MIT OCW Classical Mechanics: Simple Harmonic Motion and Driven Oscillations

MIT OpenCourseWare presents a lecture on simple harmonic motion and energy concepts. The session introduces the harmonic oscillator, derives the standard formulas for velocity, acceleration, and energy, and examines how damping and external forcing modify the motion. Through a spring mass setup, the instructor guides viewers through equal-energy relationships and phase relationships, and the conditions for stable oscillations. The discussion moves from undamped motion to damped and driven oscillations, highlighting how energy conservation and resonance shape real world systems. The talk includes problem solving demonstrations, worked examples, and the math behind oscillatory motion.

Overview

In this MIT OpenCourseWare lecture, the instructor introduces the core ideas of simple harmonic motion (SHM) by starting with a mass attached to a spring. The discussion connects SHM to fundamental energy concepts, showing how kinetic and potential energy interchange as the mass oscillates. The instructor emphasizes that in an ideal, undamped system energy is conserved, and the motion can be described by elementary trigonometric functions. The session then extends to more realistic scenarios where damping and external forcing come into play, laying the groundwork for understanding a broad class of oscillatory phenomena encountered in physics and engineering.

Equations of Motion

The heart of the topic is the second order differential equation that governs a mass on a spring. For a simple mass-spring system with stiffness k and mass m, the natural (undamped) frequency is omega0 = sqrt(k/m). In the absence of damping, the displacement x(t) follows x(t) = A cos(omega0 t) + B sin(omega0 t), or equivalently x(t) = C cos(omega0 t + phi), where A, B, and phi encode initial conditions. The lecture derives velocity and acceleration from these expressions and shows how energy oscillates between kinetic and potential forms without net loss of energy.

Energy and Phase

The instructor highlights the energy conservation in SHM, with kinetic energy KE = 1/2 m v^2 and potential energy PE = 1/2 k x^2. The total energy E = KE + PE remains constant for the ideal oscillator, illustrating a deep link between motion and energy. The phase relationship between displacement and velocity is discussed: at maximum displacement, velocity is zero and all energy is potential, while at equilibrium velocity is maximum and energy is kinetic. This section connects the mathematics to the physical picture of the oscillating system.

Damping and Forcing

The lecture then introduces damping as a force proportional to velocity, adding a term such as -c x' to the equation of motion. The damped oscillator has a decaying amplitude, with a damped frequency omega_d = sqrt(omega0^2 - (gamma/2)^2) where gamma = c/m, and different regimes depending on damping strength (underdamped, critically damped, overdamped). The instructor then moves to driven oscillations, where an external forcing F(t) = F0 cos(omega t) drives the system. The response depends on the driving frequency, and resonance occurs when the forcing frequency matches the system’s natural tendency, producing large amplitudes under certain conditions. The practical demonstration described uses a mass on a spring with magnetic or other damping mechanisms to illustrate these concepts.

Special Cases and Solutions

The session explores several limiting cases, including the undamped limit, the critically damped limit, and small damping. It also discusses how initial conditions influence the transient response before the steady-state driven response dominates. The instructor emphasizes interpreting the solutions physically, noting how damping suppresses oscillations over time and how forcing can sustain oscillations at a chosen frequency. The derivations connect to standard SHM results, such as the form of the particular solution for the driven case and the general approach to solving second order linear differential equations with constant coefficients.

Worked Examples and Applications

A sequence of worked examples demonstrates how to compute amplitudes, phases, and energy contributions given initial position and velocity, damping, and forcing parameters. The instructor illustrates how to identify regimes like resonance and how the amplitude depends on system parameters. These examples bridge theory with practical intuition, highlighting the relevance of SHM to real-world engineering systems such as vibration analysis, measurement devices, and energy transfer mechanisms.

Conclusion and Next Steps

The lecture concludes by underscoring the ubiquity of simple harmonic motion in physics and engineering, and by outlining how the concepts generalize to more complex oscillatory phenomena. Viewers are encouraged to build on these foundations to explore coupled oscillators, multi-degree-of-freedom systems, and non-linear effects, which appear in a wide range of physical problems and technological applications.