Problem No$.1(40$ points$)$

Consider the system shown in Figure $1.$ Using any analytical dynamics method covered in class $($i$.$e$.$ Lagrange's equation, Newton, etc. but just one method$)$ and determine the equations of motion $($put the final equations of motion in matrix form$).$ The fixed T $-$bar assembly rotates about the vertical axis and has a moment of inertia J$.$ The bar connecting the two collars has length $L$ and a mass of $4m.$ The two collars with mass $m$ can move along the T$-$bar assembly $($one in the vertical direction and the other in the horizontal direction$).$ The T$-$bar assembly is acted on by a nonstationary torsional load $M(t)$ resulting in a rotation given by ${}^{}.$

Figure $1$: System set$-$up for Problem $1.$