equilibrium state or trying to get to one. The study of small motions around a stable equilibrium state is called Vibrations.
1.2.1 Equilibrium of a Simple Pendulum
We start by considering the simple pendulum5 shown in Figure 1.5. Using the angle
Figure 1.5 A simple pendulum.
Once started in motion the pendulum will swing about the point of connection to the ground. In the case of the simple pendulum there is no mechanism for removing energy from the system as it swings (i.e. no friction or other forces that do work) so the motion, once started, will persist.
The motion will depend on the way in which it is started. That is, if the pendulum is rotated to some arbitrary starting angle,
The question we ask now is Are there initial values of
Consider Equation 1.44 under the conditions that there is an initial angle
Since
The total range of
There are formal methods for testing the stability of the equilibrium states but that we leave to courses on control systems. It is sufficient for us to be able to see that the state where the pendulum stands upright is unstable and the pendulum will try to get to the stable equilibrium position where
The vibrations question is What will be the response of the system for small motions away from the stable equilibrium condition where
1.2.2 Equilibrium of the Bead on the Wire
We now return to our continuing example problem – the bead on a rotating semicircular wire as shown in Figure 1.1. The equation of motion (see Equation 1.23) is
(1.46)
We look for solutions where the angle
The equilibrium condition is a group of constant terms summing up to zero that becomes an identity for us. We will see this group of terms again when we write the equation of motion for small motions around equilibrium and every time we see it, we will be able to set it equal to zero.
With some simple factoring out of terms, we get
(1.48)
This expression will hold for two cases:
. This is satisfied when and when . These correspond to the bead being directly below point and directly above point respectively. Being above point is, of course, physically impossible for the semicircular wire but would be possible for a complete hoop.
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