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The harmonic oscillator model is very important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations. Harmonic oscillators occur widely in nature and are exploited in many manmade devices, such as clocks and radio circuits. They are the source of virtually all sinusoidal ...
A mass m attached to a spring of spring constant k exhibits simple harmonic motion in closed space. The equation for describing the period: = shows the period of oscillation is independent of the amplitude, though in practice the amplitude should be small. The above equation is also valid in the case when an additional constant force is being ...
These equations represent the simple harmonic motion of the pendulum with an added coupling factor of the spring. [1] This behavior is also seen in certain molecules (such as CO 2 and H 2 O), wherein two of the atoms will vibrate around a central one in a similar manner. [1]
Harmonic motion can mean: the displacement of the particle executing oscillatory motion that can be expressed in terms of sine or cosine functions known as harmonic motion . The motion of a Harmonic oscillator (in physics), which can be: Simple harmonic motion; Complex harmonic motion; Keplers laws of planetary motion (in physics, known as the ...
The descriptor "harmonic" in the name harmonic function originates from a point on a taut string which is undergoing harmonic motion.The solution to the differential equation for this type of motion can be written in terms of sines and cosines, functions which are thus referred to as harmonics.
In mathematics, a number of concepts employ the word harmonic. The similarity of this terminology to that of music is not accidental: the equations of motion of vibrating strings, drums and columns of air are given by formulas involving Laplacians ; the solutions to which are given by eigenvalues corresponding to their modes of vibration.
In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions . There are two kinds: the regular solid harmonics R ℓ m ( r ) {\displaystyle R_{\ell }^{m}(\mathbf {r} )} , which are well-defined at the origin and the irregular solid harmonics I ℓ m ( r ...
The Schrödinger equation for a particle in a spherically-symmetric three-dimensional harmonic oscillator can be solved explicitly by separation of variables. This procedure is analogous to the separation performed in the hydrogen-like atom problem, but with a different spherically symmetric potential V ( r ) = 1 2 μ ω 2 r 2 , {\displaystyle ...