A simple pendulum consists of a massive bob attached by a string or light rod to a fixed pivot point, whereas a physical pendulum can be any rigid body attached to a fixed pivot. The equation of motion for a physical pendulum involves its moment of inertia about the pivot and the distance from the pivot to its centre of mass, but the form of the equation is the same as that for the simple pendulum, as are all features of the motion. For both, the restoring force is proportional to the sine of the angular displacement from equilibrium. For small angles, the sine of an angle is equal to angle itself (in radians), and the restoring force is proportional to the displacement (as it was for the spring). This is the condition for Simple Harmonic Motion - motion where the frequency of oscillation is independent of the amplitude.
In the example we study here, the pendulum bob of a simple pendulum is connected to the pivot by a light rod rather than a string so it can reach large amplitudes. The bob starts from the equilibrium point with velocities that range up to twice the minimum initial velocity required for it to reach a point vertically above the pivot. (This velocity is found directly using conservation of energy). In the upper half of the velocity range, therefore, the motion is circular, not oscillatory. In the graph, the initial velocity of the bob is plotted horizontally, and the period of the motion is plotted vertically (with the SHM period at the midpoint of the scale). The motion is shown at intervals at the left (in real time for an 80-cm rod). The SHM region shows up as an initial flat region in the plot. When the bob reaches greater amplitudes, the restoring force does not maintain proportionality to the displacement and the bob takes longer to reverse direction.
The important thing to realize about this example is that we have been able to solve a problem that has no easy analytic solution, and we did it using a program no more complicated than the one we used for the spring.