Edward F. Redish
(a) The two orthonormal unit vectors, form a basis for a two-dimensional vector space. Two other orthonormal unit vectors, also form a basis for the same 2D space. In these two bases, a vector is represented by the expressions and respectively. If show how you would express the new coordinates, x' , y' , in terms of the old coordinates, x, y .
(b) Two equal masses are connected to each other and to two external walls by non-identical springs. The displacements of the two masses of the system are described by a two-component vector, . The eigenvectors for the normal modes of the system are and . If the initial displacements of the masses at t = 0 are y1(0) = 10 cm and y2(0) = -5 cm, how can the initial displacements of the masses be written as combinations of the normal modes?
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Last revision 2. December, 2005.