Solutions To Pendulum Libration Problems

 Problem:

The motion of the  pendulum below is studied as part of a celestial mechanics course, where   c  is the critical torque for which  f =  p /2.

a) Write an appropriate equation for the critical torque value using the parameters: m (for pendulum mass), R for radius of motion, and g (acceleration of gravity).

Solution:

Use the torque eqn. (See Aug. 12 post): 

Torque  t  =  FR  = mg R

b) If the value of  c   is exceeded then the applied torque becomes larger than the restoring torque. Write an inequality expressing this condition.

Solution:

N >  mg R sin  f (for all angles f )

c) Write out one  plausible form for the integral: òÖA dt.

Solution :

ò ÖA dt = ò d f/ [Ö1 - k sin  ² f ]

We have the case for k = 1 which means that as f  ®  3p /2

ò ÖA dt =  ò d f/ [Ö1 -  sin  ² (3p /2 )]

d) To have libration we need k > 1. Write an appropriate integral for this.

Solution:

We need k > 1  and sin f »   f 

 Then we can use:  f  =  p/50 = 0.063 = sin f = sin (p/50)

We can therefore use:

ò d f/ [Ö1 - 1.2  sin  ² (p/50) ]

e) Given the condition sin f »   f   show the motion would be simple harmonic and write an equation for the characteristic frequency,

Solution:

Given sin f »   f  we have an approximation that linearizes the problem by making the torque proportional to the displacement so simple harmonic motion results and: f = f o sin wt. Then the characteristic frequency is: w o    = Ö ( g / R)

Given:  f^" =   - w o ² f =  ( g / R)  f

e.g.  d ² f / dt ² =   - k sin  f = - w o ² f  ( i.e. k  =  w o ² )