Simple Harmonic Motion

SIMPLE HARMONIC MOTION

There are certain oscillations that can be described as simple harmonic motion or SHM. This is the name given to oscillatory motion for a system whenever the net force can be described by Hooke’s law. This is referred to as a simple harmonic oscillator. As long as there is no damping of the motion by friction or other forces, this type of oscillator will oscillate with equal displacement on either side of the equilibrium position. The maximal displacement is referred to as the amplitude, identified by the letter X. The units of amplitude are the same as they are for displacement, which are meters for objects like a spring but, for sound oscillations, the units will be in pressure units. The amplitude is related to the energy stored in the oscillation.

An object that is attached to a spring which is sliding along a frictionless surface is a simple harmonic oscillator. The amplitude will be X and the period will be T. The maximum speed occurs as it passes through its equilibrium point. The stiffer the spring, the smaller the period will be. The greater the mass of the object, the greater is the period.

The period and the frequency are, as mentioned, related to one another but neither are related to the amplitude of the spring. A guitar string will make the same sound regardless of how hard it is struck. Because the period will be constant, a simple harmonic oscillator can be used to run a clock. The period will be related to the stiffness of the system because of its force constant. High stiffnesses have high force constants and a smaller period.

In addition, more massive systems will increase the period (think of a heavier person bouncing more slowly on a diving board). Mass and the force constant are the only factors that affect the period and the frequency of things in simple harmonic motion. This is defined according to figure 95:

Figure 95.

Simple harmonic motion can be described in terms of waves. In fact, all simple harmonic motion is intimately related to the sine and cosine waves. For instance, if the restoring force of a spring or an oscillating object can be described exclusively by Hooke’s law, then the wave will be a sine function. In such cases, the displacement over time in any simple harmonic motion is related to the equation described in figure 96:

Figure 96.

The equations seen in figure 96 show the differences in the velocity, displacement, and acceleration over time. At the start of motion, the velocity is negative because the system is moving back toward the equilibrium point. The equations for velocity and acceleration are no different from Newton’s second law, which is Force = mass times acceleration or the force constant multiplied by distance and divided by mass. Figure 97 shows the acceleration and velocity of wave forms:

Figure 97.

What this means is that the velocity will be maximum at the point near the zero point on the wave or when the object is crossing over the equilibrium point. The acceleration will be opposite to the displacement and will be directly proportional to the displacement. The maximal V or Vmax will be the square root of the ratio of the force constant multiplied by the displacement divided by the mass. This is seen in figure 98: