harmonic oscillator

Particle of mass m attached to a helical spring of spring constant k

In the study of simple harmonic motion, we saw that this is a periodic and oscillatory motion. Therefore, it is worth remembering that an oscillatory movement is any movement in which the same situation is repeated at equal intervals of time.

Therefore, we can characterize a harmonic oscillator as the device in the figure above. In it we have a body of mass m supported on a frictionless surface, attached to an ideal helical spring, whose spring constant is k . The oscillator is in equilibrium at position O , that is, the spring is in its natural state.

If we apply an external force on the body, trying to stretch or compress the spring, and then release this body, we will see that the mass begins to perform an MHS whose period is T . Assuming that there are no dissipative forces, the value x of the displacement performed is called the amplitude ( a ) of the MHS. The rectilinear trajectory of the body is oriented; and the point O , of equilibrium, is its origin.

Therefore, we can obtain at point A with the spring stretched x = +a and with the spring compressed at point B, x = -a . The  applied force is, at each instant, equal, in absolute value, to the elastic force , expressed by

el   = -kx (Hooke’s law)

The minus sign means that the elastic force is restorative , that is, it is always oriented towards the equilibrium position O.

Note that, in the equilibrium position, that is, when x = 0 , the elastic force is zero; and at the extremes A and B , it assumes the maximum value in modulus. How:


T being the period of the MHS, and starting to count the time ( t = 0 ) from the extreme point B , the following figures represent the positions of the particle at each quarter period, until it is completed.

According to the figure above, we have:

1) t=0 ⇒x=-a (v=0)

5) t=T⇒x=-a (v=0)

At the extreme points, the velocity is zero, because the particle is changing its direction; and in the equilibrium position, the speed is maximum.

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