题目内容
(12分)如图(a)所示,A、B为水平放置的平行金属板,板间距离为d(d远小于板的长和宽).在两板之间有一个带负电的质点P.已知若在A、B间加电压U0,则质点P可以静止平衡.现在A、B间加上如图(b)所示的随时间t变化的电压u.在t=0时质点P位于A、B间的中点处且初速度为零.已知质点P能在A、B之间以最大的幅度上下运动而又不与两板相碰,求图(b)中u改变的各时刻t1、t2、t3及tn的表达式.(质点开始从中点上升到最高点,及以后每次从最高点到最低点或从最低点到最高点的过程中,电压只改变一次.)![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241502542412030.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241502542412030.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824150254350970.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241502544591403.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241502545371486.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241502548491398.png)
试题分析:由已知可得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824150254974731.png)
如右图所示:为使质点在两板间以最大幅度运动,t=0开始应使质点先匀加速上升d/4到Q,再匀减速上升到M。设匀加速、匀减速的时间均为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824150255083398.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824150255224680.png)
得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824150255271760.png)
从上极板运动到下极板过程中,质点先匀加速下降d/2到P,再匀减速下降到N。设匀加速、匀减速的时间均为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824150255333424.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241502555981486.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824150255645682.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824150255801636.png)
按照题目要求,可知
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824150254350970.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241502559571407.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241502560821503.png)
同理,以后每相邻的t之间的时间间隔是
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824150256441854.png)
所以有:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241502565971167.png)
点评:难题。物理图像是表达物理过程、规律的基本工具之一,用图像反映物理过程、规律,具有直观、形象的特点,带电粒子在交变电场中运动时,受电场力作用,其加速度、速度等均做周期性变化,借助图像来描述它在电场中的运动情况,可直观展示物理过程,从而获得启迪,快捷地分析求解.
![](http://thumb2018.1010pic.com/images/loading.gif)
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