https://changzh.me/en/tags/simple-harmonic-motion/Recent content in Simple Harmonic Motion on Blog of BrianHugoenSat, 11 Jul 2026 11:00:00 +0800https://changzh.me/en/posts/simple-harmonic-motion/Sat, 11 Jul 2026 11:00:00 +0800https://changzh.me/en/posts/simple-harmonic-motion/<p>High-school physics tells us that the displacement in simple harmonic motion (SHM) is a sinusoidal function. But textbooks never explain <em>why</em> — they assume a solution of the form $x(t)=\sin(\cdots)$, substitute it into the equation, verify that it works, and call it a day. That is a <strong>working backwards from the unknown to the known</strong> approach, and it has always left me uneasy.</p> <p>So I decided to do the opposite: starting from Hooke&rsquo;s law as the sole axiom, I would analyse the evolution of velocity step by step and see whether I could <strong>derive the unknown from the known</strong> — letting that sine (or cosine) emerge on its own. This post records my two attempts: the first using brute-force infinite subdivision, the second using matrices and geometry. From complicated to clean, I ended up colliding head-on with Euler&rsquo;s formula.</p>