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 _________.

A thin uniform rigid bar of length $$L$$ and mass $$M$$ is hinged at point $$O,$$ located at a distance of $${L \over 3}$$ from one of its ends. The bar is further supported using springs, each of stiffness $$k,$$ located at the two ends. A particle of mass $$m = {m \over 4}$$ is fixed at one end of the bar, as shown in the figure. For small rotations of the bar about $$O,$$ the natural frequency of the systems is

A solid disc with radius a is connected to a spring at a point $$d$$ above the center of the disc. The other end of the spring is fixed to the vertical wall. The disc is free to roll without slipping on the ground. The mass of the disc is $$M$$ and the spring constant is $$K$$. The polar moment of inertia for the disc about its centre is $$J = {{M{a^2}} \over 2}$$

The natural frequency of this system in rad/s is given by