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A thin rod of length 2b and mass m is suspended by its two ends with two identical vertical springs (force constant k) that are attached to the horizontal ceiling. Assuming that the whole system is constrained to move in just the one vertical plane, find the normal frequencies and normal modes of small oscillations. Describe and explain the normal modes. [Hint: It is crucial to make a wise choice of generalized coordinates. One possibility would be, r, (phi) and (alpha), where r and (phi) specify the position of the rod's CM relative to an origin half way between the springs on the ceiling, and (alpha) is the angle of tilt of the rod. Be careful when writing down the potential energy.]

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