An approximate solution to the equation of motion for large-angle oscillations of the simple pendulum with initial velocity

An analytic approximation of the solution to the differential equation describing the oscillations of a simple pendulum at large angles and with initial velocity is discussed. In the derivation, a sinusoidal approximation has been applied, and an analytic formula for the large-angle period of the simple pendulum is obtained, which also includes the initial velocity of the pendulum. This formula is more accurate as compared to most of what has previously been published, and gives the period with an accuracy better than 0.04% for angles up to pi/2, and within 0.2% for angles up to 2.8 radians. The major advantage of the present derivation of the expression for the pendulum period is probably the simplicity, which makes the formula useful for analysing pendulum experiments with initial velocities in introductory physics labs. For a given set of initial conditions, the formula also predicts a critical velocity, at which the period of the pendulum becomes infinite as the pendulum will exactly come to rest at the upper, unstable equilibrium. In the small-angle regime, the formula becomes equivalent to the result for the period of the linear pendulum. For initial angles up to pi/2, the sinusoidal approximation of the solution is rather good, but deviation is increasingly observed at larger angles, as the motion of the pendulum becomes anharmonic.