Answer to Question #95893 in Mechanics | Relativity for Malex

Question

Describe the equation mx + cx + kx = 0?. And does this equation has any implications on the wave motion?.

Answer 1

The equation

$$mx+cx+kx=x(m+c+k)=0,$$

in fact, does not have any common widely known physical sense. Thus, it does not have any implications on the wave motion. However, if we consider the equation of damped free vibrations of a system with viscous damping, it may look similar to yours:

$$m\ddot{x}+c\dot{x}+kx=0,$$

where

m - mass of a body oscillating on a spring, kg;

c - viscous damping coefficient, kg/s;

k - spring constant, N/m.

The critical damping coefficient is defined as

$$c_c=2m\sqrt{\frac{k}{m}}.$$

The oscillations are heavily damped when $c>c_c,$ critically damped at $c=c_c$ and lightly damped if $c<c_c$. When the system experiences heavy and critical damping, it comes to the state of rest without oscillations, and light damping makes the oscillating body asymptotically decrease its amplitude.