Since the motion of the mass is only in an arc, we can treat this problem as a rotational problem. The net (and only) torque acting on the mass is due to the component of the weight along the direction of the arc. (The tension in the cord cannot exert any torque on the bob since it acts through the center of rotation.)
This equation is not equivalent to the second order differential equation for SHM. However, when q is small (say less than 10o) then the value of sin(q) and q are nearly the same provided q is measured in radians.
This equation is structurally equivalent to the differential equation for SHM.
By matching terms we see that
and
Note that one has to distinguish between the angular frequency w of the SHM which is constant and the instantaneous angular velocity w' due to the rotational motion of a system which changes with time.
If the pendulum were pulled to the right and released then the phase angle would equal +p/2, the same as the simple mass on a spring example.
For small angles we have found that a simple pendulum should undergo SHM with a constant period T , that does not depend upon the mass or the amplitude of swing, but only upon the length of the cord and the acceleration of gravity. This is one of the reasons that pendulums have been used as clocks. On a grandfather clock, you can adjust the location of the mass to regulate the time kept by the clock.
