Problem $1(28$ points$)$

@ A uniform rigid rod of mass $m=1kg$ and length $L=1m$ is hinged at its center and laterally supported at one end by a spring of spring constant $k=300\frac{N}{m}.$ The natural frequency $-10$

$20$

$30$

$40$

@ A mass of $1$ kg is attached to two identical springs each with stiffness $k=20kN,$ shown in the figure. Under frictionless condition, the natural frequency of the system in Hz is close to:

$32$

$23$

$16$

$11$

@ The natural frequency of the spring mass system shown in the figure is close to

$8$ Hz

$10$ Hz

$12$ Hz

$14$ Hz

@ A machine of $250$ kg mass is supported on springs of total stiffness $100k\frac{N}{m}.$ Machine has an unbalanced rotating force of $350$ N at speed of $3600$ rpm $.$ Assuming a damping factor of $0\mathrm{.}15,$ the value of $Tr($transmissibility$)$:

$0\mathrm{.}0531$

$0\mathrm{.}9922$

$0\mathrm{.}0162$

$0\mathrm{.}0028$

Please solve all of them