Cycloid
Experimental study of the cycloid
Author:
Title 4
Learning objectives :
This activity allows students to discover the mathematical curve that is the cycloid and its geometric properties. It develops the ability to analyze complex movement using digital tools.
Concepts covered
Cycloid; Rotational and translational movement; Periodicity; Curve configuration; Kinematics of Compound Motion
What students will do :
The student studies the trajectory of a point located on the edge of a moving bicycle wheel using the FizziQ kinematic analysis module. From a real video or from the application library, the student performs precise pointing of the movement frame by frame then analyzes the shape of the curve obtained, its periodicity characteristics and its maximum height relative to the ground.
What is required :
Smartphone with the FizziQ application; Video of a moving bicycle or use of the 'Cycloid' video from the FizziQ library; Experience notebook for analysis of results
Scientific background :
The cycloid is a mathematical curve described by a point fixed on the circumference of a circle which rolls without sliding on a horizontal line. Parametrically, for a circle of radius r, it is expressed by: x(t) = r(t - sin t) and y(t) = r(1 - cos t). This curve has several remarkable properties: its maximum height is equal to 2r (twice the radius of the circle) and its horizontal period is equal to 2πr. Historically, it has fascinated mathematicians like Pascal and Huygens for its properties of brachistochrone (fastest descent curve) and tautochrone (descent time independent of the starting point). The FizziQ kinematic analysis tool makes it possible to experimentally verify these theoretical properties by following the real trajectory of a point on a wheel point by point. When the tracking point is moved towards the inside of the wheel (on a spoke), the resulting curve becomes a shortened cycloid (or trochoid). At the center of the wheel, the movement becomes a simple uniform rectilinear translation. This experiment perfectly illustrates the composition of a rotation and translation movement, a fundamental concept in kinematics.