Reference no: EM132439862

Questions -

(a) Why is it that the eigenvalues of L^{^}_x and L^{^}_y are ±1? (Hint: Consider L^{^}_x o L^{^}_x and L^{^}_y o L^{^}_y.)

(b) Show that L^{^}_x and L^{^}_y commute.

(c) The result of Part (a) implies that we can use L^{^}x to divide the normal modes of the same frequency into 2 classes, one is symmetric under the operation of L^{^}x, i.e. L^{^}x ψ = ψ, and the other which is antisymmetric under the operation of L^{^}x, i.e., L^{^}xψ = -ψ. Similarly, one can say the same with L^{^}y. Using the result of Part (b), we thus arrive at the conclusion that the normal modes can be divided into four classes, each possessing a definite symmetry under the operation of either L^{^}x or L^{^}y. In this part through Part (e) we will work out explicitly the class which satisfies L^{^}xψ = ψ and L^{^}yψ = ψ. Explain why the solutions in this class must be of form ψ = (ξ, η, η, ξ, η, η).

(d) Referring to the figure above, explain why the equations of motion for Balls A and B are, respectively,

M d²ξ/dt² = -k₁(ξ- ξ) - k₂(ξ - η) - k₂(ξ - η) = -2k₂(ξ-η),

m d²η/dt² = -k₂(η - ξ).

(e) Find the normal mode frequencies and their associated normal modes (ξ, η) of the equations of Part (d).

(f) Write down the equations of motion for the class of solutions satisfying L^{^}xψ = ψ and L^{^}y ψ = - ψ.

(g) Find the normal mode frequencies and their associated normal modes (ξ, η) of the equations of Part (f).