# Oscillations and Simple Harmonic Motion

### Simple Oscillating Systems

We begin our study of oscillations by examining the general definition of an oscillating system. From this definition we can examine the special case of harmonic oscillation, and derive the motion of a harmonic system.

#### Definition of an Oscillating System

So what exactly is an oscillating system? In short, it is a system in which a particle or set of particles moves back and forth. Whether it be a ball bouncing on a floor, a pendulum swinging back and forth, or a spring compressing and stretching, the basic principle of oscillation maintains that an oscillating particle returns to its initial state after a certain period of time. This kind of motion, characteristic of oscillations, is called periodic motion, and is encountered in all areas of physics.

We can also define an oscillating system a little more precisely, in terms of the forces acting on a particle in the system. In every oscillating system there is an equilibrium point at which no net force acts on the particle. A pendulum, for example, has its equilibrium position when it is hanging vertical, and the gravitational force is counteracted by the tension. If displaced from this point, however, the pendulum will experience a gravitational force that causes it to return to the equilibrium position. No matter which way the pendulum is displaced from equilibrium, it will experience a force returning it to the equilibrium point. If we denote our equilibrium point as x = 0 , we can generalize this principle for any oscillating system:

In an oscillating system, the force always acts in a direction opposite to the displacement of the particle from the equilibrium point.

This force can be constant, or it can vary with time or position, and is called a restoring force. As long as the force obeys the above principle, the resulting motion is oscillatory. Many oscillating systems can be quite complex to describe. We shall focus on a special kind of oscillation, harmonic motion, which yields a simple physical description. Before we do so, however, we must establish the variables that accompany oscillation.

#### Variables of Oscillation

In an oscillating system, the traditional variables x , v , t , and a still apply to motion. But we must introduce some new variables that describe the periodic nature of the motion: amplitude, period, and frequency.

#### Amplitude

A simple oscillator generally goes back and forth between two extreme points; the points of maximum displacement from the equilibrium point. We shall denote this point by x m and define it as the amplitude of the oscillation. If a pendulum is displaced 1 cm from equilibrium and then allowed to oscillate we can say that the amplitude of oscillation is 1 cm.

#### Period and Frequency

In simple oscillations, a particle completes a round trip in a certain period of time. This time, T , which denotes the time it takes for an oscillating particle to return to its initial position, is called the period of oscillation. We also define another concept related to time, frequency. Frequency, denoted by ν , is defined as the number of cycles per unit time and is related to period as such:

 ν = 1/T

Period, of course, is measured in seconds, while frequency is measured in Hertz (or Hz), where 1 Hz = 1 cycle/second. Angular frequency defines the number of radians per second in an oscillating system, and is denoted by σ . This may seem confusing: most oscillations don't engage in circular motion, and can't sweep out radians like in rotational motion. However, oscillating systems do complete cycles, and if we think of each cycle as containing 2Π radians, then we can define angular frequency. Again, angular frequency for oscillations may seem a bit odd for now, but it will make more sense when we compare oscillations and circular motion. For now, we can relate our three variables dealing with the cycle of oscillation:

 σ = 2Πν =

Equipped with these variables, we can now look at the special case of the simple harmonic oscillator.

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