The Wilberforce pendulum (American Journal of Physics, 59, 32-38 (1991)), is illustrated in Figure 3. The pendulum has two modes of oscillation: vertical and torsional motion. The Lagrangian of the system is L=m()+(-kz² - 180² - 20 Where m and I are mass and rotational inertia of the bob, k and 8 are the longitudinal and torsional spring constants, and ɛ is the coupling constant between the modes. Typical values (m=0.5 kg, 1-10 kg.m2, k=5 N/m, 8 = 103 N.m, and x = 102 N. Find the equation of motion and write a program to compare z(t) and e(t) using rk4. 1 2eqution Show that when the longitudinal frequency, f. =√, equals the torsional frequency, fo= 2π == 2π m the motion periodically alternates between being purely longitudinal and purely torsional. FIGURE 2