2. In the cart and pendulum system shown, the pendulum is made of a small sphere of mass m and a light rod of length / which rotates freely about O. (25 points) a) Using x and as the generalized coordinates and assuming small angle 0, derive the linear equations of motion of the system. b) For M3m = 3 kg, l = 1.2 m, and k = 40 N/m, determine the natural frequencies and amplitude ratios of the system. Answer: The linearized equations of motion are: ((M +m)x +ml + kx = 0 mlx + ml + mgle = 0 M wwwww l 10 m 3. Consider the cart and pendulum system in problem 2. A harmonic force F(t) = Fo sin 5t is applied to the cart as shown. (25 points) a) Using the parameters given in problem 2-b determine the range of the forcing amplitudes Fo for which the amplitude of oscillations of the pendulum 8. b) It is intended to use the pendulum as a vibration absorber for the system. If Fo = 1 N, and using the parameters given in problem 2-b (except 1), determine the length 1 for which the cart does not move, i.e., X = 0. Determine the amplitude of oscillations of the pendulum. Is the small angle assumption still valid? k M wwwwww x l m F(t) Fo 2.577 N 10.392 m 1 0 = 9.81 rad or 5.84 degrees