A vibratory system consists of mass $m$, a vertical spring of stiffness $2k$ and a horizontal spring of stiffness $k$. The end $A$ of the horizontal spring is given a horizontal motion $x_A = a \sin \omega t$. The other end of the spring is connected to an inextensible rope that passes over two massless pulleys as shown.

Assume $m = 10 kg$, $k = 1.5$ kN/m, and neglect friction.

The magnitude of critical driving frequency for which the oscillations of mass $m$ tend to become excessively large is _____ rad/s (answer in integer).

A spring mass damper system (mass m, stiffness k, and damping coefficient c) excited by a force F(t) = B sin ωt, where B, ω and t are the amplitude, frequency and time, respectively, is shown in the figure. Four different responses of the system (marked as (i) to (iv)) are shown just to the right of the system figure. In the figures of the responses, A is the amplitude of response shown in red color and the dashed lines indicate its envelope. The responses represent only the qualitative trend and those are not drawn to any specific scale.

Four different parameter and forcing conditions are mentioned below.

(P) c > 0 and $ω=\sqrt{k/m}$

(Q) c < 0 and ω ≠ 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 →) of the parameter and forcing conditions to the responses?

Consider a forced single degree-of-freedom system governed by $\rm \ddot x(t) + 2 ζ ω_n \dot x (t) + ω_n^2 x(t) = ω_n^2 \cos (ω t)$, where ζ and ω_n are the damping ratio and undamped natural frequency of the system, respectively, while ω is the forcing frequency. The amplitude of the forced steady state response of this system is given by [(1 − r²)² + (2ζr)²]^-1/2, where 𝑟 = ω/ωn. The peak amplitude of this response occurs at a frequency  ω =  ω_p. If  ω_d denotes the damped natural frequency of this system, which one of the following options is true?

The radius of gyration of a compound pendulum about the point of suspension is $$100$$ mm. The distance between the point of suspension and the centre of mass is $$250$$ mm. Considering the acceleration due to gravity as $$9.81$$ m/s², the natural frequency (in radian/s) of the compound pendulum is _________.