## Quick Reference

*Oscillations in which the amplitude decreases with time. Consider the equation of motion *mẍ=−kx−cẋ*, where the first term on the right-hand side arises from an elastic restoring force satisfying Hooke's law, and the second term arises from a resistive force. The constants *k* and *c* are positive. The form of the general solution of this linear differential equation depends on the auxiliary equation *mα*^{2}+*cα*+*k*=0. When *c*^{2} < 4*mk*, the auxiliary equation has non-real roots and damped oscillations occur. This is a case of weak damping. When *c*^{2}=4*mk*, the auxiliary equation has equal roots and critical damping occurs: oscillation just fails to take place. When *c*^{2}>4*mk* there is strong damping: the resistive force is so strong that no oscillations take place.

http://www.lon-capa.org/~mmp/applist/damped/d.htm An applet exploring the effects of changing parameters on damped oscillations

*Subjects:*
Mathematics.