Simple harmonic motion

We say that a body describes a simple harmonic motion ( MAS ) in the direction of the X axis when it is subjected to a force of the type: F = -kxi ; where k is a constant and x = 0 is the position in which the force is canceled or the equilibrium position. According to the previous equation, the force opposes the body’s displacement from the equilibrium position.

According to Newton’s second law:
Therefore, the equation of a MA S in the direction of the X axis is: It can be easily verified that the solution for the equation of a MAS is: Where: x is the elongation or distance of the body to the equilibrium position in the instant t; xo is the amplitude of movement or maximum elongation ; ω is a constant called the angular frequency ; (ωt-φo) is the phase of the movement ; and φo the initial phase . The angular frequency value (ω) is:

Making the first derivative of x as a function of t we obtain the speed ; and drifting again, the acceleration of the body:

Which, together with the solution for the equation of a MAS, shows the periodic character of the MAS. Therefore, if we represent the same magnitudes in the same graph in the case of the initial phase zero: position (x), speed (dx / dt) and acceleration; as in the following image. We can see that they do not reach the maximum or minimum values ​​in the same instant. The speed is canceled when the elongation is maximum, and vice versa; because they present a phase difference of 90º (they are said to be in quadrature). The elongation and the acceleration present null and maximum values ​​of opposite sign at the same instants because their phase difference is 180º (they are said to be in position).

Furthermore, it can also be seen how the three graphs are repeated exactly every 360º, that is, every T = 2π / ω seconds, which constitutes the period of the movement.

MAS is closely related to uniform circular motion . As we can see in the following image, where it is shown how the projection on a diameter of the position of a particle that describes a circular motion obeys the equation of the x position. That is, the projection onto a diameter of the position of a particle rotating at a constant angular velocity describes a MAS.

Forces recovered in the form F = -kxi, which originate a MAS, are presented, for example, by stretching or compressing a spring (Hooke’s law) or by slightly displacing a pendulum from its equilibrium position .

The force of the type F = -kxi is conservative and corresponds to a potential energy such that: -dEp = Fdr , therefore: Taking Ep = 0, for x = 0, the integration constant is zero and We are left with: This last equation is the value of the potential energy of a simple harmonic oscillator that is displaced a distance x from the equilibrium position. Then the power energy as a function of time is: The kinetic energy of the oscillator is: The total mechanical energy , which is obtained by adding the potential and the kinetics: E = Ec + Ep is:


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