基于运动学视角的车轮悖论探秘

Exploration of Wheel Paradox from the Perspective of Kinematics

为了揭示车轮悖论产生的原因并对其作出合理的解释,以在水平面上滚动的圆轮为研究对象,构建同心圆下边缘在水平面上碾压的运动模型,应用点的运动学、点的合成运动和刚体平面运动的理论研究了车轮在水平面上滚动时其上任一点的运动规律,从运动学角度对亚里士多德车轮悖论进行探秘。研究发现,只有当圆轮滚过一圈时轮上各点才具有相同的位移,且此时圆轮上各点位移的大小仅与轮的运动形态有关;圆轮上各同心圆的下边缘在水平面上所碾过的长度仅与圆轮滚过的角度θ及圆轮的运动形态有关。当半径为R的圆轮在水平面上作纯滚动且转过角度θ时,圆轮上各同心圆在水平方向所碾过的长度均为Rθ;当圆轮边滚边滑时各同心圆在水平方向所碾过的长度也均相等但都大于Rθ;同心圆随着圆轮滚动的过程中,在任意时刻同心圆的下边缘点相对于地面有滑动,这是同心圆随圆轮滚过一圈时其在水平方向上碾过的长度不等于该同心圆周长的根本原因。

In order to clarify the causes of the wheel paradox and provide a reasonable explanation for it,a motion model of the lower edge of a concentric circle rolling on a horizontal plane was constructed by using a circular wheel as the research object,and the kinematics of points,the synthetic motion of points and the plane motion of rigid bodies were applied to study the motion law of any point in the wheel when it rolled on the horizontal plane and explore the Aristotelian wheel paradox from the kinematic point of view.It was found that only when the wheel rolled through one revolution did each point in the wheel have the same displacement,and the size of the displacement of each point in the wheel at this time was only related to the motion pattern of the wheel.The length of the lower edge of each concentric circle in the wheel on the horizontal plane was only related to the angleθwhich the wheel rolled and the movement pattern of the wheel.When a wheel with radius R rolled purely on the horizontal plane and turned the angleθ,the length of each concentric circle in the wheel at the horizontal direction was Rθ.When the wheel was rolling and sliding around,the length of each concentric circle running over in the horizontal direction was also equal,but the both were greater than Rθ.As the concentric circle rolled with the wheel,the sliding point of the lower edge of the concentric circle relative to the ground at any time was the fundamental reason why the length of the concentric circle rolling horizontally when the concentric circle rolling over the circle was not equal to the circumference of the concentric circle.