Problem $2.$ The figure below shows a pulley system consisting of a mass, an inextensible $($cannot

change in length$)$ cable, two massless pullies and two brackets. In this system, the brackets are

made of soft materials. Thus, they can deform axially. Therefore, we model each of these brackets

as spring with stiffness $k.$ Find:

a$.$ The equation of motion of the system.

b$.$ The natural frequency of the system.

This problem is a bit more involved than what it first seems. You effectively have three vertical

motions allowable:

$x(t)$: the vertical motion of the box

$y(t)$ : the vertical motion of the left pulley

$z(t)$ : the vertical motion of the right pulley