Consider the vertical spring-mass system illustrated in Figure (PageIndex{1}). Figure (PageIndex{1}): A vertical spring-mass system. When no mass is attached to the spring, the spring is at rest (we assume that the spring has no mass).

A realistic mass and spring laboratory. Hang masses from springs and adjust the spring stiffness and damping. You can even slow time. Transport the lab to different planets. A chart shows the kinetic, potential, and thermal energy for each spring.

When the mass is in motion and reaches the equilibrium position of the spring, the mechanical energy of the system has been completely converted to kinetic energy. All vibrating systems consist of this interplay between an energy storing component and an energy carrying (``massy'') component.

Vibratory systems comprise means for storing potential energy (spring), means for storing kinetic energy (mass or inertia), and means by which the energy is gradually lost (damper).The vibration of a system involves the alternating transfer of energy between its potential and kinetic forms.

The basic vibration model of a simple oscillatory system consists of a mass, a massless spring, and a damper. If damping in moderate amounts has little influence on the natural frequency, it may be neglected. The system can then be considered to be conservative.

In this section we will examine mechanical vibrations. In particular we will model an object connected to a spring and moving up and down. We also allow for the introduction of a damper to the system and for general external forces to act on the object.

Waves and Wavelike Motion. The motion of a mass attached to a spring is an example of a vibrating system. In this Lesson, the motion of a mass on a spring is discussed in detail as we focus on how a variety of quantities change over the course of time.

We introduce a one-dimensional coordinate system to describe the position of the mass, such that the (x) axis is co-linear with the motion, the origin is located where the spring is at rest, and the positive direction corresponds to the spring being extended. This "spring-mass system" is illustrated in Figure (PageIndex{1}).

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 …

The spring is supposed to obey Hooke's law, namely that, when it is extended (or compressed) by a distance x from its natural length, the tension (or thrust) in the spring is kx, and the equation of motion is m x & & = − kx . This is simple harmonic motion of period 2π/ω, where ω2 = k/m.