A pendulum is manifestly an object in a stable equilibrium
position so we should expect it to support oscillations. To determine
whether there are harmonic oscillations we have to look for
a linear restoring force. We denote the position of the pendulum along the circle of arc by, s, where we choose s=0 to
correspond to the equilibrium. In terms of the angle, 
which
the cord of the pendulum makes with the vertical direction
we have
There is no acceleration along the direction of the cord so we
have a tangential net force:
We see that the restoring force in general is not linear in the
displacement. This is only so for small angles where
In that linear limit we have :
where
Inserting this last expression in our general expression for
the resonance frequency 
gives us
From which we can derive the period of the simple pendulum:
This is the expression we started off this course by
arguing for on the basis of dimensional analysis. We have a few
different length simple pendulums set up and compare the predicted to measured pendulum periods to see that the formulae actually
are consistent with observations.