题目内容
如图所示,ABC和DEF是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.现有一质量为m=1kg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,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241557171279966.png)
(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h.(取g=10m/s2)
(3)若小球自H=0.3m处静止释放,求小球到达F点对轨道的压力大小.
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241557171279966.png)
(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h.(取g=10m/s2)
(3)若小球自H=0.3m处静止释放,求小球到达F点对轨道的压力大小.
(1)0.2m (2)0.1m (3)65N
试题分析:(1)小球从ABC轨道下滑,机械能守恒,设到达C点时的速度大小为v,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824155717267787.png)
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/20140824/20140824155717330711.png)
联立以上两式并代入数据得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824155717517555.png)
则H至少要高0.2m
(2)若h<H,小球过C点后做平抛运动,设球经C点时的速度大小为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824155717689322.png)
竖直方向:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824155717829622.png)
水平方向:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824155718016477.png)
由机械能守恒有:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824155718125826.png)
联立以上三式解得h=0.1m
(3)若小球自H=0.3m处静止释放, 设球经F点时的速度大小为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824155718250350.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241557184221059.png)
在F点有
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824155718547911.png)
联立以上两式解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824155718687647.png)
则小球到达F点对轨道的压力大小为65N
点评:关键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/20140824/20140824155717330711.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
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