题目内容
如图所示,ABC和DEF是固定在同一竖直平面内的两条光滑轨道,其中ABC的末端水平,DEF是半径为r=0.4m的半圆形轨道,其直径DF沿竖直方向,C、D可看作重合。现有一可视为质点的小球从轨道ABC上距C点高为H的地方由静止释放。(取g=10m/s2)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241527139203458.jpg)
(1)若要使小球经C处水平进入轨道DEF且能沿轨道运动,H至少要有多高?
(2)若小球静止释放处离C点的高度h小于(1)中H的最小值,小球可击中与圆心等高的E点,求此h的值。
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241527139203458.jpg)
(1)若要使小球经C处水平进入轨道DEF且能沿轨道运动,H至少要有多高?
(2)若小球静止释放处离C点的高度h小于(1)中H的最小值,小球可击中与圆心等高的E点,求此h的值。
(1)0.2m (2)0.1m
试题分析:(1)小球从ABC轨道下滑,机械能守恒,设到达C点时的速成大小为v,
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824152713998779.png)
小球能在竖直平面内做圆周运动,则在D点必须满足:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824152714138693.png)
联立以上两式并代入数据得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824152714201555.png)
(2)若
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824152714263461.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824152714357314.png)
竖直方向:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824152714435624.png)
水平方向:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824152714513470.png)
由机械能守恒得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824152714606809.png)
联立以上三式并代入数据得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824152714716507.png)
点评:本题是圆周运动结合平抛运动的题型,要知道小球能在竖直平面内做圆周运动,在圆周最高点必须满足
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824152714138693.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
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