题目内容
(12分)如图所示,半径分别为R和r的甲、乙两个光滑的圆形轨道安置在同一竖直平面上,轨道之间有一条水平轨道CD相通,一小球以一定的速度先滑上甲轨道,通过动摩擦因数为μ的CD段,又滑上乙轨道,最后离开两圆轨道,若小球在两圆轨道的最高点对轨道的压力都恰好为零,试求CD段的长度.
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250018042152771.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250018042152771.png)
CD段的长度
。
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825001804231799.png)
试题分析:设小球通过C点时的速度为
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825001804231339.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825001804246298.png)
根据小球对轨道压力为零有
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825001804262743.png)
取轨道最低点所在水平面为参考平面,由机械能守恒定律有
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250018044961053.png)
联立①②式,可得
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825001804511620.png)
同理可得小球通过D点时的速度
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825001804527601.png)
设CD段的长度为
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825001804543272.png)
由动能定理有
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250018045581012.png)
解得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825001804231799.png)
全程列式动能定力也得分。
![](http://thumb2018.1010pic.com/images/loading.gif)
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