摘要
本文应用曲率半径的数学公式及部分椭圆知识推导出椭圆曲率半径的4个表达式;将斜面上的匀速率圆周运动在水平面内的投影,得到一变速率的椭圆运动;利用投影后的速度、加速度矢量及法向加速度分量公式,推算出椭圆上任一位置的曲率半径;利用行星椭圆轨道运动时的能量和角动量守恒,推算出行星的速率和法向加速度分量,进而从动力学的角度推算出椭圆上任一点的曲率半径。文章回顾了力学中曲率半径的由来,论证了它的几何定义式与代数表达式的等价性,显示出曲率半径这个数学量已经有机地融进牛顿力学体系,无论从运动学还是动力学的角度去推算轨迹的曲率半径均是可行的。文末给出用力学知识推算轨迹曲率半径的一般思路和建议。文中所提及的方法、建议及部分结论或可为相关内容的教学与研究提供一些参考。
Four expressions of curvature radius of ellipse are derived by using the mathematical formula of curvature radius and some elliptic knowledge.The uniform velocity circular motion on the inclined plane is projected in the horizontal plane,and a variable velocity elliptical motion is obtained.The curvature radius of any position on the ellipse is calculated by using velocity and acceleration vector after projection,and the formula of normal acceleration component.By using the conservation of energy and angular momentum in the elliptical orbit motion of a planet,the velocity and the normal acceleration component of the planet are calculated,and then the curvature radius of any point on the ellipse is calculated from the point of view of dynamics.This paper reviews the origin of curvature radius in mechanics,demonstrates the equivalence between its geometric definition and algebraic expression,and shows that the mathematical quantity of curvature radius has been integrated into Newton mechanics system organically.It is feasible to calculate the curvature radius of a trajectory from both kinematics and dynamics perspectives.At the end of this paper,the general idea and suggestion of calculating the radius of curvature of trajectory by mechanical knowledge are given.The methods,suggestions and some conclusions mentioned in the article may provide some references for the teaching and research of related contents.
作者
邵云
SHAO Yun(School of Electronic Engineering, Nanjing Xiaozhuang College, Nanjing Jiangsu 211171)
出处
《物理与工程》
2019年第6期103-107,111,共6页
Physics and Engineering
基金
南京晓庄学院优秀教学团队建设项目(4187061)资助
关键词
椭圆
曲率半径
法向加速度分量
投影
行星
ellipse
radius of curvature
normal acceleration component
projection
planet
