A mass M is fixed at the right-angled vertex where a massless rod of length ? is connected to a very long massless rod (see Fig). A mass m is free to move frictionlessly along the long rod. The rod of length ? is hinged at a support, and the whole system is free to rotate, in the plane of the rods, about the support. Let ? be the angle of rotation of the system, and let x be the distance between m and M. Find the equations of motion. Find the normal modes when ? and x are both very small.

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M m