Repeat Problem 5, using Lagrange's equations.

Consider the mechanical system shown in Figure 5.92, in which a simple pendulum is pivoted on a cart of mass \(m\) and is constrained to rotate in a vertical plane. The pendulum consists of a point mass \(M\) concentrated at the tip of a massless rod of length \(L\). Assume that the friction between the cart and the horizontal surface cannot be ignored. Denote the displacement of the cart as \(x\) and the angular displacement of the pendulum as \(\theta\). Draw the necessary free-body diagram and kinematic diagram, and derive the equations of motion for small angles.

FIGURE 5.92 Problem 5. 0000 m L x M