10. Oscillations

Oscillations Simple Harmonic Motion (Free Oscillations) – Displacement of the object is proportional to its acceleration, with the acceleration pointing towards the equilibrium position. Horizontal Spring Mass Vertical Spring Mass Pendulum Note: Equilibrium is NOT at unstretched position Assume very small displacement. Arc length ≈ horizontal x SHM Quantities and their variations with respect to displacement and time. Quantity Variation w.r.t Displacement Variation w.r.t Time Equation Graph Equation Graph Acceleration Max. acceleration 2ax  2 00ax      0 0 a a sin t a a cos t     Velocity Max. velocity 22 0v x x  00vx      0 0 v v cos t v x sin t     Displacement N.A N.A.     0 0 x x sin t x x cos t     Kinetic Energy   2 2 2 k0 1E m x x2      22 k0 22 k0 1E mv cos t2 1E mv sin t2     Potential Energy 22 p 1E m x2      22 p0 22 p0 1E mv sin t2 1E mv cos t2     Total Energy k ,max p,max 22 0 TE E E 1 mx2    Constant value throughout time Angular Frequency 2 2ftT      Period, T Frequency, f 1f T Equilibrium Also the unstretched position 1 2 3 4 5 Unstretched position Equilibrium x0 x0 1 2 3 4 5 x0 x0 1 2 3 4 5 a ao -ao t v -vo t vo x xo -xo t

Damped Oscillations – Loss of total energy gradually with time  Amplitude decreases Types of Damped Oscillations Graphs Light Damping or Underdamped  Still oscillating (negative and positive x)  Period is more than undamped case, but remains constant throughout oscillation Heavy Damping or Overdamped  Takes a long time (or never) reach equilibrium Critical Damping  Returns to equilibrium in the fastest possible time Resonance – When driving frequency matches the natural frequency, maximum energy is transferred to the system, hence system oscillate at the maximum amplitude. Frequency Response Curve More Damping  All amplitudes of oscillations decrease  Resonance frequency shift left Less Damping  All amplitudes of oscillations increase  Resonance frequency shift right Ideal case with no damping  All amplitudes of oscillations increase  Reach theoretical infinite amplitude at resonance frequency Use for 1. Car Suspensions 2. Analog meters 3. Door closers