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SHM · Lecture 23 of 26

Underdamped, overdamped and critically damped motion

By Lalit, IIT Madras

Underdamped, overdamped and critically damped motion in JEE SHM, sorted by the roots of one quadratic, with x(t) worked out for each case.

Watch lecture 23

A chalkboard titled 'Overdamped: large b' shows a block of mass m on a spring fixed to a wall, with notes that going out the velocity points right while the resistance points left, coming back the velocity points left while the resistance points right, so the resistance opposes the motion both ways and the energy keeps falling.

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What this lecture covers

Substituting a trial solution turns the damped equation into a quadratic whose roots split the motion into three cases by the damping factor. Complex roots, handled with e to the power i θ, mean underdamping: a decaying oscillation whose amplitude has a half-life and a mean life. Two real roots mean overdamping, a slow creep back with no to-and-fro. Equal roots mean critical damping. Worked blocks give x(t), the last one with its graph and farthest point.

In this lecture

  1. Play from From the damped equation to its roots
  2. Play from Three cases from the roots
  3. Play from Underdamped: complex roots
  4. Play from Half-life and mean life of amplitude
  5. Play from Worked problem: the underdamped block
  6. Play from Overdamped: no to-and-fro motion
  7. Play from Worked problem: the overdamped block
  8. Play from Critically damped: one repeated root
  9. Play from Worked problem: critically damped block
  10. Play from The three cases together, and a recap