Analysis and Discussion on the Time for a Small Ball to Detach From a Smooth Hemisphere

This paper primarily addresses the time of a small ball sliding off a smooth hemispherical surface. The kinematic equations and sliding time of the ball are obtained through Newton's second law with Python's numerical and visualization associate. At the same time, the sliding time can be directly derived analytically using the work-energy principle. The results show that the numerical calculations and analytical solutions are completely equivalent. For the ball, the top of the hemisphere is an unstable equilibrium position. When the ball is subjected to a slight perturbation, whether the initial position deviates from the top of the hemisphere or the ball slowly slides down from the top, the velocity and position of the ball when it slides down the surface tend towards constant values. However, as the perturbation increases, the sliding time decreases exponentially, which is due to the non-linear nature of the net external force acting on the ball on the hemispherical surface.