1. Two Coupled oscillators (60 pts) Three ideal, massless springs and two masses are attached to each other as shown in figure 1 The extreme on the left hand side and on the right hand side are attached to fix walls. Neglect the effect due to gravity and consider k1, k2, k12, m1, m2 > 0. K1 m1 K12 mz Kz X1 X2 Figure 1: System of three ideal, massless springs with two masses. k1, k2, k12, m1, m2 > 0 (a) Prove that the system in figure [ always has two well defined normal modes. (30 points) (b) What is the angular frequency of the oscillation for each normal mode if m, = m2 = m and k1 = k2 = k # k12? (15 pts) (c) What are the two normal modes if m, = m2 = m and ki = k2 = k # k12? Is the Center of Mass a normal mode? (10 pts) (d) What are the answers to the last two questions if ki = k2 = k12 = k, and m1 = m2 = m? (5 pts) Clarification added on 12/05: answers to 1b) and Ic) if k1 = k2 = k12 = k, and m1 = m2 = m.Optional: context and insightful tips (1) If it helps, notice that the case of my = m2 = m and k1 = k2 = k # k12 can be solved on its own without solving the general case. Similarly with the simplest case: m1 = m2 = m and ki = k2 = k12 = k. Also, remember that a normal mode is a linear combination of the positions of each mass such that it exhibits (uncoupled) SHO. (2) General case: If the system has two real valued angular frequencies, then due to linear algebra results learned elsewhere, the system has two linearly independent normal modes. Thus, what is needed is to show that there are in fact two real- valued frequencies. (3) In order to prove that, we do not need to explicitly derive the actual frequen cies, but show that they exist. A suggestion: once you write the equations of motion for the two masses (i.e., the equations for their accelerations), rewrite the system as: (:) - (2 B) ( 1) (1) For some A, B, C, D. Then, derive the two eigenvalues in terms of A, B, C, D (without using the actual expression in terms of the parameters of the prob- lem). You should end up with a second order polynomial whose discriminant is (. . . )2 + some product, and the product is clearly positive when evaluated in 2a 2022: Midterm 1 terms of the parameters of the problem. At this point, you'll have proven that the roots, i. e. the eigenvalues, are real. (4) Next, you need to show that both eigenvalues are negative, so that indeed one has two real-valued angular frequencies (i.e., the usual '-w2'). One of the eigenvalues is clearly negative from the general expression of the two roots and the signs of A and D. There's only one root that is not so clear. You have to show that the contribution from the '+ ' (in the roots of the second order polynomial) is not large enough to make this root positive. Once more, work as much as you can with A, B, C, D only. At the very end, substitute the actual expression of A, B, C, D in terms of the parameters of the system to prove that this root is also negative for any m1, m2, k1, k2k1 12. In this way you have proven the existence of two well defined angular frequencies faster and more easily than getting their actual explicit expression. In fact, the question does not ask about the actual expression of the angular frequencies in the general case. Of course, if you want to, you can derive the explicit expressions for the angular frequencies, but the important point is to prove that they are real-valued