Solution of characteristic equation of motion for the free hanging pendulum

          p6_9

Assume a solution of the form   p7_2a
p7_2b
p7_2c


           p7_3

           p7_4

           p7_5

           Let  p7_6

           p7_7a            p7_7b


thetat  =  p7_8
 
 =  p7_9
 
 =  p7_10
 
 =  p7_11
 
 =  p7_12
 

Here p7_asub1, p7_asub2, and B are constants dependant upon initial conditions. In general the response  thetat   will be damped sinusoidal oscillations with a frequency omegd2p  and damping envelope of  dampenvl
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