Simple Harmonic Motion Frequency

The frequency of simple harmonic motion like a mass on a spring is determined by the mass m and the stiffness of the spring expressed in terms of a spring constant k ( see Hooke's Law):

Mass on Spring Resonance

A mass on a spring has a single resonant frequency determined by its spring constant k and the mass m. Using Hooke's law and neglecting damping and the mass of the spring, Newton's second law gives the equation of motion: The solution to this differential equation is of the form: which when substituted into the motion equation gives: Collecting terms gives B=mg/k, which is just the stretch of the spring by the weight, and the expression for the resonant vibrational frequency: This kind of motion is called simple harmonic motion and the system a simple harmonic oscillator.

Mass on Spring: Motion Sequence

A mass on a spring will trace out a sinusoidal pattern as a function of time, as will any object vibrating in simple harmonic motion. One way to visualize this pattern is to walk in a straight line at constant speed while carriying the vibrating mass. Then the mass will trace out a sinusoidal path in space as well as time.

Show energy transformation involved in this motion

Energy in Mass on Spring

The simple harmonic motion of a mass on a spring is an example of an energy transformation between potential energy and kinetic energy. In the example below, it is assumed that 2 joules of work has been done to set the mass in motion.