题目内容
⑴为使小球能通过O点,而且只能通过一次,试问μ值应在什么范围?
⑵在上述条件下,小球在O点左方的停住点与O 点的最大距离是多少?
(已知弹簧的弹性势能
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120234842499.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241202348581816.gif)
(1)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120234873700.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120234889315.gif)
(1)本题要求小球停在O点或O点左方,必有:
即:
此为μ的上限
设小球到达左方最远点距O点的距离为
,因最后停止点不能越过O点,满足:
……………………………②
因B为最远点有:
………………………②
得:![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120235076577.gif)
代入(1)得:![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120235107946.gif)
令:![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120235123939.gif)
得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120235154531.gif)
由此中知μ的取值范围为:![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120234873700.gif)
(2)要让小球在左方最远点B处停处,则必有:
……………………………③
由②③得![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120234889315.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120234936620.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120234967527.gif)
设小球到达左方最远点距O点的距离为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120234982198.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120235060762.gif)
因B为最远点有:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120235060824.gif)
得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120235076577.gif)
代入(1)得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120235107946.gif)
令:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120235123939.gif)
得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120235138547.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120235154531.gif)
由此中知μ的取值范围为:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120234873700.gif)
(2)要让小球在左方最远点B处停处,则必有:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120235185479.gif)
由②③得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120234889315.gif)
![](http://thumb2018.1010pic.com/images/loading.gif)
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