球体水中垂直移动做功的数学模型

主要研究球体水中垂直移动的做功问题。根据均匀质球体的密度与水密度之间的数量关系,给出了3种情况并分别进行讨论,建立球体水中垂直移动做功的数学模型,得出了不同密度球体出水或入水做功与球的密度之间的函数关系。当球的密度与水的密度相同时,其出水做功与水的密度成比例关系;当球的密度小于水的密度时,所做的功只与水密度有关,并且其出水做功与入水做功相等。与当球的密度大于水的密度时,用2种方法进行分析,得出的结果相同,即所做的功与水密度球密度都有关。通过分析,归纳出3种情况下提球出水做功的不同表达式的分段函数,求出球下沉的平衡位置的方程,得到压球入水做功的表达式以及压球与提球做功相互比较的结果。

Mathematical model of work done in ball’s vertical movement in water

To have studied the quantity relationships of reactive works of homogeneous balls in different densities,which are pulled out of the water or pushed into the water,and established model of vertical movement in the water,we certify the functions about the ball s’ densities and the work different densities balls to be done when they are pulled out of the water.When the densities of the ball are the same of the water,the work is irrelevant with both the densities of the ball and the water.When the density of the ball is less than the water,the work is only relevant with the density of the water.When the density of the ball is greater than the water,the work is relevant with both the densities of the ball and the water.The formula of the ball in a balance location sinking in the water is derived when the density of the ball is less than water.By comparing the work done in the process of pushing the ball to that of pulling the ball,it is proved that when the densities of the ball and the water are similar,the work in process of pushing the ball into the water closes up to zero,and the works done in pushing and pulling the ball are the same.The different expressions of piecewise function with taking the ball of water have been induced with three conditions.Give the function of the balance position with the ball sinking,and we can get the expression and results of the ball acting in the water compared with each other.