题目内容
(14分)如图所示,装置BO′O可绕竖直轴O′O转动,可视为质点的小球A与两细线连接后分别系于B、C两点,装置静时细线AB水平,细线AC与竖直方向的夹角θ=37º。已知小球的质量m=1kg,细线AC长L=1m,B点距C点的水平和竖直距离相等。(重力加速度g取10m/s2,sin37º=0.6,cos37º=0.8)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250002472142034.png)
(1)若装置匀速转动的角速度为ω1时,细线AB上的张力为0而细线AC与竖直方向的夹角仍为37°,求角速度ω1的大小;
(2)若装置匀速转动的角速度为ω2时,细线AB刚好竖直,且张力为0,求此时角速度ω2的大小;
(3)装置可以以不同的角速度匀速转动,试通过计算在坐标图中画出细线AC上张力T随角速度的平方ω2变化的关系图像
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250002472303443.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250002472142034.png)
(1)若装置匀速转动的角速度为ω1时,细线AB上的张力为0而细线AC与竖直方向的夹角仍为37°,求角速度ω1的大小;
(2)若装置匀速转动的角速度为ω2时,细线AB刚好竖直,且张力为0,求此时角速度ω2的大小;
(3)装置可以以不同的角速度匀速转动,试通过计算在坐标图中画出细线AC上张力T随角速度的平方ω2变化的关系图像
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250002472303443.png)
(1)
rad/s (2)![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000247261908.png)
(3)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/2014082500024727715432.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000247245504.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000247261908.png)
(3)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/2014082500024727715432.png)
试题分析:(1)细线AB上张力恰为零时有
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250002472921072.png)
解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250002473081694.png)
(2)细线AB恰好竖直,但张力为零时,由几何关系得 :
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000247323650.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000247339502.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250002473551001.png)
此时
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000247261908.png)
(3)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000247386931.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000247401696.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000247401227.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250002474171002.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000247433525.png)
细线AC上张力的水平分量等于小球做圆周运动需要的向心力
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000247448839.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000247464635.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000247479455.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000247479864.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000247464635.png)
综上所述
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000247386931.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000247526592.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000247542430.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000247557873.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825000247557466.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250002475736673.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
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