Problems for Intermediate Methods in Theoretical Physics 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.