题目内容
如图甲所示,在同一竖直平面内的两正对的相同半圆光滑轨道相隔一定的距离,虚线沿竖直方向,一小球能在其间运动,今在最高点与最低点各放一个压力传感器,测试小球对轨道的压力,并通过计算机显示出来,当两个半圆形轨道间距离x变化时,测得两点压力差与距离x的图像如图乙所示,取g=10m/s2,不计空气阻力,求:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/2014082415092893712551.png)
(1)小球的质量为多少?
(2)若小球在最低点B的速度为20m/s,为使小球能沿轨道逆时针运动起来,x的最大值为多少?
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/2014082415092893712551.png)
(1)小球的质量为多少?
(2)若小球在最低点B的速度为20m/s,为使小球能沿轨道逆时针运动起来,x的最大值为多少?
(1)
(2)![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824150929140508.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824150929000605.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824150929140508.png)
试题分析:(1)设轨道半径为R,取最低点为零势能面,由机械能守恒定律:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241509291871132.png)
对B点:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824150929296903.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824150929421169.png)
对A点:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824150929530924.png)
由(1)(2)(3)式得:两点的压力差:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241509295921215.png)
由图象的截距得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824150929670546.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824150929000605.png)
(2)因为图线的斜率
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824150929998749.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824150930138514.png)
在A点不脱离的条件为:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824150930357595.png)
由(1)、(6)、(7)式得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824150929140508.png)
点评:本题学生明确小球在做竖直方向的圆周运动,会分析其在最高点和最低点的向心力,清楚小球能做完整的圆周运动所满足的速度条件。
![](http://thumb2018.1010pic.com/images/loading.gif)
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