GATE ME 2022 Set 2 — Question 40

A spring mass damper system (mass $m$, stiffness $k$, and damping coefficient $c$) excited by a force $F(t) = B \sin \omega t$, where $B$, $\omega$ and $t$ are the amplitude, frequency and time, respectively, is shown in the figure. Four different qualitative responses of the system (amplitude $A$ of the response versus time $t$, starting from rest) are described below. They indicate only the qualitative trend of the envelope of the oscillation.(i) The envelope grows from zero but with decreasing slope and levels off to a constant steady amplitude.
(ii) The envelope grows linearly with time without bound (two straight lines diverging from the origin).
(iii) The amplitude rises and falls periodically: the envelope is a slowly varying periodic wave (beats).
(iv) The envelope grows with increasing slope (exponentially) without bound.Four different parameter and forcing conditions are mentioned below.
(P) $c > 0$ and $\omega = \sqrt{k/m}$
(Q) $c < 0$ and $\omega \neq 0$
(R) $c = 0$ and $\omega = \sqrt{k/m}$
(S) $c = 0$ and $\omega \cong \sqrt{k/m}$Which one of the following options gives correct match (indicated by arrow $\rightarrow$) of the parameter and forcing conditions to the responses?