Free Sliding Bead A bead is constrained to slide along a rod of length L. The rod is rotating in a vertical plane with a constant angular speed, a), about a pivot in the middle of the rod. The pivot allows the bead to freely slide along the rod, i.e. the pivot does not impede the movement of the bead. Let r03) denote the distance of the bead away from the pivot where Ht) can be positive or negative. W {6' Equation of Motion Applying Newton's second law provides a balance of forces due to gravity, friction, centripetal acceleration, and linear acceleration. The equation resulting from these forces is d2?\" dr 2 in? + ,8 E mm 1' = mg sin(wr) where m is the mass ofthe bead, [3 is the coefficient ofviscous damping, w is the constant speed of angular rotation, g : 9.81 'm/s2 is the acceleration due to gravity, and r is the distance between the pivot and the bead. The rod is initially horizontal, and the initial conditions for the bead are r(0) : re and r'(0) : v0. Consider the frictionless rod, i.e. E : 0. The equation ofmotion becomes d2?\" 2 _ In? mm 1' = mg sm(mt) with g : 9.81 m/s2 and a constant angular speed a). The rod is initially horizontal, and the initial conditions for the bead are 1(0) : re and r'(0) : 190. You will need to write an Improved Euler Method system solver to nd r(t) and 19(5) A) Numerically solve for r(t) when a) = 2, r0 = 0, and v0 = 2.40, 2.45, 2.50. Solve in the time interval . 1 1 1 t E [0,5]. Use step Slzes h = E, E' E and compare your results. Also, compare your best numerical answers with your analytic answers from Problem 1 part E). B) Numerically solve for t) when a) : 2, r0 : 0, and v0 is selected to give simple harmonic motion, i.e. 1 1 1 Problem 1 part B. Use small step sizes, e.g. h : , , , etc. Solve for the longest time interval 512 2048 8192 that provides reasonable values for r(t). Compare your results to the analytic solution that gives simple harmonic motion. What does this demonstrate about numerical solutions