The amplitude of a simple harmonic oscillator is defined as the maximum displacement from its equilibrium position.

A simple harmonic oscillator is a system that moves back and forth around an equilibrium position at a constant frequency. The amplitude, denoted by $A$, represents the furthest distance from equilibrium that the oscillator reaches. This value is influenced by the initial conditions of the system, such as the initial displacement and initial velocity.

For instance, consider a mass attached to a spring oscillating back and forth. The displacement of the mass from the equilibrium position can be described by the equation:

In this equation:

• $A$ is the amplitude,
• $\omega$ is the angular frequency,
• $t$ is time, and
• $\phi$ is the phase angle.

The angular frequency $\omega$ is determined by the mass of the object and the spring constant $k$, and is given by the formula:

where:

• $k$ is the spring constant, and
• $m$ is the mass of the object.

To ascertain the amplitude of the oscillator, we must consider the initial conditions. For example, if the mass is initially displaced by a distance of $x_0$ and then released from rest, the amplitude can be expressed as:

Alternatively, if the mass is given an initial velocity of $v_0$ and released from the equilibrium position, the amplitude is calculated using:

In conclusion, the amplitude of a simple harmonic oscillator is determined by the maximum displacement from equilibrium, which is fundamentally shaped by the initial conditions of the system.