题目内容
如右图,质量分别为m和M的两个星球A和B在引力作用下都绕O点做匀速圆周运动,星球A和B两者中心之间的距离为L.已知A、B的中心和O三点始终共线,A和B分别在O的两侧.引力常数为G.
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408242147113014428.jpg)
(1)求两星球做圆周运动的周期;
(2)在地月系统中,若忽略其它星球的影响,可以将月球和地球看成上述星球A和B,月球绕其轨道中心运行的周期记为
.但在近似处理问题时,常常认为月球是绕地心做圆周运动的,这样算得的运行周期记为
.已知地球和月球的质量分别为
和
.求
与
两者平方之比.(结果保留3位小数)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408242147113014428.jpg)
(1)求两星球做圆周运动的周期;
(2)在地月系统中,若忽略其它星球的影响,可以将月球和地球看成上述星球A和B,月球绕其轨道中心运行的周期记为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824214711831319.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824214713017365.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824214713828763.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824214714624713.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824214713017365.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824214711831319.png)
(1)
(2) ![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824214718258713.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408242147174941181.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824214718258713.png)
试题分析:(1)设两个星球A和B做匀速圆周运动的轨道半径分别为r和R,相互作用的引力大小为F,运行周期为T.根据万有引力定律有
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408242147190541041.png)
由匀速圆周运动的规律得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824214719959748.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824214721566875.png)
由题意得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824214722564399.png)
联立①②③④式得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408242147174941181.png)
(2)在地月系统中,由于地月系统旋转所围绕的中心O不在地心,月球做圆周运动的周期可由⑤式得出
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408242147241401261.png)
式中,M′和m′分别是地球与月球的质量,L′是地心与月心之间的距离.若认为月球在地球的引力作用下绕地心做匀速圆周运动,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408242147253561177.png)
式中,T2为月球绕地心运动的周期.由⑦式得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408242147264021025.png)
由⑥⑧式得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824214727540951.png)
代入题给数据得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824214718258713.png)
点评:关键是对公式的灵活掌握,万有引力这一快的物理公式比较多,注意公式的选择以及变形
![](http://thumb2018.1010pic.com/images/loading.gif)
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