题目内容
如图所示,在水平面上有两条光滑的长直平行金属导轨MN、PQ,导轨间距离为L,导轨的电阻忽略不计,磁感应强度为B的匀强磁场垂直于导轨所在平面。质量分别为
的两根金属杆a、b跨搁在导轨上,接入电路的电阻均为R。轻质弹簧的左端与b杆连接,右端被固定。开始时a杆以初速度
向静止的b杆运动,当a杆向右的速度为v时,b杆向右的速度达到最大值
,此过程中a杆产生的焦耳热为Q,两杆始终垂直于导轨并与导轨良好接触。求当b杆达到最大速度
时
(1)b杆受到弹簧的弹力。
(2)弹簧具有的弹性势能。![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241142479763187.jpg)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114247914387.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114247929204.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114247945312.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114247945312.gif)
(1)b杆受到弹簧的弹力。
(2)弹簧具有的弹性势能。
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241142479763187.jpg)
(1)![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114247992658.gif)
(2)
。
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114247992658.gif)
(2)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114248007961.gif)
(1)设某时刻a、b杆速度分别为v和
,经过很短的时间
,a杆移动距离为
,b杆移动距离为
,回路面积改变量![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114248101558.gif)
由法拉第电磁感应定律,回路中的感应电动势![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114248210509.gif)
[或直接写为
]
回路中的电流![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114248257329.gif)
b杆受到的安培力![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114248273423.gif)
当b杆的速度达到最大值
时,b杆的加速度为0,设此时b杆受到的弹簧弹力为
,由牛顿第二定律得![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114248335283.gif)
联立以上各式解得![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114247992658.gif)
(2)以a、b杆和弹簧为研究对象,设弹簧弹性势能为
,由能量转化与守恒定律![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114248382981.gif)
故
。
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114247945312.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114248039214.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114248054340.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114248085373.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114248101558.gif)
由法拉第电磁感应定律,回路中的感应电动势
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114248210509.gif)
[或直接写为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114248241522.gif)
回路中的电流
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114248257329.gif)
b杆受到的安培力
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114248273423.gif)
当b杆的速度达到最大值
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114247945312.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114248304220.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114248335283.gif)
联立以上各式解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114247992658.gif)
(2)以a、b杆和弹簧为研究对象,设弹簧弹性势能为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114248366229.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114248382981.gif)
故
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114248007961.gif)
![](http://thumb2018.1010pic.com/images/loading.gif)
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