匀速转动的水平光滑直管内小球的离心运动研究

Study on Centrifugal Motion of a Small Ball in a Horizontal Smooth Straight Tube Rotating at a Uniform Speed

本文分别在转动参考系和地面惯性系中,研究绕端点匀速转动的水平光滑直管内小球由相对静止开始的离心运动,得到严格的运动方程、速度方程、轨迹方程,以及它们在直管转过半圈后(即θ>π时)的近似表达式。结果显示,当θ>π时,小球的径向与横向速度趋于相等,其轨迹是与转速无关的对数螺线,且小球受到管壁的横向支持力FN趋于离心力的2倍。利用极坐标系,证明在地面参考系中FN做功的一半转化为横向动能,另一半转化为径向动能,从而小球的横向与径向动能(或速度)在θ>π时趋于相等。小球做极径急剧增加的对数螺线离心运动的原因在于离心力随角度θ呈指数函数增加;在其他初始条件下直管转过半圈后小球的运动状况与本文情况类似。

In this paper,the centrifugal motion of a small ball in a horizontal smooth straight pipe with uniform rotation around the end point is studied in the rotating reference frame and the ground inertial frame,respectively.The strict motion equation,velocity equation and trajectory equation are obtained,and their approximate expressions are obtained when the straight pipe rotates half a circle i.e.θ>π.The results show that whenθ>π,the radial and transverse velocities of the ball tend to be the same.Its trajectory is a logarithmic spiral independent of the rotational speed,and the transverse supporting force FN of the ball by the pipe wall tends to be twice of the centrifugal force.In polar coordinate system,it is proved that half of the work done by FN is converted into transverse kinetic energy and the other half is converted into radial kinetic energy in the ground reference system.Thus,the transverse and radial kinetic energy(or velocity)of the ball tends to be equal whenθ>π.At the end of the paper,it is pointed out that the reason why the small ball does the logarithmic spiral centrifugal motion with sharp increase of polar diameter is that the centrifugal force increases exponentially with the angleθ,the motion of the small ball under other initial conditions is similar to that in this paper.