Energy in Simple Harmonic Motion 🌟

In simple harmonic motion (SHM), energy keeps shuffling between kinetic and potential forms while the total stays fixed—like coins moving between two pockets but never leaving your jeans! 😄

1. Kinetic Energy (K) 🚀

K comes from speed (v) and peaks at the center (x = 0). It is

$$K = \tfrac12 m v^2 = \tfrac12 k A^2 \omega^2 \sin^2(\omega t + \phi)$$ :contentReference[oaicite:0]{index=0}

  • Zero at extreme positions where the object turns around.
  • Repeats every \(T/2\) seconds because the sine-square term does two full wiggles in one period. 😉 :contentReference[oaicite:1]{index=1}

2. Potential Energy (U) 🏔️

Spring stretch or compression stores energy:

$$U = \tfrac12 k x^2 = \tfrac12 k A^2 \cos^2(\omega t + \phi)$$ :contentReference[oaicite:2]{index=2}

  • Zero at the center, maximum at the extremes (\(x = \pm A\)).
  • Also repeats every \(T/2\) seconds, but it is out of step with K—when one climbs, the other slides. 🔄 :contentReference[oaicite:3]{index=3}

3. Total Mechanical Energy (E) 💡

Add K and U and all the time-dependence cancels:

$$E = K + U = \tfrac12 k A^2$$ :contentReference[oaicite:4]{index=4}

This constant value shows conservation of mechanical energy in SHM.

4. How the Swap Happens 🔄

  • Center (x = 0): all energy is kinetic, speed tops out. :contentReference[oaicite:5]{index=5}
  • Extremes (\(x = \pm A\)): speed drops to zero; energy sits entirely as potential. :contentReference[oaicite:6]{index=6}
  • Between these points, one form climbs while the other falls, but their sum stays unchanged. 🔑
  • K and U stay positive and peak twice each period. :contentReference[oaicite:7]{index=7}

5. Quick Example 🌟

Block of mass \(1\,\text{kg}\) on a spring (\(k = 50\,\text{N m}^{-1}\)), amplitude \(A = 0.10\,\text{m}\).

  1. Angular frequency: $$\omega = \sqrt{\tfrac{k}{m}} = 7.07\,\text{rad s}^{-1}$$ :contentReference[oaicite:8]{index=8}
  2. When the block sits \(0.05\,\text{m}\) from center:
    • Speed \(v = 0.61\,\text{m s}^{-1}\).
    • \(K = 0.19\,\text{J}\).
    • \(U = 0.0625\,\text{J}\).
    • Total \(E = 0.25\,\text{J}\).
    :contentReference[oaicite:9]{index=9}
  3. Total energy equals \(\tfrac12 k A^2 = 0.25\,\text{J}\), matching the sum above—nice check! ✔️ :contentReference[oaicite:10]{index=10}

6. Double-Spring Special 🎸

Two identical springs pulling a mass from opposite sides give a net force \(F = -2 k x\), so the period shrinks to

$$T = 2\pi \sqrt{\tfrac{m}{2k}}$$ :contentReference[oaicite:11]{index=11}

This trick often appears in problems that ask “What happens if you add another spring?”.

High-Yield Ideas for NEET 🎯

  • Memorize the energy formulas \(K\), \(U\), and constant \(E\).
  • Remember: K and U repeat every \(T/2\), while displacement and speed repeat every \(T\).
  • At center → all \(K\); at ends → all \(U\).
  • Energy graphs: K and U double peaks per cycle, E is flat.
  • Multi-spring setups change the effective spring constant, so expect different periods. 😎