Two masses \(m_{1}\) and \(m_{2}\), each connected by two springs of stiffness \(k\), are connected by a rigid massless horizontal rod of length \(l\) as shown in Fig. 5.22.

(a) Derive the equations of motion of the system in terms of the vertical displacement of the C.G. of the system, \(x(t)\), and the rotation about the C.G. of the system, \(\theta(t)\).

(b) Find the natural frequencies of vibration of the system for \(m_{1}=50 \mathrm{~kg}, m_{2}=200 \mathrm{~kg}\), and \(k=1000 \mathrm{~N} / \mathrm{m}\). Assume \(l=1 \mathrm{~m}\).

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k m x(t) x(t) Rigid, massless rod 8(1) m2 x2(1) 7777 FIGURE 5.22 Two-mass-spring system.