题目内容
(20分)如图所示,光滑绝缘水平面的上方空间被竖直的分界面MN分隔成两部分,左侧空间有一水平向右的匀强电场,场强大小
,右侧空间有长为R=0.114m的绝缘轻绳,绳的一端固定于O点,另一端拴一个质量为m小球B在竖直面内沿顺时针方向做圆周运动,运动到最低点时速度大小vB=10m/s(小球B在最低点时与地面接触但无弹力)。在MN左侧水平面上有一质量也为m,带电量为
的小球A,某时刻在距MN平面L位置由静止释放,恰能与运动到最低点的B球发生正碰,并瞬间粘合成一个整体C。(取g=10m/s2)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250030031271949.png)
(1)如果L=0.2m,求整体C运动到最高点时的速率。(结果保留1位小数)
(2)在(1)条件下,整体C在最高点时受到细绳的拉力是小球B重力的多少倍?(结果取整数)
(3)若碰后瞬间在MN的右侧空间立即加上一水平向左的匀强电场,场强大小
,当L满足什么条件时,整体C可在竖直面内做完整的圆周运动。(结果保留1位小数)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003064684.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003080330.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250030031271949.png)
(1)如果L=0.2m,求整体C运动到最高点时的速率。(结果保留1位小数)
(2)在(1)条件下,整体C在最高点时受到细绳的拉力是小球B重力的多少倍?(结果取整数)
(3)若碰后瞬间在MN的右侧空间立即加上一水平向左的匀强电场,场强大小
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003142637.png)
(1)
(2)18倍 (3)
或![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003189595.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003158674.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003174536.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003189595.png)
试题分析:(1)对
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003205300.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003220309.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003252904.png)
解得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250030032671079.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003205300.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003220309.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003314728.png)
解得共同速度:
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003345572.png)
设整体C在最高点速度为
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003361348.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250030033761148.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003158674.png)
(2)由牛顿第二定律,在圆周运动最高点:
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003423959.png)
解得受到的拉力:
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003439605.png)
(3)MN右侧空间加上一水平向左的匀强电场
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003470372.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250030034861178.png)
设合力方向与竖直方向间的夹角为
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003501297.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250030035171073.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003548494.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250030035643385.png)
P和Q即为等效的最低点和最高点
整体C做完整圆周运动的条件是:在Q点即等效最高点绳的拉力满足:
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003579479.png)
设此时整体C在Q点的速度为
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003610323.png)
即:
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003626898.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003642662.png)
设整体C在最低点的速度大小为v1,由动能定理:
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250030036731465.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003688639.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003205300.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003220309.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003735710.png)
若碰后整体C方向向左,取最小
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003751673.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003751682.png)
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003782654.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003174536.png)
若碰后整体C方向向右,取最小
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003813681.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003829666.png)
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003844659.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003189595.png)
所以,
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003860280.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003174536.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003003189595.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
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