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M 4 (50 points) - Differential Equations Figure 2: Pendulum A simple pendulum consists of a mass, M kilograms, attached to the end of a light rod of length L meters, whose other end is fixed. If we let 8 radians denote the angle the rod is displaced from the vertical at time t, we can fin motion of the pendulum using Newton's second law. (a) Obtain the equation of motion of the pendulum using force balance and Newton's second law (Hint: Consider also the tangential component of the force.). Please note that equation of motion is supposed to be a function with respect to time. (b) Once you have obtained the equation of motion, you will encounter a NON EAR D.E. However, if you recall the MacLaurin series expansion, you can linearize the D.E. Convert the Non-Linear D.E. into simple linear D.E. by using series expansion approximation. (c) Using above information, determine the resulting motion of the pendulum if the length of the pendulum is 0.5 m which is displaced an angle 0.1 rad from the equilibrium position and released from rest. Sketch the graph of the motion equation for necessary time interval.