Question

This time, you are asked to analyze the time dependent behavior of two masses (m, and m.) connected by a massless spring. You may assume that the spring is linear, has a spring constant k and a free length of L. That is if the spring is stretched to length L' > Lit exerts a compressive force of magnitude (L' L). However, if compressed, ie., L' <Lit exerts an expansion force of magnitude (L-1). In Newtonian Mechanics, motion of the 2 masses on a horizontal, frictionless plane can be expressed as: mis = k(x2 - 1-L) Im, iz = -k(t, - *;-) For x; < x2. Recall also that is To determine the time dependent position of each masses, we need to solve the above set of equations, which may be written better in the following form in Linear Algebra: m dr no 9]()-I :O)-* 16:2)+(35 This is equivalent to solving for the vector v(t) X2 a) Rewrite the equations in the following form: dez (t) = Mu(t) + w Identify the 2x2 matrix M and the time independent vector w. b) Find the eigenvalues i, > i, and their corresponding eigenvectors y, and of the M matrix. c) We know that any vector in the space can be written as a linear combination of, and us (they form a basis). Therefore, we may write v(t) = a(t), +b(t)uz, where aft) and b(t) represent the time dependent expansion coefficients. Obtain the set of equations a(t) and (t) satisfy. (NOTE: This is the crucial step, you should be able to obtain equations that are de-coupled). d) Show that (by proper normalization of the eigenvectors) a(t) represents the time dependent location of the center of mass of the two objects. e) Show that (by proper normalization of the eigenvectors) b(t) represents the-time dependent- distance between the two objects. Answers to parts dande involve the eigenvector of the adjoint of M, namely M. f) Given the initial conditions x:(0) = 0.x2(0) = 1.5L and (0) = 1,0) = 0 (that is initially they are at rest), determine a(t) and b(t). (NOTE: Because our equation was of 2M order in time, we needed to provide 2 sets of initial conditions). e) You should observe an oscillatory motion within the entire motion. Analyzing the frequency of this oscillation, recover the concept of "reduced mass". h) Given the initial conditions (0) = 0,x,(0) = 1.5L, determine the necessary initial conditions to be imposed on (and (0), so that a(t) = constant, Les, the center of mass is at rest. Can you identify its physical meaning?