The Simple Pendulum

A pendulum keeps the same rhythm for big swings and small ones because of a trick of geometry. The force pulling the bob back toward vertical is a piece of gravity, mg sin(theta). For small angles, sin(theta) is almost exactly equal to theta itself, measured in radians. That turns the restoring force into something proportional to displacement, not some messier function of angle, which is exactly the condition for simple harmonic motion. Once the force looks like Hooke's law, the period formula falls out the same way it does for a spring, and it only involves L and g, not the amplitude. That is why a gentle swing and a wider one started at a small angle tick at the same rate.

Look at the forces at different points in the swing and the picture clarifies. At the endpoints, the bob is momentarily at rest, so there is no centripetal acceleration; tension just balances the component of gravity along the string. At the bottom, the bob moves fastest and travels on a curved path, so the string must supply extra force to produce centripetal acceleration toward the pivot, on top of countering gravity. That is why tension peaks at the bottom and is smallest at the endpoints, even though gravity itself never changes.

This topic sets up the rest of oscillations: the pendulum is simple harmonic motion in disguise, obeying the same energy story of kinetic and potential energy trading back and forth, and the same period independence from amplitude that springs show. Push the angle too far, though, past small-angle territory, and sin(theta) and theta pull apart. The restoring force is weaker than the linear approximation predicts, so the bob takes longer to complete each swing, and the real period grows past 2 π √(L/g).