In orbit
Relationship between centripetal acceleration and rotational speed
Author:
Title 4
Learning objectives :
This activity allows students to experimentally verify the relationship between centripetal acceleration and angular velocity. It makes the link between the theoretical equations of circular motion and real measurements.
Concepts covered
Uniform circular movement; Centripetal acceleration; Angular velocity; Frenet’s landmark; Relationship between vector quantities
What students will do :
The student holds his smartphone at arm's length in a vertical position and turns around while recording the normal acceleration with the FizziQ accelerometer. After having made a few complete rotations, the student analyzes the acceleration graph to calculate the average centripetal acceleration and the angular velocity then experimentally verifies the relationship a = ω²r.
What is required :
Smartphone with the FizziQ application; Clear space to turn safely; Tape measure to measure arm length; FizziQ experience notebook
Scientific background :
In uniform circular motion, any point follows a circular path at constant speed. Although the speed is constant in norm, its direction changes continuously, which implies an acceleration perpendicular to the trajectory, directed towards the center of the circle: this is centripetal acceleration. This acceleration is linked to the tangential speed v and the radius r by the formula a = v²/r. In terms of angular velocity ω (in rad/s), this relationship becomes a = ω²r, where ω = 2πf with f the rotation frequency in Hz. The smartphone's accelerometer measures this acceleration when it is held perpendicular to the plane of rotation (in the Frenet frame, normal acceleration corresponds to centripetal acceleration). For this experiment, the radius corresponds to the length of the outstretched arm (typically 60-70 cm). The angular velocity can be determined by counting the number of revolutions made during the duration of the recording, or by analyzing the periodicity of the acceleration signal. Sources of error include: radius variation during movement, inaccuracy in smartphone orientation, and irregularity in rotation speed. This experiment illustrates a fundamental principle of celestial mechanics: it is this same centripetal acceleration that keeps the planets in orbit around the Sun, although in this case it is produced by gravitational force.