Vibrations and Repeating Motion

Many systems in nature repeat themselves in cycles, like swings and springs.

Simple Harmonic Motion (SHM)

SHM is a type of repetitive motion where restoring forces are proportional to displacement: \( F = -kx \).

• Position as a function of time: \( x(t) = A\cos(\omega t + \phi) \)
• \( \omega = \sqrt{\frac{k}{m}} \) for a mass-spring system

Real-World Relevance

SHM explains how clocks keep time, how musical instruments create sound, and how buildings sway during earthquakes.

The mathematical beauty of SHM lies in how predictable and regular these oscillations are!