# What are harmonic oscillators and physical systems?

My question is how we describe a harmonic oscillate. Wikipedia says, "a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x." My question is, how is the harmonic oscillator a "system"? I thought that the HO was the mechanism, such as a spring or a pendulum. If not, does this mean that the HO is spring + the weight, and the pendulum + the bob, not one or the other? Also, as an extension of this, when we talk about Hooke's law, ##F = -kx##, what is being displaced? Does ##x## measure the displacement of the spring itself or the weight at the end? If it describes the weight, then then does this equation refer to two objects, i.e., F refers to the force caused by the spring while x represents the displacement of some weight at the end of a spring? Does the equation ultimately refer to the spring imposing the force, or does it describe the motion of the weight that the spring manipulates? Also, how does all of this relate to physical systems?

jfizzix
Gold Member
The system is a harmonic oscillator.

All springs (for small displacements) are harmonic oscillators, but not all harmonic oscillators are springs (for example, pendulums for small displacements are also harmonic oscillators).

A harmonic oscillator is a description of a system that behaves so that when it is displaced from equilibrium, it experiences a force in the opposite direction to the displacement that is proportional to the amount of displacement.

In Hooke's law, $x$ is the change in total length of the spring from equilibrium. The force is the force the spring exerts (in compression or in tension) on the objects it is attached to at the ends.

Physical systems are objects (or groups of objects) that we can observe or measure.

We describe systems based on the sorts of laws they seem to obey. A spring is a system that we describe as (behaving like) a simple harmonic oscillator. A glass of water under normal conditions is a system that we describe as a Newtonian fluid. The distribution of electrons in a white dwarf star form is a system that we describe as a relativistic electron gas.

The names we give for describing systems are just shorthand for the laws that they obey. So, as some more examples, Kerr black holes, Rutherford scattering processes, and two-qubit systems tell us more about the models we use to describe the systems than the systems themselves.

Philip Wood
Gold Member
does this mean that the HO is spring + the weight, and the pendulum + the bob
Yes.
Does xx measure the displacement of the spring itself or the weight at the end?
There's no difference, as the end of the spring is attached to the weight (or mass).
Does the equation ultimately refer to the spring imposing the force, or does it describe the motion of the weight that the spring manipulates?
Which equation? $F=-kx$? With the minus sign, the equation applies to the mass, F being the force exerted on it by the spring. Without the minus sign it applies to the spring, F being the force exerted on it by the mass.