Sound and Oscillation

# Period and elastic constant

In wave, when studying simple harmonic motion, whose representation is given by the acronym MHS, we saw that the period of the MHS is the shortest time required for a body (or particle) to complete one revolution. In simple harmonic motion, the period is independent of the amplitude, that is, it can be said that it depends only on the mass m of the particle and the spring constant ( k ) of the spring.

It is known that the scalar acceleration of a particle in simple harmonic motion, at a position x, can be determined by the following equations:

Making the relationship of equality between these two equations, we can determine the value of the spring constant of the spring, thus we have: ( k is the spring constant )

Another relationship that we can make, equating the two acceleration equations, is as follows:

As ω = 2π/T, we have:

At times we may come across a system performing a simple harmonic motion associated with more than one spring, that is, it may be associated with two springs. In this way, two elastic constants with different values ​​will appear, k1 and k2. The spring constants of springs can be associated in series or in parallel. Let’s see the figure below.

The figure below shows an association of springs in series.

In the figure above we can see that the equivalent spring e has a spring constant given by the following equation:

The figure below shows a parallel association of springs

According to the figure above, we see that the equivalent spring, in the case of parallel association, has spring constant ke defined by the following relationship:

We have to pay attention to the fact that the period of oscillation of the SHM does not depend on the type of association that the springs present. Therefore, we have that the period is given by the following equation: