Circular arc pendulum: experiment and motion\'s analysis

In this work, a simple rocking pendulum, in a circular arc shape, is presented. The idea is to put a rigid arc on a clean flat surface of glass and leave it to oscillate under the effect of a little impulse. Then, the periodic time of this arc pendulum's motion is experimentally determined. The mathematical analysis of the arc's motion is derived based on the principle of the energy conservation. The arc's motion is found to be expressed by an equation of a simple harmonic motion. It is found that the motion's periodic time depends on the gravitational acceleration, arc's radius of curvature, the arc's central angle and the arc's thickness. Nine rigid circular arcs having the same radius of curvature and thicknesses, but are different in the central geometrical angle, are used to experimentally verify the proposed model. The measured and the calculated periodic times of all of them are in a good agreement. The dependence of the periodic time of the oscillating arc on its radius, thickness and central angle is illustrated. In spite of its simplicity and easiness of realisation, this experiment offers demonstration of a good example of rocking sensation and experimental verification of the calculated arc's moment of inertia.