Targ S.M. 1988 D5 option 08

The mechanical system consists of stepped pulleys 1 and 2 weighing P₁ and P₂ with step radii R₁=R; r₁=0.4R; R₂=R; r₂=0.8R; R=0.25m (the mass of each pulley is considered uniformly distributed along its outer rim), loads 3; 4 and a continuous homogeneous cylindrical roller 5 weighing P₃; P₄; P₅ respectively. Loads slide on planes without friction, and rollers roll without slipping. In addition to the forces of gravity, a constant force F acts on the pulleys during their rotation, constant moments of resistance equal to M₁ and M₂, respectively. Compose the Lagrange equation and determine the value indicated in the table.