For the elastic pendulum discussed in Example 5.14, and shown in Figure 5.19, derive the governing equations of motion using Lagrange's equation.

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Example 5.14 Hamilton's Principle for the Derivation of the Equations of Motion of an Elas- tic Pendulum Derive the equation of motion of the spring-pendulum system depicted in Figure 5.19. Demonstrate the utility of Hamilton's principle for the derivation of the equations of motion of complex dynamic systems. The system is assumed to have no dissipation. elle E a m Figure 5.19: A spring pendulum.