Problem $2.$ For the system shown in the figure:

$2\mathrm{.}1($Modeling$)$: Using the Newton's second law of motion, and derive the equation of motion for the system neglecting the friction and viscos damping. Then, derive a transfer function model from the input force $f$ to output position $x$:$P(s)=x\frac{s}{F}(s).$

$2\mathrm{.}2($Stability analysis$)$: Investigate if the system is open$-$loop stable or not. For this part, create a transfer function model for the system in MATLAB, and obtain its unit impulse response. This simulates an initial shock to the system. If the system's output stays bounded or returns to zero over time, the system is stable. If it keeps going toward $+/-$ infinity it is unstable.

What is your guess about the response of the system before checking it with MATLAB?

Use MATLAB's "impulse" command to obtain the system's unit impulse response:

sys Num, Den