Simple Pendulum Theory The simple pendulum is an idealization of a real pendulum. It consists of a point mass, m, attached to an infinitely light rigid cord of length L that

is itself attached to a frictionless pivot point. If displaced from its vertical equilibrium position, this idealized pendulum will oscillate with a constant amplitude forever. There is no damping of the motion from friction at the pivot or from air. Newton's second law provides the equation of motion: mL d²0 dt² = -mg sin 0, where is the angular displacement of the pendulum from the vertical position and g is the acceleration due to gravity. This equation may be simplified if we assume that amplitude of oscillation is small and that sin 0 ≈ 0. The modified equation of motion is d²0 g dt² + 0 = 0. The solution to this equation may be written as 000 sin(wt + (po), where is the angular amplitude of the swing, 2 @ = √ L is the angular frequency, and po is the initial phase angle whose value depends on how the pendulum was started, i.e., its initial conditions. The period of the motion, in this linearized approximation, is given by which is a constant for a given pendulum. L g T = 2√ ENGA1002 FT L mg