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SHM · Lecture 21 of 26

Damped oscillations: why the amplitude dies out

By Lalit, IIT Madras

Damped oscillations in SHM for JEE: the damping force F = −bv, an amplitude falling as e^(−bt/2m), and critical damping at b² = 4mk.

Watch lecture 21

A chalkboard timeline from 0 s to 15 s in three 5 s steps, each multiplying the amplitude by 0.9 (A₀, 0.9 A₀, 0.81 A₀, 0.729 A₀), beside the boxed result α = 0.9³ = 0.729.

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What this lecture covers

Falling amplitude means falling energy, taken out by a non-conservative force like air drag. With F = −bv, Newton's law gives m x'' + b x' + kx = 0, whose solution has amplitude A₀ e^(−bt/2m) and ω' below ω0. Energy falls as e^(−bt/m), faster than the amplitude, and critical damping at b² = 4mk ends the oscillation. Numbers for one block give halving times, and a drop to 0.9 A₀ in 5 s is solved by equal fractional loss in equal times.

In this lecture

  1. Play from A falling amplitude is falling energy
  2. Play from The damping force F = −bv
  3. Play from Damped block: equation and solution
  4. Play from Smaller ω; energy falls faster
  5. Play from Critical damping, b² = 4mk
  6. Play from The numbers of one damped block
  7. Play from Time for amplitude or energy to halve
  8. Play from The average life time of the amplitude
  9. Play from The 0.9 question: equal fractional loss
  10. Play from Recap: damped oscillations