Abstract: The motion of a simple pendulum with varying length coupled with a vertical motion is investigated, the second-order differential equation set of the motion is obtained in terms of plane polar coordinates. The accurate numerical solutions of the angular displacement, the length and its rate, and the path are given with the numerical method. To the approximation for small angle, the first-order analytical expression of the angular displacement and its rate, and the second-order analytical expression of the length and its rate are derived by the iteration method. The numerical results obtained by the analytical expressions are in consistence with that of accurate numerical solutions at the first some periods.

Key words: simple pendulum with varying length, plane polar coordinates, accurate numerical solutions, approximation for small angle, analytical expression