Simple Harmonic Motion

Differential equation and analytical solutions of simple harmonic motion in springs and pendulums.

General idea of the topic

Simple harmonic motion is the basic model of oscillation: the restoring force is proportional to the displacement and of opposite sense, F = −kx. From that follows an acceleration proportional to the position and a sinusoidal solution, x = A cos(ωt + φ), with ω = √(k/m).

The topic writes the equation of motion and its solution, analyzes the exchange between kinetic and potential energy, whose sum is constant and equal to ½kA², and explains the phase, which determines at what point of the cycle one starts.

What you will truly understand

By the end you will be able to derive position, velocity and acceleration as functions of time, compute the amplitude and phase from the initial conditions and obtain the speed at any point using energy.

You will also be able to relate harmonic motion to the projection of uniform circular motion.

Recommended learning route

1. Equation of Motion in Simple Harmonic Motion: differential equation x'' + ω²x = 0, solution, period and frequency.
2. Energy in Simple Harmonic Motion: kinetic and potential energy over the cycle and conservation of the total energy.
3. Phase: phase constant, initial conditions and comparison between out-of-phase motions.

Physical criterion you must acquire

The essential criterion is that the period does not depend on the amplitude: it depends only on k and m. The maximum velocity is Aω and is reached at equilibrium, and the maximum acceleration is Aω² at the extremes. As a check, the total energy ½kA² must coincide with ½mv²max.

Connection with real problems

• springs and suspensions.
• clocks and chronometers.
• waves and sound.
• oscillating circuits.

Final goal of the topic

The final goal is to describe any simple harmonic motion with confidence and obtain position, velocity, acceleration and energy at each instant.