Pen Falling Off a Finger

A pen falling off an infinitesimally thin finger located at the origin. The colored lines show the pen at various times, and the two black curves trace its ends.



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coefficient of friction — the kinematic coefficient of friction between the finger and the gliding pen
initial distance and angle — the initial distance of the center of mass of the pen from the origin and the initial angle of the pen to the vertical axis
initial velocities — the initial speed of the center of mass moving from the origin and the angular velocity of the pen
Hold a pen on the tip of one finger and let go. Depending on the angle of the pen and the distance of the center of mass of the pen from the supporting finger, the pen will either slide off or rotate to fall off.
This Demonstration carries out this experiment theoretically by solving the corresponding equations of motion of classical mechanics. Compared to a physical experiment this is more precise, reproducible, and instructive. An arXiv entry by Veniamin A. Abalmassov and Dmitri A. Maljutin is available with more details.
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