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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.

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.

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.

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.

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