1. **General formulas. In this problem, you will relate the general formulas given above to the case of two identical masses connected by three identical springs that is described in Sec. 11.2. (a) Consider the general solution (before any initial conditions are specified). What are n, q, A, and Afor this system? What are the wx? (You do not need to solve for any of these quantities-just figure out the correspondence between the quantities in the book and the general formalism.) (b) For initial conditions, assume both carts begin in their equilibrium positions, that cart 1 starts from rest, and that cart 2 has speed vo to the right. What are the /x and ix? Write explicit formulas for (t), &(t), ri (t), and x2(t). (c) Use a computer to graph the solution for r, (t) and I2(t). Take vo = k/m = 1 and 0 Akjqi(t ), (3) j=1 or, when written in matrix form for all of the normal modes, E = Aq, (4) where A is the matrix of normal coordinates (eigenvectors) for the system. To relate the generalized coordinates q'to the normal modes, invert Eq. 4, (5) where A-is the inverse matrix of the square matrix 1. The constants Sk = /x + in, are determined by the initial conditions. From the above relations, it is easy to show that they are HK = Axja; (0) (6) and VK 1 Akja; (0 ), (7) Wh 1 1 where q;(0) is the initial position of the jth generalized coordinate and 4,(0) is its initial ve- locity. With fx and v's known, the normal modes are completely determined. Substitution into Eq. 5 gives the equations of motion in terms of the original generalized coordinates