题目内容
如图甲所示,有一装置由倾斜轨道AB、水平轨道BC、竖直台阶CD和足够长的水平直轨道DE组成,表面处处光滑,且AB段与BC段通过一小圆弧(未画出)平滑相接。有一小球用轻绳竖直悬挂在C点的正上方,小球与BC平面相切但无挤压。紧靠台阶右侧停放着一辆小车,车的上表面水平与B点等高且右侧固定一根轻弹簧,弹簧的自由端在Q点,其中PQ段是粗糙的,Q点右侧表面光滑。现将一个滑块从倾斜轨道的顶端A处自由释放,滑至C点时与小球发生正碰,然后从小车左端P点滑上小车。碰撞之后小球在竖直平面做圆周运动,轻绳受到的拉力如图乙所示。已知滑块、小球和小车的质量分别为m1=3kg、m2=1kg和m3=6kg,AB轨道顶端A点距BC段的高度为h=0.8m,PQ段长度为L=0.4m,轻绳的长度为R=0.5m。 滑块、小球均可视为质点。取g=10m/s2。求:
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/2014082500482687511100.png)
(1)滑块到达BC轨道上时的速度大小。
(2)滑块与小球碰后瞬间小球的速度大小。
(3)要使滑块既能挤压弹簧,又最终没有滑离小车,则滑块与PQ之间的动摩擦因数μ应在什么范围内?(滑块与弹簧的相互作用始终在弹簧的弹性范围内)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/2014082500482687511100.png)
(1)滑块到达BC轨道上时的速度大小。
(2)滑块与小球碰后瞬间小球的速度大小。
(3)要使滑块既能挤压弹簧,又最终没有滑离小车,则滑块与PQ之间的动摩擦因数μ应在什么范围内?(滑块与弹簧的相互作用始终在弹簧的弹性范围内)
(1)
(2)
(3)![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825004826938635.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825004826891575.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825004826907557.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825004826938635.png)
试题分析:(1) 设滑块与小球碰撞前瞬间速度为
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825004826953325.png)
有
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825004826969854.png)
得
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825004826891575.png)
(2) 设小球在最高点的速度为v,由图乙可知小球在最高点时受到的拉力
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825004827000511.png)
由牛顿第二定律,有
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825004827016866.png)
设小球碰撞后瞬间速度为
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825004827031308.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250048270471067.png)
联立①②③并代入数据,解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825004826907557.png)
(3)滑块与小球碰撞过程满足动量守恒:
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825004827094716.png)
得碰撞后
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825004827109362.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825004827125544.png)
滑块最终没有滑离小车,滑块和小车之间必有共同的末速度
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825004827141386.png)
由滑块与小车组成的系统动量守恒:
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825004827172819.png)
①若
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825004827187301.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250048272031314.png)
联立④⑤⑥得
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825004827219494.png)
②若
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825004827187301.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250048272651345.png)
④⑤⑦得
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825004827281515.png)
综上所述,得
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825004826938635.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
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