Velocity & Acceleration in Simple Harmonic Motion

Velocity & Acceleration in Simple Harmonic Motion 🚀

Quick Recap of SHM ➡️

  • Displacement: \(x(t)=A\cos(\omega t+\phi)\)
  • Amplitude \(A\) is the farthest distance from the mean position.
  • Angular frequency \(\omega=\dfrac{2\pi}{T}\), where \(T\) is the period.

Instantaneous Velocity 🏃‍♂️

  • In a circle of radius \(A\) the speed is \(v=\omega A\).
  • Along the line of oscillation: \(v(t)=-\,\omega A\sin(\omega t+\phi)\) :contentReference[oaicite:0]{index=0}
  • The minus sign shows the direction is opposite to the positive \(x\)-axis whenever \(\sin(\omega t+\phi)\) is positive.
  • Maximum speed: \(v_{\text{max}}=\omega A\).

Instantaneous Acceleration ⚡

  • Differentiating once more (or using the reference circle) gives
  • \(a(t)=-\,\omega^{2}A\cos(\omega t+\phi)=-\,\omega^{2}x(t)\) :contentReference[oaicite:1]{index=1}
  • Acceleration always points toward the centre (restoring direction).
  • Maximum magnitude: \(a_{\text{max}}=\omega^{2}A\).

Restoring Force 💪

Newton’s second law links force and acceleration:

\(F(t)=m\,a(t)=-\,m\omega^{2}x(t)=-\,k\,x(t)\), where

  • \(k=m\omega^{2}\) and \(\displaystyle \omega=\sqrt{\dfrac{k}{m}}\). :contentReference[oaicite:2]{index=2}
  • The direct proportionality \(F\propto -x\) is the hallmark of a linear harmonic oscillator.

Phase Relationships 🔄

  • Velocity lags displacement by \(90^{\circ}\) (\(\pi/2\) rad).
  • Acceleration lags displacement by \(180^{\circ}\) (\(\pi\) rad). :contentReference[oaicite:3]{index=3}
  • Thus, when \(x\) is at its extreme, \(v=0\) and \(a\) is at its maximum toward the centre.

Worked Example 🎯

For the motion \(x=5\cos\,[2\pi t+\pi/4]\) (SI units):

  1. At \(t=1.5\;\text{s}\): displacement \(x=-3.535\;\text{m}\).
  2. Speed \(v=22\;\text{m s}^{-1}\).
  3. Acceleration \(a=140\;\text{m s}^{-2}\). :contentReference[oaicite:4]{index=4}

High-Yield Ideas for NEET 🌟

  1. \(a(t)=-\omega^{2}x(t)\) — acceleration is directly proportional and opposite to displacement.
  2. Restoring force \(F=-k x\) with \(k=m\omega^{2}\) and \(\displaystyle \omega=\sqrt{k/m}\).
  3. Maximum quantities: \(v_{\text{max}}=\omega A\) and \(a_{\text{max}}=\omega^{2}A\).
  4. Phase differences: velocity \(90^{\circ}\), acceleration \(180^{\circ}\) behind displacement.
  5. All three—\(x(t)\), \(v(t)\), and \(a(t)\)—are sinusoidal with identical periods.

Keep practicing—SHM problems ♥ love appearing on exams! 😊