题目内容
据美联社2002年10月7日报道,天文学家在太阳系的9大行星之外,又发现了一颗比地球小得多的新行星,而且还测得它绕太阳公转的周期为288年。若把它和地球绕太阳公转的轨道都看作圆,问它与太阳的距离约是地球与太阳距离的多少倍?(最后结果可用根式表示)
解:设太阳的质量为M;地球的质量为m0,绕太阳公转周期为T0,与太阳的距离为R0,公转角速度为ω0;新行星的质量为m,绕太阳公转周期为T,与太阳的距离为R,公转角速度为ω。则根据万有引力定律和牛顿定律,得:
![](http://thumb.1010pic.com/pic1/upload/papers/g04/20111217/201112171414245461196.gif)
![](http://thumb.1010pic.com/pic1/upload/papers/g04/20111217/201112171414245461139.gif)
![](http://thumb.1010pic.com/pic1/upload/papers/g04/20111217/201112171414246091004.gif)
![](http://thumb.1010pic.com/pic1/upload/papers/g04/20111217/20111217141424796966.gif)
由以上各式得![](http://thumb.1010pic.com/pic1/upload/papers/g04/20111217/201112171414251091141.gif)
已知T=288年,T0=1年,得
。
![](http://thumb.1010pic.com/pic1/upload/papers/g04/20111217/201112171414245461196.gif)
![](http://thumb.1010pic.com/pic1/upload/papers/g04/20111217/201112171414245461139.gif)
![](http://thumb.1010pic.com/pic1/upload/papers/g04/20111217/201112171414246091004.gif)
![](http://thumb.1010pic.com/pic1/upload/papers/g04/20111217/20111217141424796966.gif)
由以上各式得
![](http://thumb.1010pic.com/pic1/upload/papers/g04/20111217/201112171414251091141.gif)
已知T=288年,T0=1年,得
![](http://thumb.1010pic.com/pic1/upload/papers/g04/20111217/201112171414251251151.gif)
![](http://thumb2018.1010pic.com/images/loading.gif)
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