety of motions, but is typified by the oscillation of a mass on a spring when it is subject to the linear elastic restoring force given by Hooke's law. The motion is sinusoidal in time and demonstrates a single resonant frequency. Other phenomena can be modeled by simple harmonic motion, including the motion of a simple pendulum, although for it to be an accurate model, the net force on the object at the end of the pendulum must be proportional to

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When lake surface waters above the temperature of maximum density 4◦C cools, the verysurface fluid parcels become denser than their neighbours below. The latter process leads to a gravitationally unstable density distribution which may trigger ‘free convection’, i.e. the upper denser waters look for their equilibrium position somewhere down in the water column. This phenomenon usually occurs at night time, when the atmosphere is colder than surface waters. However, during daytime, it may happen that the lake surface cools—releasing heat to the atmosphere—while shortwave radiation penetrates through upper waters. In this scenario, the near surface waters of the photic zone can host an unstable layer due to the net surface cooling, whereas waters beneath it experience radiative heating and gravitational stabilization. Here, we investigate the competition between shortwave radiation and surface cooling when wind is low or negligible. Thus, depending on the relative intensity of surface cooling and shortwave radiation, the upper water column undergoes different gravitationally unstable density distributions and potentially convective regions. The objective of this Master Project is to investigate, describe, and characterise such convective regimes. To do so, the heat equation controlling temperature evolution of the upper water column subject to surface cooling and penetrative radiative heating is investigated. In parallel with in-situ observations, mathematical and numerical modelling allow to understand the phenomenon taking place. The results allow to infer how the relative importance of surface cooling and penetrating shortwave radiation may affect vertical heat and mass transfer within the lake and between the lake and the atmosphere.

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The intrinsic structures of biomolecules in the gas phase may not reflect their native solution geometries. Microsolvation of the molecules bridges the two environments, enabling a tracking of molecular structural changes upon hydration at the atomistic level. We employ density functional calculations to compute a large pool of structures and vibrational spectra for a gas-phase complex, in which a doubly protonated decapeptide, gramicidin S, is solvated by two water molecules. Though most vibrations of this large complex are treated in a harmonic approximation, the water molecules and the vibrations of the host ion coupled to them are locally described by a quantum mechanical vibrational self-consistent field theory with second-order perturbation correction (VSCF-PT2). Guided and validated by the available cold ion spectroscopy data, the computational analysis identifies structures of the three experimentally observed conformers of the complex. They, mainly, differ by the hydration sites, of which the one at the Orn side chain is the most important for reshaping the peptide toward its native structure. The study demonstrates the ability of a quantum chemistry approach that intelligently combines the semiempirical and ab initio computations to disentangle a complex interplay of intra- and intermolecular hydrogen bonds in large molecular systems.

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We propose a theoretical approach to derive amplitude equations governing the weakly nonlinear evolution of non-normal dynamical systems, when they experience transient growth or respond to harmonic forcing. This approach reconciles the non-modal nature of these growth mechanisms and the need for a centre manifold to project the leading-order dynamics. Under the hypothesis of strong non-normality, we take advantage of the fact that small operator perturbations suffice to make the inverse resolvent and the inverse propagator singular, which we encompass in a multiple-scale asymptotic expansion. The methodology is outlined for a generic nonlinear dynamical system, and four application cases highlight common non-normal mechanisms in hydrodynamics: the streamwise convective non-normal amplification in the flow past a backward-facing step, and the Orr and lift-up mechanisms in the plane Poiseuille flow.

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