Question

A uniform thin rod of mass m and length 5b is suspended from the ceiling by a single string of length 2b attached to one end (this end of the rod is called P1). Thus in equilibrium the string and rod hang straight down and the lower end of the rod is a distance 7b from the ceiling (this end of the rod is called P2). The motion is constrained to be in a fixed vertical plane, so we have a two degree-of-freedom problem. We define a fixed xy coordinate system in this plane, with its origin at the point where the string is attached to the ceiling and with the y axis pointing straight down. As generalized coordinates we use xi and x2, the x coordinates of the points P, and P2 (a) Draw a sketch of the apparatus and the coordinate system, with the rod near but not at equilibrium. (b) Calculate the potential energy of the system exactly. (c) Now we study small oscillations. Calculate the M and K matrices that are used in the study of small oscillations about the equilibrium. (d) Calculate the eigenfrequencies (e) Calculate corresponding eigenvectors. You do not have to normalize them.

Answer 1

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