According to Newton's
laws the inertia force F_{I} (i.e., mass times acceleration) has to be equal to the applied force.
In our case, the applied force is the restoring force F_{R} caused by gravity G. From the
geometry of the problem (see figure), it is clear that
F_{R} = G sin
=  mg sin,
where m is the mass of the pendulum and g is the acceleration of gravity. Note that the negative sign
is caused by the fact that the restoring force F_{R} wants to bring the pendulum back to equilibrium
(i.e., = 0).
Next, we have to express the inertia force F_{I} in terms of
the angle .
Assuming a rigid pendulum (i.e., its length l
is fixed), the mass can move only on a circle with radius l. The position (i.e., the spatial coordinate)
along this circle is given by l.
Note that the angle
is measured in radians (i.e., 180° corresponds to ).
The acceleration is therefore given by l d^{2}/dt^{2}.
Thus, from Newton's law we get
ml d^{2}/dt^{2} = mg sin.
Dividing by ml and moving the term on the righthand side to the lefthand side leads to the equation of
motion of an undamped and undriven pendulum
where
(2) 
_{0} = (g/l)^{1/2}.

