Consider a linear array of N identical simple pendula of mass m and length I that are sus- pended from equal height points, evenly spaced a distance a apart. Suppose that each pendu- lum bob is attached to its two immediate neighbors by means of light springs of unstretched length a and spring constant K. The following figure shows a small part of such an array. Let x, = ia be the equilibrium position of the ith bob, for i 1,N, and let ,(t) be its horizontal displacement. It is assumed that /a I for all i. Demonstrate that the equation of motion of the ith pendulum bob is K ; = - v; + (V- - 2i + Vi+1). m Consider a general normal mode of the form ,() = [A sin(k x,) +B cos(k x,)] cos(w t- 4). Show that the associated dispersion relation is = 4 K sin (ka/2). m Suppose that the first and last pendulums in the array are attached to immovable walls, lo- cated a horizontal distance a away, by means of light springs of unstretched length a and spring constant K. Find the normal modes of the system. Suppose, on the other hand, that the first and last pendulums are not attached to anything on their outer sides. Find the normal modes of the system. wie K