题目内容
如图所示,质量分别为m和M的滑块A和B静止于光滑的水平台面上,A和B之间夹有一轻质弹簧(A、B与弹簧不连接),弹簧处于被压缩状态,有一轻质绳连接着A和B。水平台面左端连接着一半径为R=0.2m的光滑圆轨道PQ,右端为一倾角为37°的足够长的斜面CD,滑块B与斜面CD间的动摩擦因数为
。烧断两滑块间的轻质绳后,由于弹簧弹力的作用,滑块A将沿圆轨道做圆周运动,且在最高点P轨道对滑块的压力恰好等于滑块的重力。已知两滑块的质量之比为
,重力加速度g=10m/s2.求:
(1)烧断细绳,弹簧恢复原长时两滑块的速度大小;
(2)滑块B将沿斜面上滑,且从斜面底端C开始经过t=0.5s,恰好经过D点,求CD的长度。
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241151150715659.jpg)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115040288.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115055503.gif)
(1)烧断细绳,弹簧恢复原长时两滑块的速度大小;
(2)滑块B将沿斜面上滑,且从斜面底端C开始经过t=0.5s,恰好经过D点,求CD的长度。
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241151150715659.jpg)
(1)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115086491.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115102428.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115118567.gif)
(1)设弹簧恢复原长时A、B两滑块的速率分别为
和
,滑块A到达圆轨道最高点时的速率为![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115196202.gif)
滑块A在最高点,轨道对其的压力恰好与其重力相等
① (2分)
滑块A在光滑圆轨道上运动过程中机械能守恒,所以有
② (2分)
烧断细绳后,A、B两滑块在弹簧弹力作用下运动过程中动量守恒,且二者运动速度方向相反,所以有:
③ (2分)
由题给条件有:
④ (1分)
联立以上各式求解得:
,
(3分)
(2)滑块B在C点时的速度:![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115102428.gif)
滑块沿CD段向上运动时做匀减速度直线运动至静止,设其加速度大小为
,发生的位移大小为
,所用时间为
,则有
⑤ (1分)
⑥ (1分)
⑦ (1分)
联立以上三式求解得:
,
,
(1分)
返回时做初速度为零的匀加速直线运动,设其加速度大小为
,运动时间为
,位移为
,则有
⑧ (1分)
⑨ (1分)
⑩ (1分)
联立以上三式代数求解得:
(1分)
所有CD的长度为:
(1分)
说明:用其它方法计算,只要正确,同样给分
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115149199.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115180205.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115196202.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115211183.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115227128.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115242558.gif)
滑块A在光滑圆轨道上运动过程中机械能守恒,所以有
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115258770.gif)
烧断细绳后,A、B两滑块在弹簧弹力作用下运动过程中动量守恒,且二者运动速度方向相反,所以有:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115274508.gif)
由题给条件有:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115055503.gif)
联立以上各式求解得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115086491.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115102428.gif)
(2)滑块B在C点时的速度:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115102428.gif)
滑块沿CD段向上运动时做匀减速度直线运动至静止,设其加速度大小为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115352206.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115367197.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115398199.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115414724.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115430438.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115461396.gif)
联立以上三式求解得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115492488.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115508433.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115554302.gif)
返回时做初速度为零的匀加速直线运动,设其加速度大小为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115570208.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115586204.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115601200.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115617720.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115648274.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115664488.gif)
联立以上三式代数求解得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115664459.gif)
所有CD的长度为:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824115115118567.gif)
说明:用其它方法计算,只要正确,同样给分
![](http://thumb2018.1010pic.com/images/loading.gif)
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