Simple Harmonic Motion – Quick & Fun Notes 🎉

1. Periodic Motion Basics 🔄

A motion is periodic when it repeats after a fixed time \(T\): \(f(t)=f(t+T)\). Each full cycle feels like déjà vu! ⏰ :contentReference[oaicite:0]{index=0}

For a single sine or cosine, the link between the angular frequency \(\omega\) and the period is

\(T=\dfrac{2\pi}{\omega}\) ⏰ :contentReference[oaicite:1]{index=1}

Even a mix like \(f(t)=A\sin\omega t+B\cos\omega t\) stays periodic with the same \(T\). We can rewrite it as

\(f(t)=D\sin(\omega t+\phi)\)

where \(D=\sqrt{A^{2}+B^{2}}\) (overall amplitude) and \(\tan\phi=\dfrac{B}{A}\). This trick bundles many waves into one neat sine—magic! 🔮 :contentReference[oaicite:2]{index=2}

Cool fact: Every periodic signal can be built from a bunch of sine and cosine waves with different periods (the big idea behind Fourier analysis). 🤓 :contentReference[oaicite:3]{index=3}

Example Check-up 🎯

• \(\sin\omega t+\cos\omega t\): periodic, \(T=2\pi/\omega\). ✅ :contentReference[oaicite:4]{index=4}
• \(\sin\omega t+\cos2\omega t+\sin4\omega t\): periodic, still \(T=2\pi/\omega\). ✅ :contentReference[oaicite:5]{index=5}
• \(e^{-\omega t}\): fades forever—not periodic. ❌ :contentReference[oaicite:6]{index=6}
• \(\log(\omega t)\): keeps growing—also not periodic. ❌ :contentReference[oaicite:7]{index=7}

2. Simple Harmonic Motion (SHM) 🌟

SHM is the superstar of oscillations. The displacement from equilibrium follows

\(x(t)=A\cos(\omega t+\phi)\) :contentReference[oaicite:8]{index=8}

• Amplitude \(A\): farthest stretch from center. Bigger \(A\) → wider swing. 💪 :contentReference[oaicite:9]{index=9}
• Angular frequency \(\omega\): how fast the phase spins (rad/s). Link: \(\omega=\dfrac{2\pi}{T}\). ⚡ :contentReference[oaicite:10]{index=10}
• Phase \(\omega t+\phi\): tells “where in the wiggle” the particle is. The constant part \(\phi\) sets the starting point. 🎬 :contentReference[oaicite:11]{index=11}
• Speed is max at the center (\(x=0\)) and drops to zero at the ends (\(x=\pm A\)). 🏎️ → 🚩 :contentReference[oaicite:12]{index=12}

Compare & Contrast 📊

Two SHMs can share \(\omega\) and \(\phi\) but differ in amplitude—picture one tall wave and one shorter copy dancing together. 🎶 :contentReference[oaicite:13]{index=13}

They can share \(A\) and \(\omega\) yet start out of step if their phase constants differ. 🕺🕺 :contentReference[oaicite:14]{index=14}

Or they can share \(A\) and \(\phi\) while having different \(\omega\); the one with the bigger \(\omega\) squeezes more cycles into the same time. 🚀 :contentReference[oaicite:15]{index=15}

Example Quick-Test ✅/❌

• \(\sin\omega t-\cos\omega t=\sqrt{2}\sin(\omega t-\pi/4)\) → SHM with \(T=2\pi/\omega\). ✅ :contentReference[oaicite:16]{index=16}
• \(\sin^{2}\omega t=\dfrac{1}{2}-\dfrac{1}{2}\cos2\omega t\) → periodic (\(T=\pi/\omega\)) but not SHM (it doesn’t stay sinusoidal about zero). ❌ :contentReference[oaicite:17]{index=17}

3. SHM & Uniform Circular Motion 🔗

Imagine a ball whirling in a circle of radius \(A\) with steady speed. Look at its shadow on a diameter. That shadow slides back and forth exactly like SHM with the same \(\omega\), period, and amplitude. 🌗 ⇄ 🟢 :contentReference[oaicite:18]{index=18}

The initial angle of the ball becomes the phase constant \(\phi\). Tracking the circular motion gives a handy visual for every part of SHM. 📐 :contentReference[oaicite:19]{index=19}

High-Yield NEET Nuggets 🎯

1. The golden link \(T=\dfrac{2\pi}{\omega}\) (know it cold!). ⚡ :contentReference[oaicite:20]{index=20}
2. The master formula \(x(t)=A\cos(\omega t+\phi)\) and what each symbol means. 📌 :contentReference[oaicite:21]{index=21}
3. How to spot SHM: single sine or cosine with fixed \(A\), \(\omega\), \(\phi\). 🔍 :contentReference[oaicite:22]{index=22}
4. Projection of uniform circular motion gives SHM—great for drawing free-body diagrams in questions. 🎡 :contentReference[oaicite:23]{index=23}
5. Classify functions quickly (periodic vs SHM vs non-periodic) to ace “identify the motion” problems. 🚦 :contentReference[oaicite:24]{index=24}

Keep practicing, keep oscillating, and rock that NEET! 🤩