$(2)$ Lever and wheel with pulley:

Spring $k,$ lever lengths $L_1,L_2,$ lever angle $(t)($assume small angle$),$ uniform cylinder

wheel has radius $r,$ mass $m,$ rotation $(t)($not small angle$),$ wheel rotation $$ and position $x(t)$

start from zero $($initial deformation of spring is $0),$ lever rod has negligible mass and moments

of inertial, external force $f(t)$ applied to lever as shown. Find the single equation of motion of

the system in terms of $x(t).$