Graphical analysis of an oscillator with constant magnitude sliding friction

Abstract

We treat a horizontal oscillator damped by constant-magnitude sliding friction by extending the analogy between the simple harmonic motion of a mass on a spring and the uniform circular motion of a mass attached to the end of a string. In the presence of sliding friction, the motion of the mass on a spring becomes the horizontal projection of the path of a mass attached to a string winding around two nails separated by a well-defined distance; this path is a spiral consisting of connected semi-circles of diminishing radii. This graphical analysis is very simple and pedagogically useful. It can also be generalized to any oscillation affected by other forces of constant magnitude but not necessarily constant direction.