MIT OpenCourseWare: Harmonic Motion and Waves – The Simple Harmonic Oscillator and Pendulum

Overview

In this MIT OpenCourseWare session, the instructor outlines the foundational ideas of vibrations and waves, centering on the harmonic oscillator and its equation of motion. The discussion moves from a mass on a spring to a pendulum, highlighting the small angle approximation, energy relationships, and the role of angular frequency. The goal is to build intuition for how simple oscillators describe a wide range of physical systems and to prepare for more advanced topics in waves and motion.

Introduction and Course Context

This MIT OpenCourseWare lecture introduces the broad topic of vibrations and waves with a focus on the oscillator model. The instructor emphasizes that learning from fundamental examples will yield insights applicable to more complex wave phenomena. The session sets up the central equation of motion for a simple harmonic oscillator and outlines how the same framework will be extended to pendulums and other oscillatory systems.

The Spring–Mass System and the Equation of Motion

The discussion begins with a mass attached to a spring, identifying the restoring force as proportional to displacement. This leads directly to the differential equation x¨ + ω²x = 0, where ω is the angular frequency. The instructor explains how this compact form encapsulates the essential dynamics of many physical systems, linking force, mass, and stiffness in a single relationship.

Solutions and Physical Interpretation

Solving the equation yields simple harmonic motion, with solutions of the form x(t) = A cos(ωt + φ). The amplitude A, phase φ, and frequency ω determine the particle’s motion. The video emphasizes that the motion is periodic and that the energy continuously exchanges between kinetic and potential forms without damping. This section reinforces the intuition that the oscillator’s behavior is governed by the balance between inertia and restoring force.

Energy and Stability

Energy considerations are introduced to illustrate the conservation of energy in an ideal oscillator. The total energy remains constant, partitioned between kinetic energy (1/2 m v²) and potential energy (1/2 k x²), with k related to ω by ω² = k/m. This perspective clarifies why the oscillator’s frequency is determined by system properties rather than initial conditions, and it provides a natural bridge to more advanced topics like energy methods and normal modes.

From Springs to Pendulums: The Small-Angle Approximation

The lecture extends the framework to a simple pendulum, showing how for small angular displacements the restoring torque approximates to a linear term in angle. This yields a similar differential equation with an angular frequency that depends on gravity and length, illustrating how the same mathematical structure applies to different physical setups. The small-angle approximation is highlighted as a crucial step in simplifying real-world systems to analyzable models.

Angular Frequency and Practical Considerations

The concept of angular frequency is emphasized as a fundamental descriptor of oscillatory motion. The instructor notes that ω encapsulates the rate of oscillation and is determined by system properties such as mass, stiffness, gravity, and length in the pendulum case. The discussion ties together the role of ω with the energy exchange picture and the oscillation period, which is 2π/ω for an ideal system.

Conclusion and Next Steps

The session concludes with a summary of the key ideas: the x¨ + ω²x = 0 equation, the harmonic motion solution, energy balance, and the pendulum approximation. Students are encouraged to practice deriving solutions and to anticipate how these fundamental concepts extend to more complex waves and coupled oscillators in future lectures.