THE FREE HANGING PENDULUM

Combine Equations

          p6_1

          Substitute previous values for V and H:

          p6_2

          Expand:

          p6_3

          Terms cancel:

          p6_3a


          p6_3c


          p6_4a


          Note: p6_4a = 1 (Pythagorean identity)


          p6_4


          Move all terms to left side of =

          p6_5

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


          p6_6


          Divide both sides by p6_6a


          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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