碗中小球问题探讨

DISCUSSION ON THE BALL IN THE BOWL

在大学物理的教学过程中,笔者遇到一道关于小球在光滑半球形碗内运动的问题,该问题的常规解法是利用角动量分量守恒和机械能守恒进行求解。这种解法由于无法给出小球运动的细节,对于初次接触角动量概念的本科生而言难以直观而深刻地理解。本文将小球的运动在水平面内和竖直方向进行分解,系统分析了小球的受力情况、速度的变化、能量的转化等问题,并通过数值求解拉格朗日方程给出了小球的运动轨迹。我们的方法详尽地阐述了小球的运动过程,展示了一个更加直观和清晰的物理图像。

In the teaching of College Physics,authors encounter a problem about a small ball moving in a smooth hemispherical bowl.The conventional solution is to use the conservation of one component of angular momentum as well as the conservation of mechanical energy.However,this method can’t demonstrate the details about movement of the small ball.For the freshmen who study the angular momentum for the first time,they can’t understand this method intuitively and deeply.In this paper,the motion of small ball is decomposed in the horizontal plane and the vertical direction,and the force on the small ball,the change of velocity,the transformation of energy and other problems are systematically analyzed.In addition,the motion trajectory of the small ball is given by numerically solving the Lagrange equations.In this method,the small ball’s moving process is demonstrated in detail,which offers a more intuitive and clearer physical vision.