Skip to main content

Simple Harmonic Motion

In almost all of physics, we encounter a phenomenon called simple harmonic motion. It is the most fundamental type of oscillation and is found in many systems, from the motion of a pendulum to the vibrations of a guitar string.

In this mini-series, we will explore the basics of simple harmonic motion, including its definition, the mathematical model that describes it, and its applications in physics. We will primarily focus on the mathematical aspects of simple harmonic motion.

Roadmap​

  • First, we will use a spring-mass system to introduce the concept of simple harmonic motion.
  • We will solve the differential equation that describes undamped simple harmonic motion in three ways:
    1. Indirectly by analyzing the energy of the system.
    2. By using something known as an ansatz, or an educated guess.
    3. By transforming it into a system of 1st-order DEs and then using linear algebra.
  • We will then explore the damped simple harmonic motion and solve the differential equation that describes it in the same three ways.
  • Finally, we will discuss the driven simple harmonic motion and solve the differential equation that describes it.
  • In all cases, we will derive expressions for the energy of the system and analyze the behavior of the system.
  • We will explore some examples and applications of simple harmonic motion in physics, such as a pendulum, a guitar string, a U-tube, and more.