Simple Harmonic Motion (SHM) & Uniform Circular Motion 🎡↔️

1. Watching a Circle Turn into a Wiggle 🌕➜↔️

  • Tie a ball to a string and whirl it around at steady speed. Look at it from the side: its shadow glides back and forth in straight-line SHM. :contentReference[oaicite:0]{index=0}
  • If a particle moves around a circle of radius A with angular speed ω, its projection on any diameter follows

    \(x(t)=A\cos\bigl(\,ωt+φ\,\bigr)\) :contentReference[oaicite:1]{index=1}
  • Project on the other diameter and you get \(y(t)=A\sin\bigl(\,ωt+φ\,\bigr)\)—the same dance, just a quarter-cycle (π/2) ahead. :contentReference[oaicite:2]{index=2}

2. Displacement, Period, and Frequency ⏲️

A full cycle takes time \(T\), so the angular frequency is \(ω=\dfrac{2π}{T}\). That makes ordinary frequency simply \(1/T\). :contentReference[oaicite:3]{index=3}

Two motions can share amplitude A and phase \(φ\) yet look different because their ω values—and therefore their frequencies—aren’t the same. :contentReference[oaicite:4]{index=4}

3. Speed & Acceleration—The Fast Facts 🏃💨

  • The circle-runner’s speed is \(v=ωA\). :contentReference[oaicite:5]{index=5}
  • The to-and-fro speed is \(v(t)=-ωA\sin\bigl(ωt+φ\bigr)\), pointing opposite to the positive x direction when the sine is positive. :contentReference[oaicite:6]{index=6}
  • The acceleration turns out to be \(a(t)=-ω^{2}A\cos\bigl(ωt+φ\bigr)=-ω^{2}x(t)\). So acceleration always pulls the particle back toward the midpoint—perfect restoring behavior! :contentReference[oaicite:7]{index=7}

4. Quick Function Check (Example 13.3) 🔍

  • \(\sin ωt-\cos ωt\) really is SHM; rewrite it as \(\sqrt2\,\sin\bigl(ωt-\tfrac{π}{4}\bigr)\). :contentReference[oaicite:8]{index=8}
  • \(\sin^{2}ωt\) is just periodic—its lowest-energy form is \(½-½\cos2ωt\) with period \(π/ω\). :contentReference[oaicite:9]{index=9}

5. Mapping Circular Motion to SHM (Example 13.4) 🎯

  • Case (a): Radius \(A\), period 4 s, starts \(45°\) ahead. \(x(t)=A\cos\!\bigl(\tfrac{π}{2T}t+\tfrac{π}{4}\bigr)\) gives amplitude \(A\), period 4 s, initial phase \(π/4\). :contentReference[oaicite:10]{index=10}
  • Case (b): Radius \(B\), period 30 s, starts at \(90°\) and spins the other way. You can write the result as \(x(t)=B\sin\!\bigl(\tfrac{2π}{T}t\bigr)=B\cos\!\bigl(\tfrac{π}{15}t-\tfrac{π}{2}\bigr)\), giving amplitude \(B\), period 30 s, initial phase \(-π/2\). :contentReference[oaicite:11]{index=11}

6. Why SHM Isn’t Just Circular Motion 🚀❌🏹

The math link is neat, but remember: the force that keeps SHM going is a straight-line restoring force, not the centripetal pull that bends a path into a circle. :contentReference[oaicite:12]{index=12}

7. High-Yield NEET Nuggets 🥇

  1. Core Equation: \(x(t)=A\cos\bigl(ωt+φ\bigr)\), with \(ω=2π/T\).
  2. Restoring Rule: \(a(t)=-ω^{2}x(t)\)—acceleration is always opposite to displacement.
  3. Circle Trick: Any uniform circular motion’s diameter projection is SHM (great for visualizing phase, amplitude, and period).
  4. Max Values: \(v_{\text{max}}=ωA\) and \(a_{\text{max}}=ω^{2}A\).
  5. Phase Shifts: Cosine and sine forms of SHM differ by π/2—handy when adding or comparing waves.

Keep practicing, and soon SHM will feel as smooth as the swing of a playground 🛝!