题目内容
(18分)如图所示,固定的凹槽水平表面光滑,其内放置U形滑板N,滑板两端为半径R="0.45" m
的1/4圆弧面,A和D分别是圆弧的端点,BC段表面粗糙,其余段表面光滑,小滑块P1和P2的质量均为m,滑板的质量M=4m.P1和P2与BC面的动摩擦因数分别为μ1=0.10和μ2=0.40,最大静摩擦力近似等于滑动摩擦力,开始时滑板紧靠槽的左端,P2静止在粗糙面的B点。P1以v0="4.0" m/s的初速度从A点沿弧面自由滑下,与P2发生弹性碰撞后,P1停在粗糙面B点上,当P2滑到C点时,滑板恰好与槽的右端碰撞并与槽牢固粘连,P2继续滑动,到达D点时速度为零,P1与P2可视为质点,取g="10" m/s2。问:
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250034272205536.png)
(1)P2在BC段向右滑动时,滑板的加速度为多大?
(2)BC长度为多少?N、P1和P2最终静止后,P1与P2间的距离为多少?
的1/4圆弧面,A和D分别是圆弧的端点,BC段表面粗糙,其余段表面光滑,小滑块P1和P2的质量均为m,滑板的质量M=4m.P1和P2与BC面的动摩擦因数分别为μ1=0.10和μ2=0.40,最大静摩擦力近似等于滑动摩擦力,开始时滑板紧靠槽的左端,P2静止在粗糙面的B点。P1以v0="4.0" m/s的初速度从A点沿弧面自由滑下,与P2发生弹性碰撞后,P1停在粗糙面B点上,当P2滑到C点时,滑板恰好与槽的右端碰撞并与槽牢固粘连,P2继续滑动,到达D点时速度为零,P1与P2可视为质点,取g="10" m/s2。问:
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250034272205536.png)
(1)P2在BC段向右滑动时,滑板的加速度为多大?
(2)BC长度为多少?N、P1和P2最终静止后,P1与P2间的距离为多少?
(1)
(2)1.9m 0.695m
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003427235581.png)
试题分析: (1) P2在BC段向右滑动时,P1停在粗糙面B点上。将N、P1看作整体,根据牛顿第二定律得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003427251897.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250034272821837.png)
(2)设P1到达B点的速度为v,P1从A点到达B点的过程中,根据动能定理有:
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003427298942.png)
代入数据得
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003427438559.png)
因P1、P2质量相等且发生弹性碰撞,所以碰后P1、P2交换速度,即碰后P2在B点的速度为:
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003427454609.png)
设P2在C点的速度为
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003427469339.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003427500914.png)
代入数据得
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003427516617.png)
P2从B点到C点的过程中,N、P1、P2作为一个系统所受合外力为零,系统动量守恒,设P2到达C点时N和P1的共同速度为
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003427532291.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003427547940.png)
v′为滑板与槽的右端粘连前滑板和P1的共同速度.对P2从B点到C点相对地面位移为
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003427563341.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250034275941062.png)
P1 和N作为一个整体,相对地面位移为
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003427610385.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250034276251215.png)
联立⑧⑨⑩得BC长度
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003427641762.png)
滑板与槽粘连后,P1在BC上移动的距离为
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003427656313.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003427672968.png)
P2在D点滑下后,在BC上移动的距离
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003427688350.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003427719755.png)
联立得系统完全静止时P1与P2的间距
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825003427734955.png)
⑥⑧⑨⑩各2分,其余各1分
![](http://thumb2018.1010pic.com/images/loading.gif)
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