THE FREE HANGING PENDULUM

Combine Equations

          p6_1

          p6_2

          p6_3

          p6_4

          p6_5

Assume pendulum is uniform rod with moment of inertia theta
Also assume small angle approximation sin( theta ) = theta


          p6_6

          p6_7

Let   p6_8a,       p6_8b


          p6_9

This is our characteristic equation of motion for the free hanging pendulum. Compare it with the characteristic equation for the inverted pendulum.
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