Consider the mechanical system shown in Figure 5.129, in which a block of mass \(m\) is connected to a spring of stiffness \(k\), and it can slide without friction inside a vertical tube. Assume that the tube pivots at the joint O, and it can be approximated as a uniform rod of mass \(M\) and length \(L\). The unstretched length of the spring is \(L / 2\). Denote the angular displacement of the tube as \(\theta\) and the relative displacement of the block with respect to the tube as \(x\). Derive the nonlinear equations of motion by using

(a) the energy approach and

(b) the force/moment approach.

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0000 m x FIGURE 5.129 Problem 11. M, L