Consider a mass m that oscillates at the end of a spring having a spring constant
. The following second-order differential equation describes the vertical
of this spring-mass system.
Such a differential equation implies that mass
once started, will simply oscillate up and down forever! This differential equation neglects the influence of frictional forces. Let us assume that there
is a retarding force which is proportional to the velocity of motion where
is treated as a proportionality constant (
). The aforementioned differential equation now becomes the
There are three possible types of solutions which depend upon the relative size of
, including the following:
Critically damped if
Underdamped or oscillatory if
Which of the following differential equations can be used to describe one of these cases?