Answer to Question #88738 in Classical Mechanics for Shivam Nishad

Question

Write down the equation of motion of a
damped oscillator and explain the
significance of each term in the equation.
Write the conditions for heavy, critical and
weak damping.

Answer 1

Equation of motion of damped oscillator:

$$m\ddot{x} = -\gamma \dot{x} - kx$$

where m is mass, $\gamma$ is dumping constant, k is spring constant.

The $-kx$ term means spring force, applied to body, $-\gamma \dot{x}$ term means damping force, proportional to speed but applied in opposite direction.

If $\gamma^2-4km>0$ it calls heavy damping, because oscillator will exponentially relax to equlibrium position.

If $\gamma^2-4km=0$ if calls critical damping.

if $\gamma^2-4km<0$ it calls weak damping, and oscillator will move by sine law, exponentially relaxing.