Two masses, m and 2m, are suspended from a fixed height by elastic springs as shown below. Each spring has a spring constant k and motion is only in the vertical direction. g is the magnitude of the acceleration due to gravity. The displacements of the masses are labelled y1 and y2 as shown in the diagram. The natural lengths of the springs are l1 and l2.
(i) Show that Lagrange's equations for this system are:
2mÿ1 + 2ky1 - ky2 = 2mg + kl1 - kl2,
mÿ2 + ky2 - ky1 = mg + kl2.
(ii) Find the equilibrium positions of the suspended masses and show that the displacements of the masses relative to their equilibrium positions, labelled y1' and y2' (note that the primes here do not indicate differentiation), satisfy:
2mÿ1' + 2ky1' - ky2' = 0,
mÿ2' + ky2' - ky1' = 0.
(iii) Seek solutions of the form y1' = Ae^(iwt) and y2' = Be^(iwt), where A and B are constants and w is the angular frequency of a normal mode of oscillation. Show that with these solutions, the above system of equations can be written as:
Cb = 0,
where b = [A, B]^T and C is a matrix of coefficients that you must determine.
(iv) By solving det C = 0 for w, find the two angular frequencies for the normal modes of oscillation.