质点在粗糙曲面上的滑动问题研究

RESEARCH ON THE SLIDING PROBLEM OF PARTICLES ON ROUGH SURFACES

质点在光滑半球面上的运动是高中一个很好的研究问题。在本文中,我们将此问题推广到任意粗糙曲面上,基于牛顿第二定律推导出速度的显式积分形式,它在某些情况下是有解析表达式的,并且得到了一个通用的表达式来确定质点离开曲面的分离角条件。在最后通过mathematica软件详细分析了三条曲线,一个圆、一个抛物线、一个椭圆,得到了几个有趣的结论。

The motion of particles on a smooth hemisphere is a good research problem in high school.In this article,we extend this problem to any rough surface and derive the explicit in-tegral form of velocity based on Newton's second law.It has an analytical expression in some cases and obtains a universal expression to determine the separation angle condition for parti-cles leaving the surface.In the end,three curves,a circle,a parabola,and an ellipse,were analyzed in detail using Mathematica software,and several interesting conclusions were obtained.