Q$1($a$)$ The system shown in FigQ$1($a$)$ consists of two lumped translating masses, $m,$ at A and $B$ and a stepped disc of mass $2$ m rotating frecly about a horizontal axis passing through its bearing with its centroidal radius of gyration, $\frac{?}{\text{b}\text{a}\text{r}}(k)=\frac{1}{2}(r_1+r_2).$

$($i$)$ Simplify the given configuration using proper equations of motion, to a system consisting of an equivalent translating mass at $A.$ Hence find the expression of natural frequency, in unit of Hz $,$ of the system.

$($ii$)$ If now an external harmonic moment $M(t)=M_{0}sin(2{}_{n}t)$ is applied on to the stepped disk, then find the resulting motion of Block A from its mean position as a function of time, ${}_a$ being the natural frequency

Assume: $r_i=\frac{1}{2}{r}_2.$ Neglect any friction in the moving joints and all cables are inextensible being securely wrapped around the disk.

$10$