题目内容
如图所示,质量为
、电荷量为
的小球(视为质点)通过长为
的细线悬挂于O点,以O点为中心在竖直平面内建立直角坐标系xOy,在第2、3象限内存在水平向左的匀强电场,电场强度大小为
(式中
为重力加速度) 。
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241338123923517.png)
(1)把细线拉直,使小球在第4象限与x正方向成
角处由静止释放,要使小球能沿原路返回至出发点,
的最小值为多少?
(2)把细线拉直,使小球从
处以初速度
竖直向下抛出,要使小球能在竖直平面内做完整的圆周运动,则
的最小值为多少?
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824133811549337.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824133811596330.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824133811627280.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824133811674657.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824133812033273.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241338123923517.png)
(1)把细线拉直,使小球在第4象限与x正方向成
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824133812438297.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824133812438297.png)
(2)把细线拉直,使小球从
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824133812516430.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824133812563325.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824133812563325.png)
(1)
(2)![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824133812704610.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824133812641428.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824133812704610.png)
(1)要使小球释放后能沿原路返回,则小球释放后最多只能摆至第二象限细线与x轴负向成
角处(由重力与电场力的合力方向决定)。恰摆到与x轴负向成
角对应的θ即为最小。对这一过程用动能定理:
(6分)
解之得
(2分)
(2)要使小球能在竖直平面内做完整的圆周运动,只需要让小球从
处出发能沿半径为L的圆周通过y轴最高点即可。设通过y轴最高点时小球速度为![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824133812953265.png)
对这一过程用动能定理(电场力做功为零):
(6分)
在最高点由牛顿第二定理可得:
(3分)
联立解得:
(2分)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824133812750379.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824133812750379.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241338128281503.png)
解之得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824133812641428.png)
(2)要使小球能在竖直平面内做完整的圆周运动,只需要让小球从
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824133812641428.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824133812953265.png)
对这一过程用动能定理(电场力做功为零):
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824133813000906.png)
在最高点由牛顿第二定理可得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824133813031677.png)
联立解得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824133812704610.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
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