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Simple harmonic motion can be considered the one-dimensional projection of uniform circular motion. If an object moves with angular speed ω around a circle of radius r centered at the origin of the xy-plane, then its motion along each coordinate is simple harmonic motion with amplitude r and angular frequency ω.
A simple pendulum exhibits approximately simple harmonic motion under the conditions of no damping and small amplitude. Assuming no damping, the differential equation governing a simple pendulum of length l {\displaystyle l} , where g {\displaystyle g} is the local acceleration of gravity , is d 2 θ d t 2 + g l sin θ = 0. {\displaystyle ...
The motion is simple harmonic motion where θ 0 is the amplitude of the oscillation (that is, the maximum angle between the rod of the pendulum and the vertical). The corresponding approximate period of the motion is then
Figure 2: A simple harmonic oscillator with small periodic damping term given by ¨ + ˙ + =, =, ˙ =; =.The numerical simulation of the original equation (blue solid line) is compared with averaging system (orange dashed line) and the crude averaged system (green dash-dotted line). The left plot displays the solution evolved in time and ...
[6] [7] This generic equation plays a central role in the theory of critical dynamics, [8] and other areas of nonequilibrium statistical mechanics. The equation for Brownian motion above is a special case. An essential step in the derivation is the division of the degrees of freedom into the categories slow and fast. For example, local ...
The second-order cosine approximation is especially useful in calculating the potential energy of a pendulum, which can then be applied with a Lagrangian to find the indirect (energy) equation of motion. When calculating the period of a simple pendulum, the small-angle approximation for sine is used to allow the resulting differential equation ...
The notation of a harmonic oscillator can be extended to a one-dimensional lattice of many particles. Consider a one-dimensional quantum mechanical harmonic chain of N identical atoms. This is the simplest quantum mechanical model of a lattice, and we will see how phonons arise from it. The formalism that we will develop for this model is ...
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 ...