Geometric Simulation of the Seconds Pendulum: Period, Length, and Vibrating Para

This example demonstrates, via an animated geometric construction, how a “seconds pendulum” vibrates with an approximately one-second period under gravity, showing time-dependent quantities such as displacement and velocity. The main mathematical ideas include periodic motion modeled through parameters, angle measurement and transformation, vector direction handling via orthogonal unit vectors, and computations involving averaging (mean) and summation (Sum) over the motion cycle. Construction-wise, it uses Circle to define the motion path, Rotate to swing the pendulum arm around the pivot, Point and Segment for key locations and connections, UnitOrthogonalVector to obtain a perpendicular unit direction, and Line for directional relations. Conditional logic with If and time-control variables (e.g., animation and pause controls) update the state as the time parameter t advances, producing visual outputs linked to period T, amplitude amp, and velocity vel. Suitable for advanced high school through introductory college lessons, especially for teaching vibration/oscillation concepts, transformation-based geometry, and parameter-driven or discrete numerical modeling.