Angular acceleration of the non-linear pendulum

Abstract MEMs gyros, such as those in smartphones allow the angular velocity of pendulums to be precisely measured at large angles, and phase plots of the angular acceleration versus the angular displacement confirm that φ ¨ = − ω o 2 sin φ even for the non-sinusoidal motion at amplitude φ o ≈ π . For amplitudes φ o > π / 2 the angular acceleration has extrema of φ ¨ = ω o 2 at angular displacement φ o = ± π / 2 , i.e. the four times per oscillation that the pendulum is horizontal.