Question

I. COUPLED OSCILLATIONS Consider the system below with two degrees of freedom (neglect gravitation). Denote the displacement of each of the particles with respect to equilibrium by óy (t), i= 1,2. 1. Find the Lagrangian describing the system 2. Write down the coupled equations of motion for óyn (t) and by2(t) 3. Find the 2 x 2 matrices Ť and V and solve the normal mode equation for w: det(V-2T) 0. 4. Compute the form of the eigenvectors (normal modes), and write down the general solution for dy(t), i= 1,2 describing the position of each of the particles as a function of time. Your expression should involve 4 constants that are fixed by the initial conditions. 5. Find oy (t), 6y2(t) in the case where the initial velocities are zero ón (t 0) õýz(t = 0) = 0, and the initial displacements are given by óy (t 0)= 0 and 6y2(t 0) d. wimtwn

Answer 1

_ The potential for the abobe configuration is VE - mph +! K६y+ ! A591-89- mg 8ya- 7=1m salmeye- 4 = msg + m sy: + mgSa+mgey-y (£५jiza sy - key, ५ 2 coupled equation of motion F 14 - 0 mlsy) = mg =k84 - kya - Similarly for y ...m ( sy) = m3-2KBy =ksy--

vijf v - gosytSyZ [2k --- t-r kl _ak k-mwa Im01 temt u v-w² T ] = det / 2k - mw? —-kk- = (2k-m w ²) (e-man) - k²= 6 21² - 3kmw² m2 w" - k²=0 m² wh - 3k mw² tk²=0 WW2 = 3K I ISK = (31 sk Let w, = 3 Is Wars tv-w274= o for altermining eigenrector - 2x=19_(31 15) k k 1947 I -k. R-BI(5) 192 1962-+)) = azad a ( Irs) = a 2m am g ml 2

a) (²1) for wt (a) = 26 fowa Stagiunta e sy, (t) = a cosat +8) + A Los (Wetto) 8 since for Wt azt a bola, Sy(t) = a, cos (wpt78) (1-1) + a₂ cos/w-tto) It's