Simple Harmonic Motion – Force Law, Motion & Energy 🚀
1. How the motion looks in time ⏱️
- Displacement: \(x(t)=A\cos(\omega t+\phi)\) — a comfy cosine ride starting at amplitude \(A\). :contentReference[oaicite:0]{index=0}
- Speed: \(v(t)=-\omega A\sin(\omega t+\phi)\) — reaches \(\pm\,\omega A\) at the mid-points. :contentReference[oaicite:1]{index=1}
- Acceleration: \(a(t)=-\omega^{2}A\cos(\omega t+\phi)\) — maxes out at \(\pm\,\omega^{2}A\) right at the ends. :contentReference[oaicite:2]{index=2}
- The trio shares the same period \(T\); speed leads displacement by \(\pi/2\) and acceleration by \(\pi\) (one-half and one full step ahead). :contentReference[oaicite:3]{index=3}
2. Meet the restoring force 💪
The push (or rather, pull) bringing everything back is
\(F(t)=-m\omega^{2}x(t)=-k\,x(t)\)
with \(k=m\omega^{2}\) and so \( \omega=\sqrt{k/m}\). The force always points toward the centre, earning the name restoring force. Because the force depends linearly on \(x\), we call this a linear harmonic oscillator. Real-life tweaks like \(x^{2}\) or \(x^{3}\) terms turn it into a non-linear oscillator. :contentReference[oaicite:4]{index=4}
Quick example: block between two springs 🌟
- Two identical springs (each constant \(k\)) tug a block of mass \(m\) from both sides.
- A tiny shift \(x\) makes the left spring stretch and the right one compress, giving forces \(-k x\) each.
- Total force: \(F=-2k x\) — still linear, just double strong! :contentReference[oaicite:5]{index=5}
- Period of the block: \(T=2\pi\sqrt{\dfrac{m}{2k}}\) (note the factor 2 in the denominator). :contentReference[oaicite:6]{index=6}
3. Energy ping-pong 🔄
The system’s energy keeps swapping form but stays constant overall:
- Kinetic: \(K=\dfrac12 m\omega^{2}A^{2}\sin^{2}(\omega t+\phi)\) — peaks at the centre, vanishes at extremes. :contentReference[oaicite:7]{index=7}
- Potential: \(U=\dfrac12 kA^{2}\cos^{2}(\omega t+\phi)\) — the mirror image: highest at the ends, zero in the middle. :contentReference[oaicite:8]{index=8}
- Both flip every \(T/2\), so energy exchange is twice as fast as the motion itself. :contentReference[oaicite:9]{index=9}
High-yield NEET nuggets 🎯
- The signature relation \(F=-k x\) and the handy link \(\omega=\sqrt{k/m}\).
- Phase game-plan: speed leads displacement by \(\pi/2\); acceleration leads by \(\pi\). :contentReference[oaicite:10]{index=10}
- Kinetic and potential energies oscillate with period \(T/2\). :contentReference[oaicite:11]{index=11}
- Spring-block-spring combo: remember \(T=2\pi\sqrt{m/2k}\) — a favourite MCQ! :contentReference[oaicite:12]{index=12}
- Linear vs. non-linear: real oscillators stray when higher-power terms (\(x^{2},x^{3}\ldots\)) sneak into the force. :contentReference[oaicite:13]{index=13}
Keep these ideas at your fingertips, and simple harmonic motion will feel like a smooth ride! 😎
