质点沿对数螺线运动的速度和加速度  

The Velocity and Acceleration of a Particle Moving Along a Logarithmic Spiral

作　　者：陈彦 王威 CHEN Yan;WANG Wei(Yangzhou Polytechnic College,Yangzhou 225009,China)

机构地区：[1]扬州职业大学,江苏扬州225009

出　　处：《扬州职业大学学报》2023年第2期54-57,共4页Journal of Yangzhou Polytechnic College

基　　金：国家自然科学基金(12102380)。

摘　　要：根据以时间为参数的平面曲线弧微分公式和曲率计算公式,导出了质点在任意时刻围绕极点等角速度转动,并沿着对数螺线运动的线速度、切向加速度、法向加速度以及全加速度计算公式,通过加速度在极坐标系中的投影式,判断出质点在任意时刻运动的全加速度方向始终与径矢方向关于内法线轴对称且指向曲线内凹一侧。以匀速转动的水平光滑直管内小球的离心运动为例,通过不同参考系将点的简单运动与合成运动充分地融合在一起,验证了计算结果的正确性。According to the arc differential formula and the curvature formula of the plane curve with time as the parameter,the formulas for calculating linear velocity,tangential acceleration,normal acceleration and total acceleration of a particle rotating around the pole at any time and moving along the logarithmic spiral are derived.Through the projection formula of acceleration in polar coordinate system,it is judged that the direction of the acceleration velocity of the particle at any time is always symmetric with the direction of the radial vector about the inner normal and points to the concave side of the curve.To take the centrifugal motion of a small ball in a horizontal,smooth,straight tube rotating at a uniform speed as an example,the simple motion of a point and the synthetic motion are fully integrated through different reference frames,which verifies the correctness of the calculation result.

关 键 词：对数螺线 全加速度 极坐标系 等角性 变速曲线运动 

分 类 号：O311.1[理学—一般力学与力学基础]