15.2: Characteristics of Simple Harmonic Motion

The key characteristic of the simple harmonic motion is that the acceleration of the system and, therefore, the net force are proportional to the displacement and act in the opposite direction to the displacement. Additionally, the period and frequency of a simple harmonic oscillator are independent of its amplitude. For example, diving boards move faster or slower based on their thickness. A stiff, thick diving board has a large force constant, which causes it to have a smaller period, while a thin diving board moves faster and with a higher period. The period of a simple harmonic oscillator is also affected by its mass. A heavier person on a diving board bounces up and down slower than a person with a relatively lighter weight.

The simple harmonic motion of a system results in a sine and cosine waveform. As the amplitude changes, the maxima and minima of the curve also change, but the period and frequency remain unaffected. By altering the phase angle, the curve shifts toward the right for negative values and toward the left for positive values. In general, negative phase angle values shift the standard cosine curve rightward.

Consider a ruler with one end fixed to a tabletop and the other end with a mass attached. When the ruler-mass system is pulled up from the free end and released, it executes simple harmonic motion. If the displacement is plotted with respect to time, this results in a sinusoidal waveform.

Since the oscillatory motion begins when the mass is displaced from the equilibrium position, the equation for displacement can be expressed as the amplitude, A, multiplied by the cosine function of the time-dependent angular frequency and initial phase.

Phase shift describes the difference between two similar waveforms in terms of the time interval and is measured in radians.

Recall that the first derivative of the displacement function with respect to time is velocity, and the second derivative is acceleration.

In simple harmonic motion, the velocity is a sine function that lags the displacement by phase pi by 2 and is maximum at the equilibrium, whereas the acceleration is a cosine function that lags by phase pi and is maximal at the positions of maximum displacement.