The Simple Pendulum:
When the mass of the pendulum bob is concentrated at the end of a rod or string of negligible mass, one can often characterize this as a "simple" pendulum, as opposed to a distributed mass oscillating back and forth.
The restoring force is
Whether we consider the displacement to be the linear quantity s or the angular quantity theta, this restoring force is NOT proportional to the displacement as required for SHM (Hooke's Law). ("Proportional" means that a linear relationship exists; this relationship is sinusoidal.)
So pendulum motion is not SHM. But if the angle is small (less than 5o or so), the angle in radians is approximately equal to the sine of the angle. (Try it: take the sine of 0.872 rad, which is approximately 5o.) So for small angles, we replace the sine of the angle with the angle itself and we have the proportionality demanded by SHM. In this way, we get equation 12.24 on page 345, which you should realize has the same form as equation 12.15 on page 337. Therefore, both equations have the same solution, except that k/m replaces g/L. Please don't go on if you don't understand this!
Perhaps now you realize that
and since
only the period formula changes, so that
The Physical Pendulum:
A model of the physical pendulum is shown in Figure 12.11 on page 346 in Serway. You'll notice that the pivot is not the center of mass (CM) - it can't be, since if it were pivoted at the CM, no oscillation at all would result from a displacement and subsequent release.
As for the simple pendulum, Serway writes Newton's Second Law assuming that the displacement angle is small. The differential equation that results has the same form as for the spring and the simple pendulum, but the omega-squared is different:
Example 1: Measurement of g
Example 2: Speed and acceleration for a simple pendulum
Example 3: A ring pendulum