The Harmonic Oscillator
Mechanical Oscillator Classic
A mechanical oscillator is a system. A typical example is the set consisting of a body attached to a spring, set to move. This type of system is what we will approach through the study of the harmonic oscillator.
A harmonic oscillator is a system whose motion of the body that constitutes it is a simple harmonic motion. A harmonic oscillator can be schematized according to the following elements.
- a particle of mass m;
- a reference system (time and position);
- a restoring force F.
The restoring force F can be obtained through the action of a spring with spring constant k, whose effect is to make it always point to the central point O. The modulus of F will be a function of the position of the particle of mass m.
This motion is governed by Newton's second law, which algebraically for this case will be written by the differential expression:
Using the chain derivation rule, and rewriting the expression:
and, doing manipulations and the required integration operation,
an expression of the form:
the first term on the left side of this expression corresponds to the expression of kinetic energy K, and the second corresponds to the expression of potential energy V. These values vary during the path of the oscillation of the particle of our model, but their sum is constant and corresponds to the mechanical energy E of the particle.
- For any finite value of E, the particle of mass m will oscillate between points A, at x equals L and A’ at x equals -L.
- Depending on L and v, the spectrum of allowed values for E belong to a continuous set.
If we define a parameter ω such that
then we can rewrite that initial differential expression based on Newton’s first law in the form
or,
This equation is linear with constant coefficients and its general solution is known and can be written in the following form:
and that from this form it can be expanded algebraically by means of Euler's formula, and, using trigonometric identities, that another expression is arrived at in the form:
This, therefore, is the clockwise equation of motion of the particle, it has terms in sine and cosine and represents a clockwise equation of simple harmonic motion. This equation still requires two constants to be determined. In this regard, the technique to be applied is to search through the boundary conditions
results for its first constant
and for the other constant, the velocity coordinates are used, that is,
therefore, using the definition of velocity, we have
With this we have the final form of the expressions of motion, position and velocity:
With these equations we can rewrite the expression for mechanical energy E:
As an integral part of the study of harmonic motion, we have the energy relationships at critical points of motion, summarized in table 1.
Other interesting approaches to harmonic oscillator:
