倾角和摩擦系数的不均匀性对伽利略斜面实验核心结果的影响

Influence of inhomogeneity of the dip angle and the friction coefficient on the core result of Galileo’s inclined plane experiment

伽利略斜面实验的核心结果是小球走过的路程正比于时间的平方,但倾角和摩擦系数的局部不均匀性会导致局部加速度的不均匀性.本文用一个简化模型计入倾角和摩擦系数的不均匀性,数值计算发现这些不均匀性不仅引入随机误差,而且导致路程与时间偏离平方关系.如果保持倾角情况不变,减小摩擦系数使球从无滑滚动变为有滑滚动,路程与时间对于平方关系的偏离程度会增大而不是减小.这个结论虽然与直觉相反,但却与物理学史的相关研究得到的结论相符.为了排除小球被释放时的随机因素的影响,本文在实验中设计了零初速度释放小球的装置,还设计了一个更方便的测量摩擦系数的方法.

The core result of Galileo’s inclined plane experiment is the proportion relation between distance and the square of time, but the local inhomogeneity of the dip angle and friction coefficient leads to inhomogeneity of local acceleration. In this paper, a simplified model including inhomogeneity of the dip angle and friction coefficient is introduced. Numerical results show that this inhomogeneity not only induces random error but also leads to deviation of the square relation between the distance and the time. It is further found that if the dip angle keeps unchanged and the friction coefficient decreases, the ball should change from non-slip rolling to slip rolling, the deviation degree of the square relation between the distance and the time would increase rather than decrease. This result, although contrary to intuition, coincides with the result achieved in some corresponding researches on the history of physics. These theoretical results are confirmed by our experiments. In order to exclude the random factors in releasing the ball, an equipment is designed to release the ball with zero velocity. A more convenient method is also carried out to measure the friction coefficient.