题目内容
据报道,最近的太阳系外发现了首颗“宜居”行星,其质量约为地球质量的6.4倍,已知一个在地球表面质量为50kg的人在这个行星表面的重量约为800N,地球表面处的重力加速度为10m/s2。求:
(1)该行星的半径与地球的半径之比约为多少?
(2)至少要以多的速度水平抛出一物体才不至于落回地面(已知地球的第一宇宙速度为8km/h)?
(1)该行星的半径与地球的半径之比约为多少?
(2)至少要以多的速度水平抛出一物体才不至于落回地面(已知地球的第一宇宙速度为8km/h)?
(1)2:1,(2)14km/h
(1)在该行星表面处,由
得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241145466751157.png)
由万有引力定律得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114546690873.png)
则:
,![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114546831955.png)
故![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241145468462048.png)
(2)物体不落回来的条件是物体绕该行星做圆周运动,故万有引力提供向心力,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114546862960.png)
则![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114546893760.png)
故:![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241145469092155.png)
即![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241145469241129.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114546628599.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241145466751157.png)
由万有引力定律得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114546690873.png)
则:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114546706967.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114546831955.png)
故
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241145468462048.png)
(2)物体不落回来的条件是物体绕该行星做圆周运动,故万有引力提供向心力,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114546862960.png)
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114546893760.png)
故:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241145469092155.png)
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241145469241129.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
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