$6.$ A pulley with a radius $\backslash ($ r $\backslash )$ and a mass moment of inertia $\backslash ($ I $\backslash )$ is connected to a rotational spring $\backslash ($ k$\_\{\backslash$theta$\}\backslash )$ as shown in Figure Q$6.$ It supports a mass $\backslash ($ m $\backslash )$ via a spring $\backslash ($ k $\backslash ).$ Use the coordinates $\backslash ($ x $\backslash )$ and $\backslash (\backslash$theta $\backslash )$ to obtain the equation of motion of the system using the Lagrange method. What is the type of the coupling in the system?

$[8$ marks$]$

Pulley, mass moment of inertia $\backslash ($ I $\backslash )$

Figure Q$6.$ The pulley and mass system.