The assembly shown consists of two sheaves of mass m₁ = 35 kg and m₂ = 15 kg, outer groove radii r₁ = 525 mm and r₂ = 250 mm, and centroidal radii of gyration (k_O)₁ = 350 mm and (k_O)₂ = 150 mm. The sheaves are fitted to a central shaft at O with bearings which allow them to rotate independently of each other. Attached to the central shaft is a carriage of mass m₃ = 25 kg. Each sheave has an inextensible cable wrapped securely within its outer groove. Each cable is attached to a spring at one end and to a fixed support at the other end. The springs have stiffnesses k₁ = 800 N/m and k₂ = 650 N/m. By the method of this article, determine the equation of motion for the system in terms of the variable x and state the period τ for small vertical oscillations about the equilibrium position. Neglect friction in the bearings at O.

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