Problem $2(35$ points$)$:

A vibration$-$isolation block is to be installed in a laboratory so that the vibration from adjacent factory operations will not disturb certain experiments, as presented in Figure $9.$ If the isolation block weighs $100$ kg and the surrounding floor and foundation vibrate at $\frac{18}{}Hz$ with an amplitude of $0\mathrm{.}003$ m $.$

$1-$ Determine the equations of motion and natural frequency in terms of $m,c,k,Y,$ and $(7$ points$)$

$2-$ For an undamped system $(c=0),$ determine the value $k$ of the isolation system, in $\frac{N}{m},$ such that the isolation block will have a particular solution amplitude of only $0\mathrm{.}0005$ m $.(8$ points$)$

$3-$ For a damped system, with the value of $k={3\mathrm{.}468}^{**}{10}^4\frac{N}{m},$ determine the damping coefficient $c$ that will allow us to have the same particular solution amplitude $(0\mathrm{.}0005m).(10$ points$)$

$4-$ Suppose an additional component is rigidly attached to the machine. This component adds an additional mass of $50$ kg $.$ Determine the particular solution amplitude of oscillation for the undamped system $(c=0)$ with this additional mass when $k=3\mathrm{.}7029**{10}^4\frac{N}{m}.(10$ points$)$

Problem $2$

$12-$id:EOM:

$\{\begin{array}{c}\text{:}{F}_0=\frac{y}{m}\end{array}$

$2.)$

$k=m*{}_0^2$

$K=100*(\frac{18}{}{)}^2$

$k=$

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